introduction to koopman-von neumann mechanics an …

INTRODUCTION TO KOOPMAN-VON NEUMANN MECHANICS

AN HONORS THESIS

SUBMITTED ON THE FIFTH DAY OF DECEMBER, 2021

TO THE DEPARTMENT OF PHYSICS AND ENGINEERING PHYSICS

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

OF THE HONORS PROGRAM

OF NEWCOMB-TULANE COLLEGE

TULANE UNIVERSITY

FOR THE DEGREE OF

BACHELOR OF SCIENCES

WITH HONORS IN PHYSICS

BY

Daniel W. Piasecki

APPROVED:

Denys I. Bondar, Ph.D.

Director of Thesis

Ryan T. Glasser, Ph.D.

Second Reader

Frank J. Tipler, Ph.D

Third Reader

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1

Abstract

This work is to consolidate current literature on Koopman-von Neumann

(KvN) Mechanics into a simple and easy to understand text. KvN Mechanics is

a branch of Classical Mechanics that has been recast into the mathematical lan-

guage of Quantum Mechanics. KvN Mechanics utilizes a Hilbert phase space

with operators to calculate the expectation values of observables of interest (ex-

pectation values such as position, momentum, etc.) It is an important tool in

statistical physics and guide to illuminate the mysterious relationship between

Quantum and Classical Physics. A lot of important applications of KvN Me-

chanics have been developed in the last decades.

ii

Acknowledgements

Thanksgiving is approaching at the time I’m writing this, so there are some people

who will get some messages of gratitude from me a few days early.

First, I would like to thank the Tulane Summer Research Program and the Tulane

Honors Program. This project would not have been possible without their support.

Many thanks to the readers of this thesis: Dr. Ryan Glasser, Dr. Frank Tipler, and

especially my advisor for this project, Dr. Denys Bondar. I am grateful for his advice

on this paper and guidance for how I can take what I’ve learned from it and apply

it to the rest of my career, both in terms of new knowledge about the subject matter

and skills I’ve been practicing while working on this. His openness and directness

in advising me made a big difference, and I’ve learned a lot from working on this

project. Also, 80 pages is a lot to read, so I am grateful for the time all three of these

professors have dedicated to reading my work.

******************************************************************

I am also very appreciative to my friends and family for their encouragement,

prayers, and patience with me. This thesis was a huge commitment for me, and they

were respectful of my time when I was busy and curious and uplifting during my few

moments of downtime. One of them also helped look over some of it and did some

proofreading, which I also am thankful for. Thanks goes especially to my sweetheart

Clover Robichaud for her continual advice, patience, and support while the project

has been ongoing. Also, thanks goes to my good friend Kushal Singh for helping me

prepare for the Thesis Defense itself.

Finally, I am grateful to all other physicists and aspiring physicists who read this;

if you are one of these people, you are exploring a little known topic that can make

a big difference, and I appreciate that you are likely branching out from the standard

iii

topics you are already taught in class. I am hopeful that my work contributed to

their knowledge of Koopman–von Neumann Mechanics and that this will help them

expand the field of Physics. This is currently a little known branch of Physics, and the

goal of this project was to make this information more accessible to others unfamiliar

with the topic, both in Physics and otherwise. If anyone could take what I did and

have that help him or her in contributing to Physics, then I will know that I have made

a difference.

iv

Contents

1 Introduction 1

2 General Review of Quantum Mechanics 2

2.1 Fundamentals of Hilbert Space . . . . . . . . . . . . . . . . . . . . 3

2.2 Operators and Commutators . . . . . . . . . . . . . . . . . . . . . 7

2.3 Expectation Values and the Eigenvalue Problem of Quantum Mechanics 10

2.4 Review of Dirac Delta Functional . . . . . . . . . . . . . . . . . . 13

2.5 Summary of Quantum Mechanics and its Postulates . . . . . . . . . 14

2.6 The Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Introduction to KvN Mechanics 19

4 Path Integral Formulation of Quantum and Classical Mechanics 34

4.1 Feynman Path Integral of Quantum Mechanics . . . . . . . . . . . . 34

4.2 Koopman-von Neumann Classical Path Integral . . . . . . . . . . . 40

5 Classical and Quantum Comparison 46

5.1 The Phase in Koopman-von Neumann and Quantum Mechanics . . 46

5.2 Uncertainty Principle in Koopman-von Neumann and Quantum Me-

chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Measurement in Koopman-von Neumann and Quantum Mechanics . 51

5.4 The Double Slit Experiment . . . . . . . . . . . . . . . . . . . . . 61

6 Miscellaneous KvN Topics of Interest 74

6.1 Operational Dynamic Modeling . . . . . . . . . . . . . . . . . . . 74

6.2 Wigner Quasiprobability Distribution . . . . . . . . . . . . . . . . 77

6.3 Time Dependent Harmonic Oscillator . . . . . . . . . . . . . . . . 81

6.4 Aharonov–Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . 89

7 Other Research 100

v

References 104

Appendices 107

A Useful Mathematics for Hilbert Space Physics 107

B The Principle of Causality and Stone’s Theorem 110

C Exponated Operators and the Propogator 111

vi

List of Figures

1 This is the set up of the double slit experiment. Middle wall is at

y = yM and rightmost wall is at y = yR. Adapted from Mauro 2002 114

2 a. Left: Graph of KvN probability distribution. b. Right: Graph of

QM probability distribution. Adapted from Mauro 2002. . . . . . . 114

3 a. Left: Example graph of a single variable gaussian curve. b. Right:

Example graph of a double variable gaussian curve. Adapted from

Wolfram Alpha 2021. . . . . . . . . . . . . . . . . . . . . . . . . . 114

4 Aharanov-Bohm Experiment set up. Reproduced from Aharonov and

Bohm 1959. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

vii

1 Introduction

Quantum Mechanics (QM) is known for its unique mathematical structure unlike that

traditionally found in other branches of Physics. In QM, one has operators acting on

the wavefunction in complex Hilbert Space in order to compute what the expectation

values of observables seen in the laboratory are (Shankar 1988). Classical Mechanics

(CM), on the other hand, is based on continuous, real valued functions of spatial and

time dependent variables. One merely has to plug in the values of the variables at

a particular point, and one knows the value of the function of interest. QM makes

probabilistic predictions of the quantum particle in question, whereas CM makes

deterministic predictions of all classical behavior.

The large differences in mathematical form has historically made it difficult to

compare QM to CM (Mauro 2003). There have been many attempts to make QM

appear more mathematically similar to CM (Mauro 2003). For instance, the Born

and Madelung approach to QM has a modified Hamilton-Jacobi equation as its basis

(Mauro 2003). But there have been few attempts to make CM more similar in form

to QM. This changed in the 1930’s, however, when a form of statistical CM was

devised by Bernard Koopman and John von Neumann that had the same operatorial

mathematical structure as QM (Koopman 1931; von Neumann 1932). Koopman-von

Neumann Mechanics (KvNM), as it later became known, was subsequently used by

mathematicians to formulate ergodic theory, while physicists mostly forgot about the

existence of this field. Currently, there is a large renewal of interest by the physics

community in this relatively obscure field of study.

In this Thesis I intend to articulate many significant contributions of KvNM to

physics in a straightforward and easy to understand fashion. I am writing this text

in such a way that even an undergraduate with limited training in Physics could un-

derstand. I will point out significant topics of research in KvNM and how they are

changing physics as we know it. By making this topic more accessible to a variety of

readers, I hope that many will find it of interest and continue to build upon this novel

and fascinating avenue of study.

1

KvNM is statistical CM cast in the language of complex wavefunctions in Hilbert

Space. Although wavefunctions in Hilbert Space are thought to be tools of QM, they

can be used to exactly model the statistical properties of purely classical systems.

Classical wavefunctions have been used to study a wide range of physical phenom-

ena. For example, they have been used in studying chaotic systems in quantum and

classical theory (Wilkie and Brumer 1997), been used to develop a model of stochas-

ticity from determinism (McCaul and Bondar 2021), been used to study the inter-

play of connected quantum and classical systems (Bondar, Gay-Balmaz, and Tronci

2019), been used to study free fall and Fischer Information (Sen, Dhasmana, and

Silagadze 2020), been used to reproduce equations governing spin 1/2 relativistic

particles (Cabrera et al. 2019), among many other areas of research.

Of great interest, a field like KvNM might be able to decipher the true meaning

of QM, which has been hotly debated for over a century.1

2 General Review of Quantum Mechanics

What follows is a brief review of Quantum Theory. QM depends on operations on

the vector |ψ〉 in Hilbert Space. The wavefunction |ψ〉 is a complete description of

the state of the system in QM (Shankar 1988). It contains probabilistic information

of the system’s behavior.

It has a number of convenient properties, such as the fact that it makes switching

between one set of bases and another is a rather straightforward task. Hilbert Space

useful for all sorts of mathematical procedures. It is, however, not the typical tool of

CM.1Despite a century of progress, QM has no consensus interpretation of its equations.

‘Shut up and calculate’ rings as true today as it did a century ago. Feynman gives us a glimmer ofhope for the situation, however, as it always takes several generations to understand a great, mysterioustruth: “We have always had a great deal of difficulty understanding the world view that quantummechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point thatthis stuff is obvious to me... You know how it always is, every new idea, it takes a generation or twountil it becomes obvious that there’s no real problem.” (Feynman 1982) Perhaps we’re getting closerto an understanding and perhaps a budding field like KvNM can hold clues to the answer.

2

2.1 Fundamentals of Hilbert Space

To understand QM, we must first be comfortable with the concept of a Hilbert Space.

A Hilbert Space is an infinite dimensional vector space with a well defined inner

product. One can conceive of the Hilbert Space as a coordinate system with an infinite

number of perpendicular axes as a crude model. For QM, the Hilbert Space we use

not only contains real numbers, but has also been extended to the complex numbers.

Although there are Hilbert Spaces that only contain real numbers, we will not need

to concern ourselves with those when working with QM or KvNM.

To see why Hilbert Spaces are so useful in QM, imagine a generic function f(x)

which is continuous on its domain (Shankar 1988). Since any continuous function

can be represented by an infinite number of points, we can represent any continuous

function in Hilbert Space with state vectors. Each point along a generic function

f(x) can be represented by a vector in Hilbert Space. Because there is an infinite

number of points along the function f(x) (even in a finite segment), we should not

be surprised it will take an infinite number of vectors to represent f(x).

This collection of vectors can be represented in an orthogonal basis (i.e., each

vector is perpendicular to any other vector in the collection or set, as there are an

infinite number of perpendicular axes in Hilbert Space.) A vector can be represented

with the following bra-ket notation (original to Dirac):

~v =

v1

v2

v3

=⇒ |v〉 =

v1

v2

v3

, 〈v| =(v∗1 v∗2 v∗3.

)

where |v〉 is the ket vector and 〈v| is the bra vector (Shankar 1988). The asterisk

(*) represents the complex conjugate. Vectors like ~v are ambiguous, because it is

not often clear if to write them as columns (as depicted above) or rows of numbers.

However, the Dirac bra-ket notation overcomes the ambiguity of typical vector nota-

tion in physics, by having that the ket is always equivalent to a column and the bra is

equivalent to a row of values. Hilbert Space utilizes bra-ket notation in its infinite di-

3

mensional mathematics. A traditional Hilbert Space ket might have more than three

entries therefore; it might be represented with an infinite number of entries.

With this in mind, the continuous function f(x) would be represented in Hilbert

Space as:

f(x) =⇒ |f(x)〉 =

f(x1)

f(x2)

f(x3)

...

f(xn)

...

. (1)

You can think about this representation of f(x) as the value of each point along f(x),

i.e. f(xi), having a unique independent basis vector in Hilbert Space multiplying it.

So according to the rules of Linear Algebra you may expand |f(x)〉, for example:

|f(x)〉 =

1

0

0

...

0

...

f(x1) +

0

1

0

...

0

...

f(x2) +

0

0

1

...

0

...

f(x3) + ...+

0

0

0

...

1

...

f(xn) + ...

In the above expansion, notice that the column vectors form a unique set of linearly

independent basis vectors.

The way to switch between bras and kets in a complex Hilbert Space is to take

the transpose of the vector and complex conjugate of each of the values, like so:

4

|v〉 =

v1

v2

v3

...

⇐⇒ 〈v| =

(v∗1 v∗2 v∗3 . . .

)

The asterisks represents the complex conjugate. This operation will be represented

by †, so that |v〉† = 〈v| and vice versa. Just like any ordinary vector, bra-ket vectors

can be added component wise and multiplied together:

v1

v2

v3

...

+

w1

w2

w3

...

= |v〉+ |w〉 =

v1 + w1

v2 + w2

v3 + w3

...

= |v + w〉 , (2)

(w∗1 w∗2 w∗3 . . .

)

v1

v2

v3

...

= v1w

∗1 + v2w

∗2 + v3w

∗3 + . . . = 〈w|v〉 ,

where 〈w|v〉 is a generalized “dot product” or inner product for the vector space. Two

vectors are said to be orthogonal if their inner product is 0. So in order for two vectors

|v〉 and |w〉 in Hilbert Space to be orthogonal, we must have 〈w|v〉 = 0. Also, like

for any common vector, you can rescale a vector by a complex constant c to stretch

or shrink its magnitude: |v′〉 = c |v〉.

The vectors additionally obey the following properties:

(〈w|v〉)† = 〈v|w〉 ,

|c1v + c2w〉† = (c1 |v〉+ c2 |w〉)† = (c1 |v〉)† + (c2 |w〉)† = c∗1 〈v|+ c∗2 〈w| .

5

Since continuous functions like f(x) can be represented in Hilbert Space, we can

recast continuous functions of QM into the Hilbert Space formalism. We will shortly

see why this is useful to do.

The wavefunction ψ is the backbone of the quantum mechanical description of

reality. The wavefunction ψ is an equation of state, i.e., it provides us with a descrip-

tion of the state of the system (Shankar 1988). 2The wavefunction got its name from

the fact that its form resembles that of a wave equation. In configuration space, the

wavefunction is often written as:

ψ(q1, q2, ..., qn, t) = R(q1, q2, ..., qn, t)eiS(q1,q2,...,qn,t)/~, (3)

where q1, q2, ..., qn are positions in configuration space and t is time. S represents the

Action and R is the amplitude of the wave equation. In light of the above discussion

and since ψ is a continuous function, we can represent ψ as |ψ(t)〉 in complex Hilbert

Space. Schrodinger’s equation, for example, becomes:

i~∂ψ

∂t= − ~2

2m

∂2ψ

∂q2+ V (q)ψ =⇒ i~

∂

∂t|ψ(t)〉 = − ~2

2m

∂2

∂q2|ψ(t)〉+ V (q) |ψ(t)〉 .

(4)

The benefits of using Hilbert Space instead of just utilizing ψ is because Hilbert

Space provides a rich mathematical, abstract structure where various manipulations

are straightforward and easy to do (Shankar 1988; Bondar et al. 2012). We can very

easily, for instance, change our basis of interest from one to another. We can ‘de-

construct’ functions and put them in various representations. This is not a flexibility

given by simply using ψ. As we will see, changing representations is a fundamental

ingredient to QM, so the Hilbert Space form is absolutely necessary.2One way you can think of an ‘equation of state’ is, for instance, the ideal gas law, PV = NkT .

This equation will tell you what the state of the system, i.e., what the pressure P , volume V , numberof gas particles N , and temperature T , for an ideal gas is.

6

2.2 Operators and Commutators

An operator is a mathematical object that “acts on” a function, vector, or matrix to

produce a new function (the derivative), a new vector, or a new matrix. For example,

the derivative ddt

acting on a function f(t) is an operator, as is a matrix M acting on

a vector |v〉. The operator produces a new mathematical entity, such as the df(t)dt

from

f(t) or a new vector |v′〉 from the original vector (|v′〉 = M |v〉). All operators will

be represented with triangular “hats”, for example, A.

Two important types of operators are Hermitian operators and anti-Hermitian op-

erators, which obey the following rules respectively:

A† = A,

A† = −A,

where the † symbol is the transpose with complex conjugate as before. The operators

we most use in QM are Hermitian operators. Hermitian operators are incredibly

useful to utilize, because if we take the † of an expression with a Hermitian operator,

the form of the Hermitian operator is left unchanged. For example, for two vectors

|v〉 and |w〉 and Hermitian operator H:

(〈w| H |v〉)† = 〈v| H† |w〉 = 〈v| H |w〉 .

An observable in QM is a property of a system you would like to measure or observe.

Examples of observables in QM would be the position of a particle, its momentum,

its spin, etc. Observables in QM are given by Hermitian operators A acting on the

state vector |ψ(t)〉, i.e., A |ψ(t)〉. The state vector |ψ(t)〉 is a vector in Hilbert Space

that contains complete dynamical information about the system in question.

Hermitian operators relate directly to properties you measure inside the labora-

tory. For any generic operator A, there exist special types of ket vectors which when

acted on by an operator A they equal themselves rescaled by a constant. In other

7

words, they obey the mathematical form of

A |ψ(t)〉 = A |ψ(t)〉 , (5)

where ‘A’ is always a real number (Shankar 1988).

Note that this special relationship is not true for just any vector |ψ(t)〉. This is only

true for specific vectors which we will call eigenkets. The rescaling real number ‘A’

is known as the eigenvalue, and the mathematical problem to identify for an operator

A the correct eigenkets and eigenvalues is called the eigenvalue problem(Shankar

1988)3. The eigenvalue problem is central to QM because it directly relates to the

observables measured in the laboratory, as will be developed below.

Perhaps one of the most important properties of a Hermitian operator is that its

eigenvalues must necessarily always be real (Shankar 1988). Other types of opera-

tors might give, for example, imaginary eigenvalues. Hermitian operators will only

give you real numbers, which are the types of values we would measure inside the

laboratory.

Two very important operators are the ones for the observables momentum and

position. In coordinate representation (as other representations are possible), the mo-

mentum and coordinate operators are given respectively as follows:

p =~i

∂

∂q, (6)

q = q. (7)

These operators are important, because as we shall see, they are related to the average

momentum and position of interest. 4

3The function version of the eigenvalue problem is Aψ = Aψ, where ψ is called an eigenfunction.To recover the Hilbert Space version, remember the colloquial transformation ψ =⇒ |ψ(t)〉. In theliterature, terms like eigenvectors and eigenfunctions are used interchangeably, as they are technicallythe same thing, just one written in an abstract Hilbert Space and one written without.

4For example, in standard function form one can easily show that the pψ = pψ where p is themomentum expectation value measurable in the laboratory and p is the above momentum operator.Earlier, we said that a common form of ψ in configuration space is ψ = ReiS/~, where R is the

8

Commutators describe relations between operators. The principle that two num-

bers A and B obey A · B = B · A is known as the principle of commutativity.

Operators do not need to, however, obey the principle of commutativity. We might

have, for example, that AB 6= BA for two operators A and B. This might be most

explicitly clear when we think about the example of matrix multiplication. The order

matrices are multiplied in might give you different end products. We say therefore

that matrix multiplication is noncommutative.

In order to describe this property for operators, we use the idea of a commutator.

For two operators A and B, the commutator is defined as:

[A, B] ≡ AB − BA.

If the commutator equals zero, the two operators commute:

[A, B] = AB − BA = 0 =⇒ AB = BA.

However, if the commutator is nonzero, then obviously the operators do not commute.

The position and momentum operators are an example of noncommuting opera-

tors. It can be demonstrated that:

[q, p] = qp− pq = q~i

∂

∂q− ~i

∂

∂qq.

[q, p]ψ = q~i

∂ψ

∂q− ~i

∂

∂q(qψ) = q

~i

∂ψ

∂q− ~i

∂q

∂qψ − q~

i

∂ψ

∂q= −~

iψ.

Ergo, the commutator relation for q and p is

amplitude of the wavefunction and S is the Action. Let’s assume a simple, one dimensional casewhere R = R(t) and S = S(q, t). Then we would have:

pψ =~i

∂

∂qReiS/~ =

~iReiS/~

i

~∂S

∂q=∂S

∂qψ.

From CM, we know that p = ∂S∂q , so the shoe fits. Please note that this exercise is a bit of an over-

simplification. Generally, R = R(t) is not considered a legitimate choice because it can lead tononnormalizable wavefunctions (see Tipler 2010 for further discussion.) For the sake of a simpledemonstration of how operators acting on wavefunctions lead to eigenvalues measured in the labora-tory, I chose this oversimplified approach to highlight connections between ideas .

9

[q, p]ψ = −~iψ = i~ψ =⇒ [q, p] = i~. (8)

This is the famous Heisenberg Canonical Commutator (Shankar 1988). In section

5.2 we will derive the Heisenberg Uncertainty Principle from the Canonical Com-

mutator. The following are some important commutator relationships given below

with derivation for brevity:

[A, B] = −[B, A]. (9)

[cA, B] = [A, cB] = c[A, B]. (10)

[AB, C] = A[B, C] + [A, C]B. (11)

[A, BC] = B[A, C] + [A, B]C. (12)

[A, B + C] = [A, B] + [A, C]. (13)

2.3 Expectation Values and the Eigenvalue Problem of Quantum

Mechanics

What exactly are the eigenvalues that we get out of the eigenvalue problem? I have

stated before that they are linked to measured observables, but in what way exactly?

Observables can be anything you would like to measure, such as the momentum or

position of a quantum particle.

Eigenvalues are going to be the numbers you measure directly inside the labora-

tory. If you conduct an experiment on a quantum system, the eigenvalues will be the

exact numbers you record. Remember that for a Hermitian operator, its eigenvalues

10

will always be real numbers. This is critical, because any number you measure in an

experiment will be real instead of, say, complex.

Eigenvalues are connected to finding the expectation values of an experiment.

Imagine you conduct an experiment on a quantum system and you measure the po-

sition or momentum. Instead of predicting individual momenta or position, QM can

predict the weighted mean value that you should expect. If you took many measure-

ments on identical systems, and took the weighted average of all those measurements,

the value you get out of this is the expectation value.

I will demonstrate how expectation values arise in QM in the following. I will

use both the Hilbert Space formulism and the function ψ that some might be more

comfortable with. Both derivations will end up with identical conclusions.

First, start with the eigenvalue expression Aψ = Aψ and multiply both sides by

the complex conjugate, like so:

ψ∗Aψ = ψ∗Aψ = Aψ∗ψ

In Hilbert Space, the same can be achieved by multiplying the eigenvalue expression

A |ψ(t)〉 = A |ψ(t)〉 with a bra vector from the left, so that the operator is ‘sand-

wiched’:

〈ψ(t)| A |ψ(t)〉 = 〈ψ(t)|A |ψ(t)〉

For the typical function form, we are going to multiply both sides by the ‘operator’´∞−∞ dx, in order to get:

ˆ ∞−∞

ψ∗Aψdx =

ˆ ∞−∞

Aψ∗ψdx (14)

We now introduce a fundamental postulate of QM, namely, the Born Rule. The

Born Rule states that ρ = ψ∗ψ where ρ represents the probability density for contin-

uous ψ. From probability theory, we know that if a probability density is integrated

11

over all possible states, it must be equal to 1 (the renormalization condition):

ˆ ∞−∞

ψ∗ψdx = 1 (15)

In other words, the probability of something happening is certain (= 1) if all possible

outcomes occur. The equivalent renormalization condition in Hilbert Space is that

〈ψ(t)|ψ(t)〉 = 1.5

Equation (14) is defined to be the expectation value of the observable given by

operator A, as can be seen from the similarity of both:

〈f(x)〉 =

ˆ ∞−∞

f(x)ρ dx⇐⇒ˆ ∞−∞

ψ∗Aψdx =

ˆ ∞−∞

Aψ∗ψdx,

where you can see that the eigenvalueA takes on the role of f(x) in the above expres-

sion and ψ∗ψ is the probability density. Please keep in mind that there could be many

different values you can measure for observable A, so it makes sense it is equivalent

to the function f(x). Therefore the expectation value of observable A is given by

〈A〉 =

ˆ ∞−∞

ψ∗Aψdx = 〈ψ(t)| A |ψ(t)〉 . (16)

The expectation values measured in the laboratory happen to be the exact ones pre-

dicted by the above equation. The expectation value expression was strictly derived

from the eigenvalue problem and the Born Rule.5Both renormalization conditions can be shown to be equivalent. From Appendix A, we have the

important completeness theorem, which says:

I =

ˆdx |x〉〈x|

Since I is just the identity operator (operator where IA = A or I |ψ〉 = |ψ〉, just like multiplyingsomething by the number ‘1’), we can do the following operations:

〈ψ(t)|ψ(t)〉 = 〈ψ(t)| I |ψ(t)〉 =ˆ〈ψ(t)|x〉〈x|ψ(t)〉 dx

Recall from before that we have said that 〈x|ψ(t)〉 = ψ(x, t), so we get:

〈ψ(t)|ψ(t)〉 =ˆψ∗ψdx

12

2.4 Review of Dirac Delta Functional

The Dirac delta functional is important to QM for the renormalization of all the basis

vectors. For any Hilbert Space, one can find an orthogonal set of bases to represent it.

But we can do one better - one can always find an orthonormal basis. An orthonormal

basis is a basis where not only are all the basis vectors orthogonal, but they are also

normalized to unit length 1:

|A〉normalized =|A〉√〈A|A〉

(17)

Any two generic vectors |A〉 and |A′〉 in Hilbert Space are orthonormal if and only if

they satisfy

〈A|A′〉 = δAA′ =

1 if |A〉 = |A′〉

0 if |A〉 6= |A′〉

where δAA′ is the Kronecker delta. The Kronecker delta functional is however only

appropriate when dealing with strictly discrete systems. Many problems in QM, how-

ever, can be either discrete or continuous. We therefore need a functional that can

encompass both discrete and continuous systems to describe an orthonormal basis.

This is done by using the Dirac delta functional6:

. 〈A|A′〉 = δ(A− A′) (18)

The Dirac delta functional is a generalization of the Kronecker delta (Shankar 1988).

The Dirac delta functional obeys the following properties:

δ(A− A′) = 0 if A 6= A′.

ˆ b

a

δ(A− A′) dA′ = 1 if a ≤ A ≤ b.

6Please note that the A here is a generic placeholder variable with arbitrary units. It could bea placeholder for position q, momentum p, or some other measurable quantity. So you could, forexample, have 〈q|q′〉 = δ(q − q′) for position and 〈p|p′〉 = δ(p− p′) for momentum, etc.

13

The functional is zero everywhere but at A′ (Shankar 1988).

The Dirac delta functional should not be treated as a regular function. It has its

own unique properties and form, distinct from how functions behave. For example,

the area under a single point when A = A′ should be zero (the area under a point is

zero). But for the Dirac delta functional, it is precisely 1. This and other properties

of this functional highlight the fact that it should not by treated like the functions

you have seen up to this point. It is a functional, not a function. It is, however,

fundamental to the renormalization of vectors in Hilbert Space.

Other important properties of the functional we will utilize in the text of this

article include the following:

ˆδ(A− A′)f(A′)dA′ = f(A). (19)

δ(A− A′) = δ(A′ − A). (20)

d

dAδ(A− A′) = − d

dA′δ(A− A′). (21)

Some useful relations for QM:

dn

dAnδ(A− A′) =

ˆdω

2π(−iω)ne−iω(A−A′). (22)

ˆdn

dAnδ(A− A′)f(A′) dA′ = (−1)n

dnf(A)

dAn. (23)

δ(αA) =δ(A)

|α|, (24)

where α is a constant.

2.5 Summary of Quantum Mechanics and its Postulates

The postulates of QM can be reviewed as follows (Shankar 1988; Bondar et al. 2012):

14

1. The state of a quantum system (particle or collection of particles) is represented

by a vector |ψ〉 in complex Hilbert Space.

2. For any observable A, there is an associated Hermitian operator A with eigen-

value problem A |A〉 = A |A〉 where |A〉 is the specific eigenket and A is the

eigenvalue which is measured in the laboratory. Examples of common observ-

ables are position of a particle q and momentum p. The variables position q

and momentum p are represented by Hermitian operators q and p which obey

the following properties: 〈q| q |q′〉 = qδ(q−q′) and 〈q| p |q′〉 = −i~ ddqδ(q−q′).

3. Born Rule: For a continuous distribution, the probability density for a particle

is ρ = ψ∗ψ. Equivalently in Dirac Notation, it can be written that the proba-

bility density of observing A is ρ(A) ∝ 〈ψ|A〉〈A|ψ〉 = ‖ 〈A|ψ〉 ‖2 where “∝”

is the proportionality sign. More precisely we could say that the Born Rule

dictates that

ρ(A) =〈ψ|A〉〈A|ψ〉〈ψ|ψ〉

As long as the above equation meets the renormalization condition 〈ψ|ψ〉 = 1

(see equation (15) and associated footnote), the Born Rule in its final form is

often written as

ρ(A) = 〈ψ|A〉〈A|ψ〉

For a discrete system, the Born Rule is written as:

P (A) = 〈ψ|A〉〈A|ψ〉 ,

assuming renormalization, where P is the probability. For both the continu-

ous and discrete cases, the state of the system collapses from the initial state

|ψ(t)〉 to the final state |A〉 whenever a measurement of the quantum system is

undertaken. Under the Copenhagen Interpretation of QM, the collapse of the

wavefunction |ψ〉 to the specific eigenket |A〉 is instantaneous and irreversible.

If A is observed in experiment, the wavefunction collapses to the associated

15

eigenket |A〉.

4. The state space of a composite system is the tensor product of the subsystem’s

state spaces

There are a couple other important things to say about the above overview of QM.

Firstly, although not included as a postulate, the expectation value relationship (16)

plays a critical role inside of QM. They are of fundamental importance as we reviewed

in section 2.3.

Not discussed thoroughly is the centrality of the Schrodinger equation to QM.

The state of the quantum system |ψ〉 obeys the Schrodinger equation:

i~∂

∂t|ψ〉 =

[− ~2

2m

∂2

∂q2+ V (q)

]|ψ〉 = H |ψ〉 (25)

where H is the quantum Hamiltonian that bears great similarity to the classical Hamil-

tonian7, except that the momentum and position are replaced by the momentum and

position operators of QM:

H(q, p) =p2

2m+ V (q) =⇒ H(q, p) =

p2

2m+ V (q) = − ~2

2m

∂2

∂q2+ V (q)

Of important note, the Hamiltonian operator which acts on the wavefunction in equa-

tion (25) is Hermitian (Shankar 1988). Since it is Hermitian, its eigenvalues will al-

ways be real instead of complex or imaginary, as required for numbers measured in

the laboratory.

There will be a slight adjustment of notation from the previous couple sections.

Although it was convenient to previously write A |ψ〉 = A |ψ〉 as the eigenvalue

problem, it is more precise to write the eigenvalue problem as A |A〉 = A |A〉. As

described before, for an operator A there are specific eigenvalues and eigenkets that7That is, a classical Hamiltonian where the potential energy has no velocity or time dependence,

which takes on the form H(q, p) = p2

2m + V (q). Note that p is momentum and V (q) is a one dimen-sional potential energy.

16

satisfy the above equation. Although |ψ〉might be used to describe the specific eigen-

ket in the context of the eigenvalue problem, |ψ〉 is often the generic state vector of

the quantum system, so this might lead to some confusion. To highlight the fact that

only specific eigenkets |A〉 satisfy this relationship, A |A〉 = A |A〉will be used from

now on.

Two very important eigenvalue problems are q |q〉 = q |q〉 for position q and

p |p〉 = p |p〉 for momentum p. The expectation values of the position and momentum

measured in the laboratory will be the eigenvalues q and p. For any observable A,

there is a relationship A |A〉 = A |A〉 for which combined with the Born Rule axiom,

you can deduce the expectation value of A (as described in section 2.3).

One of the initially confusing things about the Dirac notation is that the symbols

inside the bras and kets are just labels. So you might see vector bras and kets with

labels such as |1st〉, |2nd〉, |I〉, |II〉, |ω = 2.5〉, etc. All of these have valid nota-

tion, as the labels simply describe the associated vector, and that is all. Usually, the

eigenvalue is used to label the associated eigenket. Hence the eigenvalue ‘A’ being

associated with eigenket |A〉 in the eigenvalue problem. It is why in equation (2), we

could make a statement such as |v〉 + |w〉 = |v + w〉, because both v and w are just

labels.

Finally, the collapse of the wavefunction describes one of the ongoing mysteries of

QM. It can be seen that the Schrodinger equation is fully deterministic from equation

(25), but the collapse of the wavefunction is (as far as we know) indeterministic and

random. There is no known formula to describe how the wavefunction collapses.

More will be said about wavefunction collapse in section 5.3 of this article, in the

context of QM and CM.

2.6 The Density Operator

Before we delve straight into KvNM, it is critical to briefly look at one more tool

that will be of use to us in this work. Instead of representing the state of a system as

a wavefunction vector in Hilbert Space |ψ〉, we can instead use something called a

17

density matrix for certain systems, represented by ρ(t) (not to be confused with the

previously used ρ for probability density.) The density matrix is defined by

ρ(t) =∑j

pj |ψj(t)〉〈ψj(t)| , (26)

for the discrete case. The density matrix is used when you are not certain of the

configuration of your system at some point in time. You can represent it as statistical

ensemble of possible wavefunction configurations |ψj(t)〉, each with probability of

occurrence pj . Note with probability, we must have

∑j

pj = 1.

We will see an example of the density matrix used in action later in section 5.3.

Density matrices are good for describing the above described so-called “mixed states”

of a system (with the “pure states” being the individual wavefunctions |ψj〉). If j = 1

in (26) the density matrix represents a pure state, but if it is greater than 1 it represents

a mixed state.

The probability of measuring A, or P (A), is given by the density matrix formu-

lation as follows:

P (A) = Tr[ρ(t) |a〉〈a|], (27)

where Tr[...] refers to the trace of the matrix. You can represent the trace by

Tr[...] =∑n

〈n| ... |n〉 , (28)

where |n〉 are arbitrary basis of your choice. This is the density matrix version of the

Born Rule. It is easy to see that this is fully equivalent to the Born Rule, for instance,

when we plug (26) with j = 1 into (27):

P (A) = 〈a|ψ1〉〈ψ1|a〉〈a|a〉 = | 〈A|ψ1(t)〉 |2, (29)

18

since 〈a|a〉 = 1 assuming orthonormality. The expectation value of operator A is

given by

〈A〉 = Tr[Aρ(t)

], (30)

where Aρ(t) form a matrix and Tr[...] refers to the trace of the matrix. You can reduce

(30) to (16) just by applying the trace formula (28), easily demonstrated for the pure

state j = 1:

Tr[Aρ(t)

]=∑n

〈n| A |ψ1(t)〉〈ψ1(t)|n〉 =∑n

〈ψ1(t)|n〉〈n| A |ψ1(t)〉 .

Applying resolution of identity (Theorem II of Appendix A) we get the standard ex-

pectation value formula:

Tr[Aρ(t)

]= 〈ψ1(t)| A |ψ1(t)〉 = 〈A〉.

Other useful properties of the density matrix:

1. Tr[ρ(t)] = 1, which is the normalization condition.

2. Tr[ρ(t)2] = 1 for pure state.

3. Tr[ρ(t)2] < 1 for mixed state.

4. ρ(t) = ρ(t)†, in other words, ρ(t) is Hermitian.

(Based on MIT OpenCourseWare 2006.) We will utilize the density operator in future

sections.

3 Introduction to KvN Mechanics

Koopman-von Neumann Mechanics is unique because it is a form of classical statis-

tical mechanics that truly takes on a lot of the mathematical structure of QM (Koop-19

man 1931; von Neumann 1932; Bondar et al. 2012). It was invented in the 1930’s by

Bernard Koopman and John von Neumann. Subsequently forgotten for some time by

most physicists, its resurgence raises intriguing questions about the nature of QM to

CM. Just like with QM, we have a set of operators that act on vectors in Hilbert Space

in order to retrieve eigenvalues of observables (Koopman 1931; von Neumann 1932;

Bondar et al. 2012). In order to have a proper Hilbert Space, we first must define the

inner product of that Hilbert Space (Koopman 1931):

〈ψ|φ〉 =

ˆdA 〈ψ|A〉〈A|φ〉 =

ˆdA ψ∗(A)φ(A).

CM always has distinct trajectories that particles travel along. In QM, we have the

Heisenberg Uncertainty Principle which states you cannot have both a well defined

position of a particle and traveling momentum (or velocity) at the same time. Un-

der the Copenhagen Interpretation, this is not due to some principle of measurement,

rather, this is a fundamental property of the fabric of reality (Shankar 1988). Reality

forbids quantum particles from having a precise momentum and position simultane-

ously. In QM, trajectories could be viewed as being “fuzzy” instead of well defined

and localized like in CM. You can no longer draw nice curves to describe how a

particle travels in space under QM.

Because of this fact, both position and momentum would have a common set of

eigenstates in KvN Hilbert Space (Bondar et al. 2013; McCaul and Bondar 2021).

The state vectors for KvN would therefore have the following properties (Koopman

1931):

|A〉 = |q, p〉 = |q〉 ⊗ |p〉 ,

q |q, p〉 = q |q, p〉 ,

p |q, p〉 = p |q, p〉 ,

20

〈q, p|q′, p′〉 = δ(q − q′)δ(p− p′),

ˆdp dq |q, p〉〈q, p| = I. (31)

The observable A’s eigenket |A〉 is the KvN ket |q, p〉 because of the common set of

eigenstates between q and p. Because it contains information about q and p, this is

providing information in phase space instead of the previously discussed configura-

tion space (Koopman 1931). Previously for configuration space we had 〈q|ψ(t)〉 =

ψ(q, t),8 whereas here we will have 〈q, p|ψ(t)〉 = ψ(q, p, t) in phase space.

In QM, we have the Schrodinger equation (25) to dictate to us the time evolution of

the quantum particle described by |ψ〉. To estimate the probability of an observable,

we utilize ρ(A, t) = ψ∗(A, t)ψ(A, t) = | 〈A|ψ(t)〉 |2 (Born Rule). In CM, we have the

following equation known as the Louiville Equation to describe classical trajectories:

∂ρ

∂t+∑(∂H

∂pj

∂ρ

∂qj− ∂H

∂qj

∂ρ

∂pj

)= 0 (32)

The Louiville Equation is important because it tells you the probability density ρ of

finding a particle in observable point ω = (q, p) in phase space. In other words, it

defines the probabilistic time evolution of an ensemble of particles. We can define

the Louiville operator to be (Mauro 2003):

L =∑(

− i∂H∂pj

∂

∂qj+ i

∂H

∂qj

∂

∂pj

)(33)

By doing so, we can rewrite equation (32) in operational terms:

i∂ρ

∂t= Lρ (34)

Notice each term in equation (33) has been multiplied by the imaginary unit i. Just8Which can be generalized into 〈q1, q2, q3, ..., qn|ψ(t)〉 = ψ(q1, q2, q3, ..., qn, t) for configuration

space.

21

like in QM, we postulate a Born Rule for this probability density. If ρ(A, t) = ψ∗ψ

(where A is the observable), then we can easily show that 9

i∂ψ

∂t= Lψ (35)

This equation, of course, bears a striking resemblance to equation (25), the Schrodinger

equation in its operatorial form. The classical wavefunction ψ here is interpreted as a

probability density amplitude, like under most interpretations of QM (Shankar 1988).

Just like with QM, KvN depends on the generic state vector |ψ(t)〉 containing statis-

tical information about the system. Colloquially, just as before, ψ =⇒ |ψ(t)〉.

An important property of KvN Mechanics is that the Louiville operator L is Her-

mitian, just like the Hamiltonian operator of the Schrodinger equation. The norm

of 〈ψ(t)|ψ(t)〉 =´dωψ∗(ω)ψ(ω) is therefore conserved, further strengthening the

Born’s Rule postulated for KvN Mechanics (Mauro 2003).

The Axioms of KvN Mechanics can be summarized as follows (based off Bondar

et al. 2012):

1. The state of a system is represented by a vector |ψ〉 in a complex Hilbert Space.

2. For any observable A, there is an associated Hermitian operator A with eigen-

value problem A |A〉 = A |A〉 where |A〉 is the specific eigenket and A is the

eigenvalue measured in the laboratory.

3. Born Rule: The probability of measuring A is given by P (A) = | 〈A|ψ(t)〉 |2.

The wavefunction |ψ〉 describing the system instantaneously collapses into the9Demonstrated as follows. If ρ = ψ∗ψ, then for a simple case with index j = 1 we plug it into

equation (34):

i∂ψ∗ψ

∂t+ i

∂H

∂p

∂ψ∗ψ

∂q− i∂H

∂q

∂ψ∗ψ

∂p= 0

i∂ψ∗

∂tψ + i

∂ψ

∂tψ∗ + i

∂H

∂p

∂ψ∗

∂qψ + i

∂H

∂p

∂ψ

∂qψ∗ − i∂H

∂q

∂ψ∗

∂pψ − i∂H

∂q

∂ψ

∂pψ∗ = 0

[i∂ψ∗

∂t+ i

∂H

∂p

∂ψ∗

∂q− i∂H

∂q

∂ψ∗

∂p

]ψ +

[i∂ψ

∂t+ i

∂H

∂p

∂ψ

∂q− i∂H

∂q

∂ψ

∂p

]ψ∗ = 0

Since generally neither ψ nor ψ∗ are zero, this means the expressions in square brackets must equalzero. The bracketed expressions are, of course, equation (35) and its complex conjugate.

22

eigenket |A〉 associated with the observed eigenvalue A.

4. The state space of a composite system is the tensor product of the subsystems’

state space

Notice, these are exactly the same postulates of QM in section 2.5, just with anything

“quantum” removed (Bondar et al. 2012). The formula for the expectation value of

the system is still the same, since Axioms 2 and 3 remain unchanged, and therefore

the derivation in section 2.3 is the same. The very important expectation value rule

(16) therefore still applies in KvNM.

Given these four ‘universal’ postulates, one can derive both QM and KvNM from

the Ehrenfest Theorems. The Ehrenfest Theorems are as follows:

d

dt〈q〉 =

〈p〉m, (36)

d

dt〈p〉 = 〈−V ′(q)〉, (37)

and are usually considered strictly within the context of QM.

Under QM, the fabric of reality forbids the mutual existence of a well defined

position and momentum. Under CM, we can know the position and momentum to an

arbitrary precision. This difference is encoded with the commutator relationship. In

section 5.3, we will show that the Canonical Commutator of QM (8) directly produces

the Heisenberg Uncertainty Principle. We can also write a commutator relationship

for CM under the axioms of KvNM. We can say:

[q, p] = 0 (38)

I shall call this the “Classical Commutator”. This commutator relationship would

produce no uncertainty relationship between the classical position q and momentum

p. Ergo, position and momentum can be arbitrarily known.

For QM, one can show that the Canonical Commutator combined with the Ehren-

fest Theorem produces the traditional Schrodinger equation with quantum Hamilto-

23

nian. We will demonstrate this as follows. According to the common set of Axioms,

we can say that the expectation value of observable A is simply 〈ψ(t)| A |ψ(t)〉 (16).

Therefore we can easily say that

〈q〉 = 〈ψ(t)| q |ψ(t)〉

〈p〉 = 〈ψ(t)| p |ψ(t)〉

Then from the Ehrenfest Theorems one can carry out the following operations:

d

dt〈ψ(t)| q |ψ(t)〉 =

1

m〈ψ(t)| p |ψ(t)〉 ,

d

dt〈ψ(t)| p |ψ(t)〉 = 〈ψ(t)| − V ′(q) |ψ(t)〉 .

We are using the Schrodinger picture of QM, so we assume time dependence

solely resides in |ψ(t)〉 and any operators are independend of time. 10 Using the

chain rule of calculus, we write:

⟨dψ(t)

dt

∣∣∣∣ q |ψ(t)〉+ 〈ψ(t)| q∣∣∣∣dψ(t)

dt

⟩=

1

m〈ψ(t)| p |ψ(t)〉 ,

⟨dψ(t)

dt

∣∣∣∣ p |ψ(t)〉+ 〈ψ(t)| p∣∣∣∣dψ(t)

dt

⟩= 〈ψ(t)| − V ′(q) |ψ(t)〉 . (39)

Stone’s Theorem (see Appendix B) dictates that there exists a unique Schrodinger-like

equation i ∂∂t|ψ〉 = G |ψ〉 where G is the Hermitian generator of motion in operator

form. We deduce there must be a G that governs the evolution of CM, just as the

Hamiltonian operator does for QM. The only thing is to figure out what this generator

G is.

For QM, we can derive what the generator of motion is by using the commutator

relationships. We can take Stone’s Theorem and then derive:10This is the opposite of the Heisenberg picture, which assumes no time dependence in the wave-

function vector ket and puts all the time dependence inside the operators!

24

∣∣∣∣dψ(t)

dt

⟩=H

i~|ψ(t)〉 ⇐⇒

⟨dψ(t)

dt

∣∣∣∣ = −〈ψ(t)| Hi~

Which means:

−〈ψ(t)| Hi~q |ψ(t)〉+ 〈ψ(t)| q H

i~|ψ(t)〉 =

1

m〈ψ(t)| p |ψ(t)〉

− 〈ψ(t)| Hi~p |ψ(t)〉+ 〈ψ(t)| p H

i~|ψ(t)〉 = 〈ψ(t)| − V ′(q) |ψ(t)〉

You can expand the vector ‘sandwich’ and rewrite it in terms of a commutator to get:

〈ψ(t)| − Hq + qH |ψ(t)〉 = −〈ψ(t)| [H, q] |ψ(t)〉 =i~m〈ψ(t)| p |ψ(t)〉

〈ψ(t)| − Hp+ pH |ψ(t)〉 = −〈ψ(t)| [H, p] |ψ(t)〉 = 〈ψ(t)| − i~V ′(q) |ψ(t)〉

Since the Ehrenfest Theorems are true regardless of the particular state of the system

|ψ(t)〉 (i.e., they are true for any state you might find), we can remove the wavefunc-

tion vector from both sides to get:

−[H, q] =i~mp

[H, p] = i~V ′(q) (40)

We will make the assumption that H = H(q, p) and utilize Theorem VI from Ap-

pendix A. By this Theorem, we have q = A1, p = A2, and H(q, p) = f(A1, A2).

Therefore:

[H, q] = [H(q, p), q] =n=2∑k=1

[Ak, q]∂H(A1, A2)

∂Ak= [q, q]

∂H(q, p)

∂q+ [p, q]

∂H(q, p)

∂p

25

[H, p] = [H(q, p), p] =n=2∑k=1

[Ak, p]∂H(A1, A2)

∂Ak= [q, p]

∂H(q, p)

∂q+ [p, p]

∂H(q, p)

∂p

A commutator with itself will always equal zero:

[q, q] = qq − qq = 0

[p, p] = pp− pp = 0

Ergo, the above expressions based in Theorem VI reduce to:

[H, q] = [p, q]∂H(q, p)

∂p

[H, p] = [q, p]∂H(q, p)

∂q

Plugging these expressions into (40) gives us:

−[p, q]∂H(q, p)

∂p=i~mp

[q, p]∂H(q, p)

∂q= i~V ′(q) (41)

From the Canonical Commutator we know that [q, p] = i~.11 This step is absolutely

critical. It is in fact the step that is going to distinguish QM from classical KvNM

- the usage of the Canonical Commutator. We are going to plug in the Canonical

Commutator to the above expressions, so that we will get:

∂H(q, p)

∂p=

p

m

11We will also use the fact that [q, p] = −[p, q]. It’s not hard to prove, just takes a couple lines ofalgebra:

[q, p] = qp− pq = −(−qp+ pq) = −(pq − qp) = −[p, q]

26

∂H(q, p)

∂q=dV (q)

dq

Theorem VII of Appendix A will be useful, as we can now translate this expression

with operators into more familiar analytical functions in order to do operations with

derivations. We will have the following analytical functions as a result of Theorem

VII:

∂H(q, p)

∂p=

p

m

∂H(q, p)

∂q=dV (q)

dq

Some basic calculus reveals:

ˆ∂H(q, p)

∂pdp = H(q, p) =

ˆp

mdp =

p2

2m+ C(q)

whereC(q) is some function solely in terms of q so that it disappears when the partial

with respect to p is taken. Taking the partial derivative with respect to q of the above

expression reveals:

∂H(q, p)

∂q=dC(q)

dq=dV (q)

dq

Ergo, C(q) = V (q) so that the final solution isH(q, p) = p2

2m+V (q), which happens

to be equivalent to the quantum Hamiltonian. Therefore, plugging this solution of H

into the above version of Stone’s Theorem we reclaim the Schrodinger equation(25):

i~∂

∂t|ψ〉 =

[ p2

2m+ V (q)

]|ψ〉

The same series of steps can be carried out to derive KvNM. The only differ-

ence in the derivation is that we will assume the Classical Commutator instead of

the Canonical Commutator. Once again we start with the Ehrenfest Theorems and

follow a similar series of steps. Starting from (39), we will utilize Stone’s Theorem

27

as i ∂∂t|ψ〉 = K |ψ〉 to get:

−〈ψ(t)| Kiq |ψ(t)〉+ 〈ψ(t)| q K

i|ψ(t)〉 =

1

m〈ψ(t)| p |ψ(t)〉

− 〈ψ(t)| Kip |ψ(t)〉+ 〈ψ(t)| p K

i|ψ(t)〉 = 〈ψ(t)| − V ′(q) |ψ(t)〉

Following a similar series of steps, we can derive that:

−[K, q] =i

mp

[K, p] = iV ′(q) (42)

Now we run into a problem. If we assume K = K(q, p) as before, then we will derive

once again (41). However, now we will utilize the Classical Commutator instead of

the Canonical Commutator, since we want well defined classical trajectories. By

imposing the Classical Commutator [q, p] = 0, we run into a contradiction, namely:

−[p, q]∂K(q, p)

∂p= (0)

∂K(q, p)

∂p= 0 =

i

mp =⇒ p = 0

[q, p]∂K(q, p)

∂q= (0)

∂K(q, p)

∂q= 0 = iV ′(q) =⇒ dV

dq= 0

Obviously, this cannot be true for all systems, because many classical systems def-

initely have a nonzero momentum. This contradiction between theory and observa-

tion suggests that we must approach this from a different angle. One such angle is to

propose two new operators θ and λ (Bondar et al. 2012). These operators obey the

eigenvalue problems, θ |θ〉 = θ |θ〉 and λ |λ〉 = λ |λ〉, by postulate 2 of the ‘universal

axioms’, but are not tied to any direct physically observable features of reality (as

noted in Ramos-Prieto et al. 2018).12 They serve as an analogue to Lagrange Multi-

pliers (McCaul and Bondar 2021), serving to constrain the realm of possible behavior

to a very specific subset.12Even though these operators are not directly physically observable, their existence has been seen

through indirect means. These operators are the famous Bopp Operators of the phase-space formula-tion of QM (Bondar et al. 2013). We will explore the Wigner Distribution in section 6.

28

For KvNM, we will postulate the following behavior for these operators:

[q, θ] = [p, λ] = i

[qc, pc] = [q, λ] = [p, θ] = [θ, λ] = 0 (43)

where i is the imaginary unit. Equations (43) make up what is known as the Koopman-

von Neumann Algebra. If we do this, then we can assume K = K(q, p, θ, λ). By

Theorem VI of Appendix A again, we will get:

[K(q, p, θ, λ), q] =n=4∑k=1

[Ak, q]∂K(A1, A2, A3, A4)

∂Ak= [q, q]

∂K(q, p, θ, λ)

∂q+[p, q]

∂K(q, p, θ, λ)

∂p

+[θ, q]∂K(q, p, θ, λ)

∂θ+ [λ, q]

∂K(q, p, θ, λ)

∂λ

[K(q, p, θ, λ), p] =n=4∑k=1

[Ak, p]∂K(A1, A2, A3, A4)

∂Ak= [q, p]

∂K(q, p, θ, λ)

∂q+[p, p]

∂K(q, p, θ, λ)

∂p

+[θ, p]∂K(q, p, θ, λ)

∂θ+ [λ, p]

∂K(q, p, θ, λ)

∂λ

Three terms from each expression will disappear because of the KvN Algebra above

and the Classical Commutator. The equations of motion will end up being:

[θ, q]∂K(q, p, θ, λ)

∂θ= − i

mp

[λ, p]∂K(q, p, θ, λ)

∂λ= iV ′(q) (44)

Once we substitute in [q, θ] = [p, λ] = i we will simply have:

∂K(q, p, θ, λ)

∂θ=

p

m

29

∂K(q, p, θ, λ)

∂λ= −V ′(q)

Utilizing Theorem VII again, we get the differential equations:

∂K(q, p, θ, λ)

∂θ=

p

m

∂K(q, p, θ, λ)

∂λ= −V ′(q)

Which we proceed to solve:

ˆ∂K(q, p, θ, λ)

∂θdθ = K(q, p, θ, λ) =

ˆp

mdθ =

pθ

m+ C(q, p, λ)

∂

∂λK(q, p, θ, λ) =

∂

∂λ(pθ

m+ C(q, p, λ)) =

∂C(q, p, λ)

∂λ= −V (q)

dqˆ∂C(q, p, λ)

∂λdλ = C(q, p, λ) = −

ˆV (q)

dqdλ = −V (q)

dqλ+ C(q, p)

We can therefore derive

K = K(q, p, θ, λ) =pθ

m− V ′(q)λ+ C(q, p) (45)

which is the important Koopman Generator of KvNM. The Koopman Generator will

become very important when we discuss the Path Integral Formulation of QM and

CM (section 4).

We can show that the Koopman Generator gives us the Louiville Equations again

and produces the KvN classical wavefunction. To show this is quite straightforward.

Since the Born Rule for KvNM postulates that ρ = | 〈q, p|ψ(t)〉 |2, let us try to use the

Koopman Generator to find an expression that governs the evolution of the classical

probability density.

First, refer to (31). From these basic properties, one can easily show the following:

〈q, p| q |ψ(t)〉 = [〈q, p| q] |ψ(t)〉 = 〈q, p| q |ψ(t)〉 = q 〈q, p|ψ(t)〉 (46)

30

〈q, p| p |ψ(t)〉 = [〈q, p| p] |ψ(t)〉 = 〈q, p| p |ψ(t)〉 = p 〈q, p|ψ(t)〉 (47)

Using Theorem IV from Appendix A, we can demonstrate that the following are true.

Since [q, θ] = i, we can set κ = 1, |A〉 = |q, p〉, and B = θ for the parameters in

Theorem IV. We can say that the following is true then:

〈q, p| θ |ψ(t)〉 = −i ∂∂q〈q, p|ψ(t)〉 (48)

Notice that although A = (q, p), we do not write a partial derivative with respect to

p, and that is because [p, θ] = 0 per the Koopman Algebra.

The same can be done with [p, λ] = i. Using Theorem IV with parameters κ = 1,

|A〉 = |q, p〉, and B = λ, we get:

〈q, p| λ |ψ(t)〉 = −i ∂∂p〈q, p|ψ(t)〉 (49)

Again, there is no partial with respect to q here, since [q, λ] = 0 per the Koopman

Algebra.

Plugging the Koopman Generator into Stone’s Theorem gives us:

i∂

∂t|ψ(t)〉 = K |ψ(t)〉 =

[ pθm− V ′(q)λ+ C(q, p)

]|ψ(t)〉

And now we will ‘sandwich’ it from the left hand side with the vector bra 〈q, p| and

work out the algebra:

〈q, p| i ∂∂t|ψ(t)〉 = 〈q, p|

[ pθm− V ′(q)λ+ C(q, p)

]|ψ(t)〉

i∂

∂t〈q, p|ψ(t)〉 = 〈q, p| pθ

m|ψ(t)〉 − 〈q, p|V ′(q)λ |ψ(t)〉+ 〈q, p|C(q, p) |ψ(t)〉

31

= −i pm〈q, p|ψ(t)〉 ∂

∂q〈ψ(t)|q, p〉 − i 〈ψ(t)|q, p〉 p

m

∂

∂q〈q, p|ψ(t)〉

+iV ′(q) 〈q, p|ψ(t)〉 ∂∂p〈ψ(t)|q, p〉+ i 〈ψ(t)|q, p〉V ′(q) ∂

∂p〈q, p|ψ(t)〉

Recognizing the involvement of chain rule applied to the wavefunction:

i∂

∂t| 〈q, p|ψ(t)〉 |2 = −i p

m

∂

∂q| 〈q, p|ψ(t)〉 |2 + iV ′(q)

∂

∂p| 〈q, p|ψ(t)〉 |2

Since | 〈q, p|ψ(t)〉 |2 = ρ, the probability density, we can see right away that this

equation is fully equivalent to the Louiville equation (32)14, and that 〈q, p|ψ(t)〉 =

ψ(q, p, t) is the previously identified KvN classical wavefunction in phase space.

In summary, KvNM is a classical theory of physics that utilizes complex Hilbert

Spaces, just like in QM. Like QM, it is probabilistic in nature, so it can only make

probabilistic predictions of classical systems. Like in QM, you have a wavefunction

that completely describes the state of the system (35). This wavefunction is in phase

space, and KvNM is a phase space theory. It operates on the same axioms as QM,

but is a fully classical theory. It is not an extension of classical theory, but fully

equivalent to it, recast in a non-traditional form. The mathematical conveniences of

QM exist in KvNM. For example, one reason QM has its odd Hilbert Space form is

to conveniently change between vector bases. One can do this in KvNM with equal

ease. Just like in QM, the expectation value of observable A is given by 〈A〉 =

〈ψ(t)| A |ψ(t)〉.

The backbone of KvNM lies in the standard Hilbert Space behaviors (31), the

Koopman Generator (45), and the Koopman Algebra (43), which lets one derive the

Louiville Equation governing classical probabilities via usage of the KvNM classical

wavefunction. One can derive KvNM from the Ehrenfest Theorems by assuming the14Recall from Classical Mechanics that

F = −∂H∂q

= −V ′(q)

dq

dt=∂H

∂p=

p

m

33

Classical Commutator. The main difference between Classical and Quantum Physics

appears to lie in the choice of the commutator for position q and momentum p (Bondar

et al. 2012).

In the pages that follow, we will demonstrate the many applications of KvNM.

Even though technically it is strictly equivalent to any classical theory, it allows for

a wide variety of applications and uses. KvNM is a very important and convenient

tool for statistical physics.

4 Path Integral Formulation of Quantum and Classi-

cal Mechanics

4.1 Feynman Path Integral of Quantum Mechanics

Feynman Path Integral is another formulation of QM that is completely equivalent to

the Schrodinger equation. It was invented by the famous physicist Richard Feynman

in 1948 and has applications many usages in QM, some which you will see in this

Thesis. Whereas the Schrodinger equation is more analogous to Hamiltonian Dy-

namics (for instance, see section 5.1 for the similarity to Hamilton-Jacobi Theory),

the Path Integral is more analogous to Lagrangian Dynamics (Shankar 1988).

Under Lagrangian Dynamics in Classical Theory, you select two points, a starting

point and an endpoint point. In the allowed time interval, the classical particle would

take the path where the function S, called the Action, would be minimized. From

CM, the term T − V is called the Lagrangian (where T is kinetic energy and V is

potential energy) and the Action is defined as the time integral of the Lagrangian:

S =

ˆ t

0

dt(T − V ) (51)

This function has to be minimized to define the classical path. The minimization rule

for classical particles - called the Least Action Principle or Hamilton’s Principle - is

often written as:

34

δ

ˆ t

0

dt(T − V ) = 0

Things are not so simple in Quantum Theory.

We begin with a simple derivation of the Feynman Path Integral to understand

the differences between QM and Lagrangian Theory. We start with the following

expression, which is simply a rewrite of the Schrodinger equation:

|ψ(t)〉 = e−i~ tH |ψ(0)〉 (52)

In Appendix C, one can see how this expression is completely equivalent to the

Schrodinger equation. One might also notice that the operator H is exponated, but

this is not unusual in QM (more information about exponated operators and what

they mean is found in the same Appendix). The term exp− i

~tH

is called the uni-

tary time-evolution operator and defines how the quantum particle will evolve in time

from its starting condition |ψ(0)〉 to its final condition |ψ(t)〉 at time t (see Appendix

B for more on unitary operators).

To begin, we insert a position q bra on both sides of (52):

〈q|ψ(t)〉 = ψ(q, t) = 〈q| e−i~ tH |ψ(0)〉 (53)

And then insert resolution of identity (Appendix A Theorem II) for initial position

q′:

〈q|ψ(t)〉 = ψ(q, t) =

ˆdq′ 〈q| e−

i~ tH |q′〉〈q′|ψ(0)〉 =

ˆdq′ 〈q| e−

i~ tH |q′〉ψ(q′, 0)

(54)

The term 〈q| e− i~ tH |q′〉 is called the kernell of the Path Integral. The kernell rep-

resents the probability amplitude while traveling from the starting point to the final

point. Think of the |q′〉 = |q0〉 as the starting point in the Lagrangian formulism and

|q〉 = |qf〉 as the final prespecified point.

35

Let ∆t be a very small change in time, where ∆t = tj+1 − tj . We will break the

total allowed time t into N equal increments, so that we have ∆t = t/N . We can

therefore write:

〈q| e−i~ tH |q′〉 = 〈q| e−

i~ (∆t1+∆t2+...+∆tN )H |q′〉 = 〈q| e−

i~∆t1H− i

~∆t2H+...− i~∆tN H |q′〉

〈q| e−i~ tH |q′〉 = 〈q| e−

i~∆t1He−

i~∆t2H · · · e−

i~∆tN H |q′〉

The subscripts on each of the ∆t are there just to make clear that we are breaking the

total time t into a large number of N increments. Since the ∆t here are very small,

we will drop the subscripts from here on out. Just keep in mind there are now a total

N exponential terms inside the above expression.

Using Theorem II of Appendix A we can insert between each exponential the

resolution of identity:

〈qf | e−i~∆tHe−

i~∆tH · · · e−

i~∆tH |q0〉 = 〈qf | e−

i~∆tHIe−

i~∆tHI · · · Ie−

i~∆tH |q0〉 =

=

ˆ ˆ· · ·ˆdqN−1dqN−2 · · · dq1 〈qf | e−

i~∆tH |qN−1〉〈qN−1| e−

i~∆tH |qN−2〉×

× 〈qN−2| · · · |q1〉〈q1| e−i~∆tH |q0〉 (55)

We will take one of the “bra-ket” sandwiches above and do some further opera-

tions on it. We can represent each “sandwich” as 〈qj+1| e−i~∆tH |qj〉 and plug in the

value of the Hamiltonian operator H:

〈qj+1| e−i~∆tH |qj〉 = 〈qj+1| e−

i~∆t[ p

2

2m+V (q)] |qj〉 = 〈qj+1| e−

i~∆t p

2

2m e−i~∆tV (q) |qj〉

From Theorem I of Appendix A we know that:

36

e−i~∆tV (q) |qj〉 = e−

i~∆tV (qj) |qj〉

Next we apply the completeness theorem with momentum p:

〈qj+1| e−i~∆t p

2

2m Ie−i~∆tV (q) |qj〉 =

ˆdp 〈qj+1| e−

i~∆t p

2

2m |p〉〈p| e−i~∆tV (qj) |qj〉

Again applying Theorem I we know:

e−i~∆t p

2

2m |p〉 = e−i~∆t p

2

2m |p〉

so that the whole thing becomes


2

2m e−i~∆tV (q) |qj〉 =

ˆdp 〈qj+1|p〉〈p|qj〉 e−

i~∆t[ p

2

2m+V (qj)]

Since QM obeys the Canonical Commutator [q, p] = i~, we know what 〈qj+1|p〉

and 〈p|qj〉 are from Theorem V of Appendix A where we set κ = ~. Plugging these

results into the above equation we get:


2

2m e−i~∆tV (q) |qj〉 =

ˆdp e−

i~∆t[ p

2

2m+V (qj)]

1

2π~ei~pqj+1e−

i~pqj =

=

ˆdp e−

i~∆t[ p

2

2m+V (qj)]

1

2π~ei~p(qj+1−qj)

This is known as a Gaussian Integral. From Theorem VIII of Appendix A, we know

how to compute Gaussian Integrals. We simply set a = i~

∆t2m

, b = i~(qj+1 − qj), and

c = − i~∆tV (qj) in this Theorem and conduct the following steps:

1

2π~

ˆdp e−

i~

∆t2m

p2+ i~ (qj+1−qj)p− i

~∆tV (qj) =1

2π~

√π

aeb2

4a+c =

37

=1

2π~

√π

( i~∆t2m

)exp

[( i~(qj+1 − qj))2

4( i~∆t2m

)− i

~∆tV (qj)

]

∴ 〈qj+1| e−i~∆tH |qj〉 =

1

2π~

√2πm~i∆t

exp

[im(qj+1 − qj)2

2~∆t− i

~∆tV (qj)

](56)

Plugging equation (56) into (55) we get the following for the total time t:

〈qf | e−i~ tH |q0〉 =

( 1

2π~

)N(2πm~i∆t

)N2

ˆ ˆ· · ·ˆdqN−1dqN−2 · · · dq1×

× exp

[N−1∑j=0

(im(qj+1 − qj)2

2~∆t− i

~∆tV (qj)

)]

This hideous looking expression can be notationally simplified a bit by treating all

the integrals with respect to qj as a product of integrals

ˆ ˆ· · ·ˆdqN−1dqN−2 · · · dq1 =

N−1∏j=0

ˆdqj

where Π just means product from j = 0, 1, ..., N − 1, similarly like Σ means sum

from j = 0, 1, ..., N − 1. Compactly written:

〈qf | e−i~ tH |q0〉 =

( 1

2π~

)N(2πm~i∆t

)N2

N−1∏j=0

ˆdqj exp

[N−1∑j=0

( i~m(qj+1 − qj)2

2∆t− i

~∆tV (qj)

)](57)

Factoring out a factor of i∆t~ from the exponent:

〈qf | e−i~ tH |q0〉 =

( 1

2π~

)N(2πm~i∆t

)N2

N−1∏j=0

ˆdqj exp

[N−1∑j=0

(i∆t~

(m(qj+1 − qj)2

2∆t2− V (qj)

))](58)

Now, let us take the limit as ∆t → 0. Since ∆t = t/N , this is equivalent to taking

N →∞.

lim∆t→0

[qj+1 − qj∆t

]2

=(dqdt

)2

(59)

38

This is simply the definition of a derivative. And:

limN→∞

N−1∑j=0

=

ˆ t

0

(60)

We end up with:

〈qf | e−i~ tH |q0〉 =

( 1

2π~

)N(2πm~i∆t

)N2

N−1∏j=0

ˆdqj exp

[i

~

ˆ t

0

dt(m

2

(dqdt

)2

− V (q))]

(61)

Since q is the position, dqdt

is obviously the velocity. A quick glance will reveal that

the exponated term m2

(dqdt

)2 is simply the Kinetic Energy T . Ergo, if you compare the

exponent to (51), the last term becomes eiS/~, which should be familiar. It is exactly

the same term that appears in equation (3), the wavefunction represented in radial

form.

We can write some of the messy, ugly terms into a nice format. We say that

limN→∞

(−i2πm∆t

)N2

N−1∏j=0

ˆdqj =

ˆDq(t)

And therefore the entire thing can be written simply as:

〈qf | e−i~ tH |q0〉 =

ˆDq(t) exp

[i

~

ˆ t

0

dt(m

2

(dqdt

)2

− V (q))]

〈qf | e−i~ tH |q0〉 =

ˆDq(t) eiS/~ (62)

Equation (62) represents the final Kernel of the Feynman Path Integral. This inte-

gral is fascinating because it says that for a fixed starting point |q0〉 and ending point

|qf〉, the quantum particle takes all possible paths inbetween within the allotted time

interval. This is like saying that if you are throwing darts at a dart board, the flying

dart takes all possible paths from your hand to the bullseye. This shocking outcome

highlights another strange property of quantum particles. This goes hand in hand

with the earlier comments about quantum particles having “fuzzy” trajectories.

39

4.2 Koopman-von Neumann Classical Path Integral

Just like a Path Integral can be formulated for Quantum Theory, one can also formu-

late a path integral for classical theory. The path integral, however, must enforce New-

ton’s Laws and particles must travel along well defined paths instead of the “fuzzy”

trajectories of QM. This classical path integral is possible thanks to the KvN for-

mulism (Mauro 2003; McCaul and Bondar 2021).

First, in order to develop the KvN Path Integral, let us develop some initial use-

ful tools. These will be ingredients we will use in a future step in making the Path

Integral. Using Theorem V of Appendix A, you can compute what expressions like

〈q, λ|q′, p〉 are. LetA = p, B = λ, and κ = 1. Utilizing the Koopman-von Neumann

Algebra (43), you can get:

〈q, λ|q′, p〉 = 〈q| ⊗ 〈λ| |q′〉 ⊗ |p〉 = 〈q|q′〉〈λ|p〉 =1√2πδ(q − q′)e−ipλ (63)

Note that there is no need to evaluate 〈q|p〉 and 〈λ|q′〉 since [q, p] = 0 and [λ, q] = 0.

From (18), recall 〈q|q′〉 = δ(q− q′). You can follow a similar procedure to calculate

the following two expressions:

〈θ, p|q, p′〉 = 〈θ| ⊗ 〈p| |q〉 ⊗ |p′〉 = 〈θ|q〉〈p|p′〉 =1√2πδ(p− p′)e−iθq (64)

〈θ, p|q, λ〉 = 〈θ| ⊗ 〈p| |q〉 ⊗ |λ〉 = 〈θ|q〉〈p|λ〉 =1

2πeipλ−iθq (65)

The three expressions above are critical ingredients which we will utilize soon.

Now we can begin with the proper formulation of the Path Integral. Similar to

(52), we will utilize

|ψ(t)〉 = e−itK |ψ(0)〉 (66)

40

with the Koopman Generator K (45) instead of the quantum Hamiltonian H .15 Since

KvNM deals with phase space, we will attack this expression with a q, p bra on the

left to get the following:

〈q, p|ψ(t)〉 = ψ(q, p, t) = 〈q, p| e−itK |ψ(0)〉 (67)

And like before, we will introduce the resolution of identity, this time with respect to

q′, p′:

〈q, p|ψ(t)〉 =

ˆdq′ dp′ 〈q, p| e−itK |q′, p′〉〈q′, p′|ψ(0)〉 =

=

ˆdq′ dp′ 〈q, p| e−

i~ tK |q′, p′〉ψ(q′, p′, 0) (68)

The KvN Kernel is 〈q, p| e− i~ tK |q′, p′〉 with |q′, p′〉 = |q0, p0〉 being the initial state

of the system in phase space and |q, p〉 = |qf , pf〉 being the final state of the system.

Again, let ∆t be a small change in time. As before in the quantum case, it is a very

small increment where we will have ∆t = t/N . We will get:

〈q, p| e−itK |q′, p′〉 = 〈q, p| e−i∆t1Ke−i∆t2K · · · e−i∆tN K |q′, p′〉 (69)

The subscripts on each of the ∆t are there just to make clear that we are breaking the

total time t into a large number of N increments. Since the ∆t here are very small,

we will drop the subscripts from here on out. Just keep in mind there are now a total

N exponential terms inside the above expression.

Using Theorem II we can insert between each exponential a completeness rela-

tionship, as before:

〈qf , pf | e−i∆tKe−i∆tK · · · e−i∆tK |q0, p0〉 =

=

ˆ· · ·ˆdqN−1dpN−1 · · · dq1dp1 〈qf , pf | e−i∆tK |qN−1, pN−1〉×

15More information about exponated operators in Appendix C.

41

Therefore, plugging into the expression (71) we get:

˘dθ dp dq dλ 〈qj+1, pj+1|θ, p〉〈θ, p|q, λ〉〈q, λ|qj, pj〉 e−i∆t

pθm ei∆tV

′(q)λ

For the expressions 〈qj+1, pj+1|θ, p〉, 〈θ, p|q, λ〉, and 〈q, λ|qj, pj〉, we substitute in our

ingredients we derived earlier (63), (64), and (65):

˘dθ dp dq dλ

1√2πδ(p−p′)eiθqj+1

1

2πeipj+1λ−iθqj 1√

2πδ(q−q′)e−ipjλe−i∆t

pθm ei∆tV

′(q)λ

Remember that according to (19), the integrals of Dirac delta functionals will reduce

to the following, neater expression:

=

¨dθ dλ

1√2πeiθqj+1

1

2πeipj+1λ−iθqj 1√

2πe−ipjλe−i∆t

pθm ei∆tV

′(q)λ

And simplifying:

=1

(2π)2

¨dθ dλ eiθqj+1−iθqj−i∆t pθm eipj+1λ−ipjλ+i∆tV ′(q)λ

Factoring out the observables θ and λ from each exponent we get:

1

(2π)2

¨dθ dλ exp

(iθ∆t

[qj+1 − qj∆t

− p

m

])exp

(iλ∆t

[pj+1 − pj∆t

+ V ′(q)])

Inserting this important expression into equation (70) we will get that:

〈qf , pf | e−itK |q0, p0〉 =

=

ˆ···ˆdqN−1dpN−1···dq1dp1

[1

(2π)2

¨dθN−1 dλN−1 exp

(iθN−1∆t

[qj+1 − qj∆t

−pjm

])×

× exp(iλN−1∆t

[pj+1 − pj∆t

+V ′(qj)])]···[

1

(2π)2

¨dθ1 dλ1 exp

(iθ1∆t

[qj+1 − qj∆t

−pjm

])×

× exp(iλ1∆t

[pj+1 − pj∆t

+ V ′(qj)])]

43

We can utilize some of the simplifying notation we used for Feynman’s Path Integral:

〈qf , pf | e−itK |q0, p0〉 =1

(2π)2N

˘ N−1∏j=1

dqj dpj dθj dλj×

× exp(i∆t

N−1∑j=0

[θj

(qj+1 − qj∆t

− pjm

)])exp

(i∆t

N−1∑j=0

[λj

(pj+1 − pj∆t

+V ′(qj))])(72)

Just as before, let ∆t→ 0 (equivalent to N →∞) to get:

lim∆t→0

qj+1 − qj∆t

=dq

dt

lim∆t→0

pj+1 − pj∆t

=dp

dt

limN→∞

N−1∑j=0

=

ˆ t

0

One more piece of notation. Just as before, let us define that for a variable f , that we

will have:

Df = limN→∞

N∏n

dfn√2π

(73)

And so the classical analog of the path integral can be written:

〈qf , pf | e−itK |q0, p0〉 =

˘DqDpDθDλ exp

(i

ˆ t

0

dt[λ(t)

(dqdt− p

m

)+θ(t)

(dpdt

+V ′(q))])

(74)

What is the relevance of this? This encodes the equations of CM (McCaul and Bondar

2021). The path integral is a kind of redundancy that can be shown to enforce the laws

of CM. We can show that it leads to a Dirac delta functional that enforces Newton’s

Laws.

Recall that the Dirac delta functional δ(A − A′) is nonzero when A = A′ but

44

zero if A 6= A′. Let us use equation (72) above, as it is written in a slightly easier

to mathematically manipulate form (following the footsteps of McCaul and Bondar

2021). Let us look at the portion of the integral with respect to θj:

ˆdθj exp

(i∆t[θj

(qj+1 − qj∆t

−pjm

)])=

ˆdθj exp

(i∆t

m

[θj

(mqj+1 − qj

∆t−pj

)])

Next, we will utilize equation (22) for our convenience. We will take the 0th

derivative of the Dirac delta functional in the formula (which is the same as taking

no derivative at all):

d0

dA0δ(A− A′) = δ(A− A′) =

ˆdω

2π(−iω)0e−iω(A−A′) =

ˆdω

2πe−iω(A−A′)

If you set θ = ω/2π, you can plug into the integral with respect to θj to get:

ˆdθj exp

(i∆t

m

[θj

(mqj+1 − qj

∆t−pj

)])=

ˆdω

2πexp

(−i∆t

m

[2πω

(pj−m

qj+1 − qj∆t

)])=

=

ˆdω

2πexp

(− iω

[2π

∆t

m

(pj −m

qj+1 − qj∆t

)])By comparing it to the Dirac delta expression above, we see right away that (A−A′) =[2π∆t

m(pj −m qj+1−qj

∆t)], and so:

ˆdθj exp

(i∆t[θj

(qj+1 − qj∆t

− pjm

)])= δ( ∆t

2πm(pj −m

qj+1 − qj∆t

))

=

= 2πm

∆tδ(pj −m

qj+1 − qj∆t

)(75)

utilizing equation (24) for the last step. This reinforces the dynamical equation pj −

mqj+1−qj

∆t= 0, which is simply the discretized version of the momentum relation,

45

p = mv. If this delta functional is plugged back into equation (74), then you can

reduce the total number of variables integrated over from four to just q and λ (McCaul

and Bondar 2021). The two variable version of the Classical Path Integral ensures

CM holds (Mauro 2003; McCaul and Bondar 2021).

The above integration over θ can also be conducted over the variable λ instead to

get

ˆdλj exp

(i∆t[λj

(pj+1 − pj∆t

+V ′(qj))])

=

ˆdω

2πexp

(−iω

[∆t

2π

(−pj+1 − pj

∆t−V ′(qj)

)])=

=2π

∆tδ(pj+1 − pj

∆t+ V ′(qj)

)(76)

This reinforces the dynamical equation pj+1−pj∆t

+ V ′(qj) = 0, which is simply the

discrete version of Newton’s 2nd Law of Motion, dpdt

= −V ′(q) = F .

All the above equations are elaborate formulations of the laws of CM. These are

different ways of enforcing the laws of CM hold true. We will go over some practical

usages of the Classical Path Integral in the pages that follow.

5 Classical and Quantum Comparison

5.1 The Phase in Koopman-von Neumann and Quantum Mechan-

ics

In QM, the interaction of the quantum phase S with the wavefunction amplitude R

gives rise to quantum mechanical behavior (Mauro 2003). Recall that one way of ex-

pressing the wavefunction ψ(t) is (3). If we plug this expression into the Schrodinger

equation (4) we get two equations:

∂S

∂t+

(∇S)2

2m− ~2

2mR∇2R + V (q) = 0 (77)

46

which is the classical Hamilton-Jacobi theory with quantum potential16, and

∂R2

∂t+∇ · (R2∇S

m) = 0 (78)

which is the continuity equation for density R2. 17 From the above expression, you

can see that the S and R are coupled together and cannot be separated from each

other in QM.

This is not the case in CM. From the KvNM equation (35), you can see if you plug

in ψ(q, p, t) = R(q, p, t)eiG(q,p,t) you will get complete decoupling of the amplitude

R from the phaseG. All one has to do is plug the phase space wavefunction ψ(q, p, t)

into (35), like so:

i∂ψ

∂t= Lψ = i

∂

∂tR(q, p, t)eiG(q,p,t) =

[− i p

m

∂

∂q+ i

dV

dq

∂

∂p

]R(q, p, t)eiG(q,p,t)

This process begets:

i∂R

∂t= LR (79)

i∂G

∂t= LG (80)

The amplitude and the phase are completely seperate. Quantum mechanics has there-

fore been described as the theory that tells how the phase interacts with the amplitude

(Mauro 2003).

If the phase and the amplitude evolve separately in KvNM, one may ask what is16As an interesting aside, one can trivially show that the quantum potential disappears if one as-

sumes the Classical Commutator. For instance, equation (77) can be rewritten since we know that(i~)2 = −~2:

∂S

∂t+

(∇S)2

2m+

[q, p]2

2mR∇2R+ V (q) = 0

If you assume the Canonical Commutator, you retrieve the Schrodinger equation as is. If you assumethe Classical Commutator, you retrieve the completely classical Hamilton-Jacobi equation. This resultis trivial, but still interesting to consider from the perspective of unity of ideas.

17Keep in mind that if ρ = ψ∗ψ, then ρ = ψ∗ψ = Re−iS/~ReiS/~ = R2.

47

the practical use of the KvNM formulism? Why the use of complex wavefunctions

if the exponated phase is completely independent of the amplitude? While it is true

that in the momentum position representation |q, p〉 of KvNM the formulism can be

redundant, one can always use the convenient properties of Hilbert Spaces to change

to other representations, which gives us new insights into CM (Bondar et al. 2012;

Bondar et al. 2013; McCaul and Bondar 2021, etc.) We will see examples of this

change in basis utilized in section 6. In order to have the most general form of CM, it

is important to write the classical wavefunction in this complex form (Mauro 2003).

5.2 Uncertainty Principle in Koopman-von Neumann and Quan-

tum Mechanics

One of the hallmarks of QM is the existence of an uncertainty principle between the

position and momentum. The uncertainty principle tells you that the more precisely

you measure where a wavefunction is localized, the less precisely you know the mo-

mentum distribution. This contrasts with CM, where you can measure the values of

both the momentum and position to an arbitrary precision at the same time. This

trade off is another re-articulation of the strange property of “fuzzy” trajectories in

QM and well-defined trajectories in CM.

One can naturally derive the Heisenberg Uncertainty Principle from the Canoni-

cal Commutator. Through the same series of steps, one can also demonstrate the lack

of an uncertainty principle between position and momentum for the KvN classical

case. The procedure is quite straightforward.

Firstly, from statistics we know that the standard deviation of some measured

quantity A is given by:

σA =√〈A2〉 − 〈A〉2 (81)

And this will be a crucial ingredient in deriving the Heisenberg Uncertainty Principle.

48

From equation (16), we know that

〈A〉 = 〈ψ(t)| A |ψ(t)〉

〈A2〉 = 〈ψ(t)| A2 |ψ(t)〉

Treat 〈A〉 as a constant. Then we can see the following important relationship:

(A− 〈A〉)2 = A2 − 2〈A〉A+ 〈A〉2,

〈ψ(t)| [A2−2A〈A〉+〈A〉2] |ψ(t)〉 = 〈ψ(t)| A2 |ψ(t)〉−〈ψ(t)| 2〈A〉A |ψ(t)〉+〈ψ(t)| 〈A〉2 |ψ(t)〉 =

= 〈ψ(t)| A2 |ψ(t)〉 − 2〈A〉〈ψ(t)| A |ψ(t)〉+ 〈A〉2 〈ψ(t)|ψ(t)〉 =

= 〈ψ(t)| A2 |ψ(t)〉 − 2〈A〉2 + 〈A〉2 = 〈A2〉 − 〈A〉2.

utilizing (16) and 〈ψ(t)|ψ(t)〉 = 1.

We have all the ingredients we need to derive the Heisenberg Uncertainty Prin-

ciple. First, let us compute the variance σ2 for both the position and momentum.

Definitionally following (81) and our above result, they would be

σ2q = 〈q2〉 − 〈q〉2 = 〈ψ(t)| (q − 〈q〉)2 |ψ(t)〉

σ2p = 〈p2〉 − 〈p〉2 = 〈ψ(t)| (p− 〈p〉)2 |ψ(t)〉

We can define a new vector to simplify our calculation. We can define |Q〉 = (q −

〈q〉) |ψ(t)〉 and |P 〉 = (p− 〈p〉) |ψ(t)〉 in order to write the variance of position and

momentum simply as

σ2q = 〈Q|Q〉

σ2p = 〈P |P 〉

49

Then we apply the Cauchy-Schwarz Inequality (Theorem III of Appendix A) which

tells us that:

σ2pσ

2q = 〈P |P 〉〈Q|Q〉 ≥ | 〈P |Q〉 |2 (82)

We know that for any complex number z = a+ bi, that

|z|2 = a2 + b2 ≥ b2

where we know from complex analysis that b = z−z∗2i

. Ergo,

|z|2 = | 〈P |Q〉 |2 ≥(z − z∗

2i

)2

=(〈P |Q〉 − 〈Q|P 〉

2i

)2

,

where

〈P |Q〉 = 〈ψ(t)| (p−〈p〉)(q−〈q〉) |ψ(t)〉 = 〈ψ(t)| [pq−q〈p〉−〈q〉p+〈q〉〈p〉] |ψ(t)〉 =

= 〈ψ(t)| pq |ψ(t)〉 − 〈ψ(t)| q〈p〉 |ψ(t)〉 − 〈ψ(t)| 〈q〉p |ψ(t)〉+ 〈ψ(t)| 〈q〉〈p〉] |ψ(t)〉 ,

and

〈Q|P 〉 = 〈ψ(t)| qp |ψ(t)〉−〈ψ(t)| q〈p〉 |ψ(t)〉−〈ψ(t)| 〈q〉p |ψ(t)〉+〈ψ(t)| 〈q〉〈p〉] |ψ(t)〉 .

With the calculations above, we plug into (82) and derive:

| 〈P |Q〉 |2 ≥(〈P |Q〉 − 〈Q|P 〉

2i

)2

=(〈ψ(t)| pq |ψ(t)〉 − 〈ψ(t)| qp |ψ(t)〉

2i

)2

,

| 〈P |Q〉 |2 ≥(〈ψ(t)| [p, q] |ψ(t)〉

2i

)2

. (83)

50

From this, it is simple to see that:

σpσq ≥| 〈ψ(t)| [p, q] |ψ(t)〉 |

2(84)

If we plug in the Canonical Commutator for [p, q] we derive the Heisenberg Uncer-

tainty Principle,

σpσq ≥ ~/2 (85)

If we plug in the Classical Commutator for [p, q] = 0 there is no uncertainty rela-

tion between the position and momentum, hence there is no theoretical restriction to

knowledge of precision of both at the same time. Depending on your choice of the

Commutator, you derive either Classical or Quantum Physics (Bondar et al. 2012).

This does not, however, mean there are no uncertainty relations for KvNM. Be-

cause the Koopman Algebra (43) tells us that [q, θ] = i and [p, λ] = i, we can develop

uncertainty relationships for q-θ and p-λ by plugging the commutators into expres-

sions analogous to (84):

σpσλ ≥| 〈ψ(t)| [p, λ] |ψ(t)〉 |

2=

1

2

σqσθ ≥| 〈ψ(t)| [q, θ] |ψ(t)〉 |

2=

1

2

These would be the ‘classical’ uncertainty relationships. Since neither θ nor λ are

directly observable (Ramos-Prieto et al. 2018), these uncertainty relations will not

impact our observations of classical systems. For KvNM, many “quantum” phenom-

ena are tucked into the not directly observable operators θ and λ.

5.3 Measurement in Koopman-von Neumann and Quantum Me-

chanics

QM is known to have the strange property that it makes probabilistic predictions

instead of deterministic ones. One strange feature related to this fact that tucked

51

into the previously mentioned postulates of QM you have a phenomenon termed the

collapse of the wavefunction. Any wavefunction can be expanded across a basis using

the completeness theorem (Theorem II of Appendix A):

|ψ〉 = I |ψ〉 ,

where I is the identity operator (analogous to the number ‘1’.) Inserting the discrete18

completeness relation we get:

|ψ〉 =∑j

|j〉〈j|ψ〉 = |A〉〈A|ψ〉+ |B〉〈B|ψ〉+ |C〉〈C|ψ〉+ · · ·, (86)

where the collection of all |j〉 vectors (j = A,B,C, ...) are your basis vectors ac-

cording to the principles of Linear Algebra and 〈j|ψ〉 are scalar values or functions

(recalling that 〈j|ψ〉 = ψ(j)). This is simply expanding the state vector |ψ〉 along a

particular set of basis vectors |j〉. A, B, C, etc., are different eigenvalues associated

with each ‘eigenbasis’, for example, A |A〉 = A |A〉.

If you measure B then the initial wavefunction |ψ〉 instantaneously collapses the

entire quantum system to state |B〉. How the state of the system |ψ〉 instantaneously

changes to |B〉 when B is measured is still a modern day quantum mystery. As

described by Postulate 3 in section 2, the probability for observing B is P (B) =

| 〈B|ψ〉 |2 for discrete systems and the probability density is ρ(B) = | 〈B|ψ〉 |2 in

continuous systems.

The strange phenomenon of collapse is preserved in KvNM. You have a KvN

classical wavefunction that again can be expanded over a particular basis of your

choice

|ψ〉 = |A〉〈A|ψ〉+ |B〉〈B|ψ〉+ |C〉〈C|ψ〉+ · · ·,18Note that in (86) I used the discrete case instead of the continuous completeness identity. This is

simply due to the ease of highlighting certain concepts. The continuous completeness identity woulddo the same thing as the discrete case, but might obscure for some readers results we will discuss.Remember that the integral is just a continuous summation of infinitely small things.

52

and that when a particular eigenvalue is observed, the system collapses to its asso-

ciated eigenbasis. The same Born Rule applies. You have the same postulate for

collapse from the set of ‘universal axioms’ .

The common behavior of the wavefunction collapsing during measurement in

QM and KvNM is mystifying, and perhaps one day further light can be shed on it.

Maybe it is a common feature of probabilistic models built in Hilbert Space, or per-

haps there will be some other explanation developed. As far as we know, the collapse

is instant, irreversible, and no causal process exists to explain it.

There is an important difference though in the collapse of the wavefunction dur-

ing measurement for the KvNM versus QM cases. The wavefunction will not col-

lapse for non-selective measurements in the KvN case but will in the QM case (Gozzi

and Mauro 2002; Mauro 2003). Non-selective measurements are measurements con-

ducted on the system where the results are not ‘read out loud’, meaning the system

would be hypothetically collapsed but we do not ‘record’, remember, or consider

which eigenbasis it collapses to. We remain completely ignorant of whether the sys-

tem state |ψ〉 hypothetically collapses into |A〉, |B〉, |C〉, or some other state. This

difference in collapse for KvN and QM wavefunctions will be demonstrated (follow-

ing the work of Gozzi and Mauro 2002; Mauro 2003).

First, let us consider the QM case. Consider a Hamiltonian H which obeys the

following eigenvalue relationships:

H |+〉 = ~ω |+〉 , (87)

H |−〉 = −~ω |−〉 , (88)

where |+〉 and |−〉 are the eigenvectors and ~ω and −~ω are the energy eigenvalues

of the operator H , respectively. 19 Let us consider an observable with operator Ω,

which has the eigenvectors |a〉 and |b〉 defined by:19This should bring to mind, for instance, that the energy of a photon is given by E = ~ω, where ω

is the angular frequency.

53

|a〉 =1√2

[|+〉+ |−〉

], (89)

|b〉 =1√2

[|+〉 − |−〉

]. (90)

For the sake of our analysis, we can choose the initial state of the system to be the

following superposition of states (Mauro 2003):

|ψ(t = 0)〉 =1

2|+〉+

√3

4|−〉 (91)

We will utilize the Schrodinger equation in the form of (52) (see also Appendix C)

order to evolve the above wavefunction, |ψ(0)〉. We want to evolve the system to

t = 2τ and see what the probability of measuring a, that is P (a), will be (Mauro

2003).

|ψ(t = 2τ)〉 = e−i~ H2τ |ψ(0)〉 = exp

[− i~H2τ

][1

2|+〉+

√3

4|−〉

]. (92)

We will utilize the very useful Theorem I from Appendix A alongside (87) and (88).

This will give us:

|ψ(t = 2τ)〉 =1

2exp

[− i~H2τ

]|+〉+

√3

4exp

[− i~H2τ

]|−〉 =

=1

2exp

[− i~

2τ(~ω)

]|+〉+

√3

4exp

[− i~

2τ(−~ω)

]|−〉 =

=1

2exp[−2iτω] |+〉+

√3

4exp[2iτω] |−〉 . (93)

To find the probability of observing a at t = 2τ we will use the Born Rule:

P (a) = 〈ψ(2τ)|a〉〈a|ψ(2τ)〉 = | 〈a|ψ(2τ)〉 |2,

where we can evaluate:54

〈a|ψ(2τ)〉 =1

2exp[−2iτω] 〈a|+〉+

√3

4exp[2iτω] 〈a|−〉 =

=1

2exp[−2iτω]

1√2

[〈+|+ 〈−|

]|+〉+

√3

4exp[2iτω]

1√2

[〈+|+ 〈−|

]|−〉 .

Using the orthonormality condition, we know that 〈+|−〉 = 0 while 〈+|+〉 = 〈−|−〉 =

1, so that the above expression reduces to

〈a|ψ(2τ)〉 =1

2√

2exp[−2iτω] +

√3

8exp[2iτω].

Therefore, taking (〈a|ψ(2τ)〉)† = 〈ψ(2τ)|a〉 and plugging into the Born Rule:

P (a) = 〈ψ(2τ)|a〉〈a|ψ(2τ)〉 =( 1

2√

2exp[2iτω] +

√3

8exp[−2iτω]

)×

×( 1

2√

2exp[−2iτω] +

√3

8exp[2iτω]

)=

=1

2√

2exp[2iτω]

1

2√

2exp[−2iτω] +

1

2√

2exp[2iτω]

√3

8exp[2iτω]+

+

√3

8exp[−2iτω]

1

2√

2exp[−2iτω] +

√3

8exp[−2iτω]

√3

8exp[2iτω] =

=1

8+

√2

4

√3

8exp[4iωτ ] +

√2

4

√3

8exp[−4iωτ ] +

3

8. (94)

There are one important trigonometric identity we can make use of. It is:

cos(θ) = (1/2)(eiθ + e−iθ) (95)

Taking the identity (95) with (94), we can calculate that

55

P (a) =1

2

(1 +

√3

4cos(4ωτ)

). (96)

The above usage of the wavefunction is for “pure states” (term defined in section 2.6).

Now, instead of letting the quantum system evolve continuous from t = 0 to

t = 2τ , let us introduce a non-selective measurement at t = τ . How will this affect

the probability of measuring a at t = 2τ? We will start with the same |ψ(0)〉. We

will, however, use the density operator formulism to describe the system (see section

2.6) instead of strictly using only the ‘pure’ wavefunction. This is because we have

here what is called a “mixed state”. When we take the measurement at t = τ , the

original system given by (91) collapses, but we do not know whether it collapses into

|a〉 or |b〉, as we do not ‘read out’ or record what the collapse leads to. We can use

(26) to write down:

ρ(τ) = pa(τ) |a〉〈a|+ pb(τ) |b〉〈b| , (97)

where pa(τ) and pb(τ) are the probabilistic weights to the ‘statistical mixture’ of

possible states |a〉 and |b〉 of the system. The factor pa(τ) refers to the probability

that we observe a at t = τ (and hence the system collapsed to |a〉) and the term pb(τ)

refers to the probability that we observe b at t = τ (and hence the system collapsed to

|b〉). Since we are ignorant of which way the wavefunction collapses, the probabilities

became weights of likelihood in (97).

We calculate what the factors pa(τ) and pb(τ) are in a similar fashion to calculat-

ing (96):

pa(τ) = 〈ψ(τ)|a〉〈a|ψ(τ)〉 =(√2

4exp[−iωτ ] +

√3

8exp[iωτ ]

)×

×(√2

4exp[iωτ ] +

√3

8exp[−iωτ ]

)=

=1

2

(1 +

√3

4

[1

2exp[−2iωτ ] +

1

2exp[2iωτ ]

])=

1

2

(1 +

√3

4cos(2ωτ)

)(98)

56

pb(τ) = 〈ψ(τ)|b〉〈b|ψ(τ)〉 =(√2

4exp[−iωτ ]−

√3

8exp[iωτ ]

)×

×(√2

4exp[iωτ ]−

√3

8exp[−iωτ ]

)=

=1

2

(1−

√3

4

[1

2exp[−2iωτ ] +

1

2exp[2iωτ ]

])=

1

2

(1−

√3

4cos(2ωτ)

)(99)

We will use (52) again as before to evolve the system then from (97) at t = τ to

t = 2τ , in the following fashion:

|ψa(2τ)〉 = e−i~ H(2τ−τ) |a〉 = exp

[− i~Hτ

]1√2

[|+〉+ |−〉

], (100)

|ψb(2τ)〉 = e−i~ H(2τ−τ) |b〉 = exp

[− i~Hτ

]1√2

[|+〉 − |−〉

], (101)

utilizing the expressions (89) and (90) above.

As before, utilizing the eigenvalue expressions for H (87) and (88), and Theorem

I we get:

|ψa(2τ)〉 =1√2

exp

[− i~τ(~ω)

]|+〉+

1√2

exp

[− i~τ(−~ω)

]|−〉 , (102)

|ψb(2τ)〉 =1√2

exp

[− i~τ(~ω)

]|+〉 − 1√

2exp

[− i~τ(−~ω)

]|−〉 . (103)

The density operator therefore at t = 2τ can be represented as the following using

the above expressions:

ρ(2τ) = pa(τ) |ψa(2τ)〉〈ψa(2τ)|+ pb(τ) |ψb(2τ)〉〈ψb(2τ)| (104)

And we can use this to evaluate the probability of measuring a by using (27):

57

P (a) = Tr[ρ(2τ) |a〉〈a|] = 〈a| ρ(2τ) |a〉〈a|a〉+ 〈b| ρ(2τ) |a〉〈a|b〉 .

Remember that 〈a|b〉 = 0 and 〈a|a〉 = 1, so:

P (a) = 〈a| ρ(2τ) |a〉 .

Plugging in (104) we get:

P (a) = pa(τ) 〈a|ψa(2τ)〉〈ψa(2τ)|a〉+ pb(τ) 〈a|ψb(2τ)〉〈ψb(2τ)|a〉 . (105)

And now we can substitute the expressions (102), (103), (98), and (99) to get after

some algebra our final expression:

P (a) =1

2

(1 +

√3

4cos2(2ωτ)

). (106)

Obviously, comparing (96) with (106) shows that they are two different equations.

This shows us that in the quantum case making a non-selective measurement will

change the outcome of a later measurement. A non-selective measurement will mod-

ify the probability of outcome at a later point in time t.

So far, what we looked at is an abstract example to demonstrate non-selective

measurements in QM. Let us look at a more concrete example in QM, and then figure

out what the implications are for KvNM (following Mauro 2003). Starting off, let us

measure the probability of a quantum particle’s position from t = 0 to t = τ .

This will be given by (26) except we are no longer dealing with a discrete case.

Since position is continuous, we replace the summation sign with an integral (as an

integral is simply continuous summation of infinitely small things). We start with a

pure state with wavefunction |ψ(0)〉 at t = 0. The probability density ρ (not to be

confused with the density matrix) of observing q at time t = τ will be given by

58

ρ(q, τ) = Tr[ρ(τ) |q〉〈q|] = |ψ(q, τ)|2, (107)

where ρ(τ) = |ψ(τ)〉〈ψ(τ)| (pure state density matrix).

Let us say that we make a non-selective measurement immediately after t = 0,

which we will represent as time t = 0+. The pure state density matrix gets replaced

with a mixed density matrix. We utilize a continuous version of (26) to write the

density matrix

ρ(0+) =

ˆdq0|ψ(q0, 0)|2 |q0〉〈q0| .

To represent a state of the system immediately before t = 0, we can write time t = 0−

and say that:

ρ(0−) = |ψ(0−)〉〈ψ(0−)|

The system goes from ρ(0−) to ρ(0+) as a non-selective measurement is made at

t = 0.

In QM, the measurement right before and the measurement right after will ob-

serve the same thing (Mauro 2003). You can convince yourself of this by computing

the following:

ρ(q, 0−) = Tr[ρ(0−) |q〉〈q|] = ψ∗(q, 0)ψ(q, 0)

ρ(q, 0+) =Tr[ρ(0+) |q〉〈q|]

Tr[ρ(0+)]= ψ∗(q, 0)ψ(q, 0)

Remember that the denominator, Tr[ρ(0+)], just represents the normalization condi-

tion (described in section 2.6). We can evolve ψ∗(q, 0)ψ(q, 0) to see that in the long

term the non-selective measurement does affect the quantum system, just like in the

above worked out example (Mauro 2003). The probability density will be the same

instantaneously right before and right after measurement, but will alter in the long

run in QM.

59

Under KvNM, however, we can observe some differences arise. If we start with

a pure state again

ρ(0−) = |ψ(0−)〉〈ψ(0−)| ,

and conduct no measurements, we will have the KvN wavefunction evolve to

ρ(q, p, τ) = Tr[ρ(τ) |q, p〉〈q, p|] = |ψ(q, p, τ)|2, (108)

similarly to the quantum case (107). Also like the quantum case, in KvNM measuring

the probability density right before and after will not alter the distribution:

Tr[ρ(0−) |q, p〉〈q, p|]Tr[ρ(0−)]

=Tr[ρ(0+) |q, p〉〈q, p|]

Tr[ρ(0+)]

However, a non-selective measurement can be shown to leave the KvN wavefunction

unaltered, whereas a quantum wavefunction will be changed due to the non-selective

measurement .

ρ(0+) =

ˆdq0dp0|ψ(q0, p0, 0)|2 |q0, p0〉〈q0, p0| .

The above probability density, taken right after the measurement, can be shown to

naturally evolve into (108), as if no measurement has taken place (Mauro 2003).

There is one final similarity worth mentioning between QM and KvNM. Under

QM, when a measurement is taken for position q, it affects the outcome of the distri-

bution of p. Perhaps, this is not surprising, considering the Heisenberg Uncertainty

Principle. Under KvNM, a measurement taken for (q, p) will alter the statistical dis-

tribution of the variable λ (Mauro 2003). This should also not be surprising because,

as we discussed in section 5.2, there exists a classical uncertainty relationship be-

tween p and λ. The effects of QM are still present in KvNM therefore, if we replace

q with (q, p) and p with λ.

60

5.4 The Double Slit Experiment

The Double Slit Experiment is at the center of Quantum Theory. QM has photons,

electrons, neutrons, and other quanta going through two tiny slits and forming an

interference pattern on the other side. Even if we send one particle at a time through

the double slit, the particle build up on the other side will reflect an interference

pattern. Even if the particle does not seem to interfere with any other particle, it still

is part of an interference pattern. This shows that the interference patterns are due to

something more fundamental in nature. 20

The wavefunction mathematically describes this strange behavior of elementary

particles, even though the exact interpretation of the wavefunction is not known.

Feynman believed that the strange behavior must lie in the complex nature of the

wavefunction (Mauro 2003). KvNM also contains complex wavefunctions in Hilbert

Space, so we might logically ask the obvious question: if KvNM has complex wave-

functions, will it successfully preserve our knowledge of CM, without interference

patterns arising? Here we will test the outcome of complex wavefunctions in classical

phase space. This analysis is based on the work of (Mauro 2003).

The double slit experiment is set up in Figure 1 above. The y axis is aligned

parallel to the direction of travel. The x axis will be the coordinate parallel to the

back wall where the interference is supposed to occur for the quantum case. In the

middle, at y = yM you have the wall with the two slits. At y = yF you have the wall

where interference occurs in the quantum case.

Under KvNM, we should expect no interference build up at the backwall at y =

yR. Under QM (and in reality), we want there to be an interference build up at the

backwall. We will start with analyzing the quantum case, and how it works. We want

a gaussian wave emerge from y = 0 and travel right until it hits the double slit wall.

Some of the wavefunction will travel through the two slits and eventually interfere at

y = yR if the system behaves quantum.20Interestingly, if you cover up on of the two slits, you will recover particle travelling in straight

trajectories without any interference pattern, just as in CM. This makes the relationship between CMand QM even more bizarre.

61

A gaussian function has the following form:

f(u) =1√

2πσ2e−(u−µ)2/2σ2 (109)

where µ is the average value µ = 〈u〉 and σ is the standard deviation, given by

σ =√〈u2〉 − 〈u〉2.

Recalling the form of a quantum wavefunction (3), we can make it a gaussian

wavefunction by setting R equal to (109) with a generic renormalization constant

N .21 Let µ = 0 in order to center the wavefunction of the double slit experiment.

The equation will have the form:

ψ(x, t = 0) = N exp

[− x2

2σ2x

]exp

[i

~S

]Note that for this wavefunction, S = 0, however. This is obvious when you consider

the definition of the Action (51) for a free particle. Since we decided that the time at

the leftmost wall is t = 0, the Action at the leftmost wall would be:

S =

ˆ 0

0

dtT = 0

Therefore, a better way to write the previous equation at t = 0 is:

ψ(x, t = 0) = N exp

[− x2

2σ2x

](110)

Following the Path Integral Formulism (54), in order to know the know the wavefunc-

tion ψ(x, t) at some later time t we will need to utilize the Kernel (57). The Kernel

〈x| e− i~ tH |x0〉 will evolve the system to the time of interest.

〈x|ψ(t)〉 =

ˆdx0 〈x| e−

i~ tH |x0〉ψ(x0, 0)

We will utilize a very special trick in order to evaluate the above expression. Since21In order to keep things nice and streamlined, the constant in front of (109) will simply be absorbed

into the Renormalization Constant.

62

we are dealing with a free particle, there are no potentials, or V (x, y) = 0 (hence no

forces or accelerations). Because of this, we do not have to use the full, intimidating

form of the Feynman Kernel (57). In this semiclassical treatment there is no acceler-

ation of the quantum particle in the y direction, and so the Kernel can be written in a

much simpler form.

Instead of taking on the form (57), it will instead have the form (56). This is

because we can treat the final position of the wavefunction as qj+1 and the initial

position as qj if there is no acceleration. The term qj+1−qj simply becomes one large

step instead of a miniscule step as before. This is analogous to how in kinematics,

when there is no acceleration, the formula for instantaneous velocity is v = ∆x/∆t,

even though in every other case the formula gives you average velocity (as it is a slope

formula). You can also think of this simplification as taking (57) and setting N = 1

(one large step due to no acceleration). Please note that this is not an approximation,

but this will give us an exact expression.

Our Kernel K (not to be confused with the Koopman Generator) is therefore:

K =

√m

2πi~(t− t0)exp

[im(x(t)− x0)2

2~(t− t0)

](111)

And then we can write the following Path Integral:

〈x|ψ(t)〉 =

ˆ ∞−∞

dx0

√m

2πi~(t− t0)exp

[im(x(t)− x0)2

2~(t− t0)

]N exp

[− x2

0

2σ2x

]

We want to evaluate the path integral up until the point it reached the middle wall with

the double slits, at time tM . Remembering that we have set the initial time t0 equal

to 0 and treating t and x(t) as constants, we can evaluate the integral. First, we will

pull out all constants and absorb them into N , the renormalization constant (recall

the renormalization condition described in section 2.3). Then we do the following

algebraic manipulations, in preparation to evaluating the integral itself:

63

exp

[im(x(t)− x0)2

2~t

]exp

[− x2

0

2σ2x

]= exp

[im(x(t)2 − 2x(t)x0 + x0

2)

2~t− x2

0

2σ2x

]=

= exp

[( im2~t− 1

2σ2x

)x2

0 −imx(t)

~tx0 +

imx(t)2

2~t

]We can then use Theorem VIII of Appendix A to evaluate the integral. Comparing

the two, we can recognize the following coefficients of the Gaussian integral:

a = −( im

2~t− 1

2σ2x

)b = −imx(t)

~t

c =imx(t)2

2~t

And so by the Theorem, the integral turns out to be:

N

ˆ ∞−∞

dx0 exp

[( im2~t− 1

2σ2x

)x2

0 −imx(t)

~tx0 +

imx(t)2

2~t

]=

= N

√π

−(im2~t −

12σ2x

) exp

( imx(t)~t )2

−4(im2~t −

12σ2x

) exp

[imx(t)2

2~t

]=

= N

√2π~tσ2

x

−imσ2x + ~t

exp

[m2x(t)2σ2

x

2~t(−imσ2x + ~t)

]exp

[imx(t)2

2~t

]

∴ 〈x|ψ(t)〉 = N

√2π~tσ2

x

−imσ2x + ~t

exp

[−1

2

mx(t)2

mσ2x + i~t

]The above wavefunction 〈x|ψ(t)〉 = ψ(x, t) is true for anytime 0 < t ≤ tM , where

tM is the time to reach the middle barrier with the two slits. This is because the above

wavefunction is that of a free particle, and once it encounters a barrier (or potential),

the particle is obviously no longer free.

Next, since the expression ψ(x, tM) gives you the quantum wavefunction at the

double slits, we have to take into consideration the portion of the wavefunction that

64

goes through the double slits. To describe the wavefunction going through the double

slit we will utilize step functions. The step functions will closely model the fact that

the wavefunction that hits the middle wall and stops will not be important in creating

any effects on the other side of the wall. Only the wavefunction that goes through the

two slits will be important in creating the diffraction pattern on the other side (Shankar

1988; Mauro 2003). The step function, also called the θ-Heaveside function, is given

by:

H(x) =

1 x > 0

0 x ≤ 0(112)

Each slit has a width of 2δ. The first slit has width ∆1 = (xA − δ, xA + δ) and the

second slit has spans ∆2 = (−xA − δ,−xA + δ) (following the notation of Mauro

2003).

Then, the wavefunction right after the two slits can be written as

ψ(x, tM+dt) = Nψ(x, tM)[C1(x)+C2(x)] = N exp

[−1

2

mx2

mσ2x + i~t

][C1(x)+C2(x)]

(113)

where the following functions can be defined:

C1(x) = H(x− xA + δ)−H(x− xA − δ) (114)

C2(x) = H(x+ xA + δ)−H(x+ xA − δ) (115)

This is done in order to filter out the effects of the wavefunction that actually goes

through the slits. We do not care for the portion of the wavefunction that does not go

through the double slit. They will produce no effects. The above equations make cer-

tain that only the wavefunction that goes through slits ∆1 and ∆2 will be studied. The

N in (113) is a renormalization constant, so that´dxψ∗ψ = 1 (the renormalization

condition described in section 2.3).

Using our Kernel for the free particle again, we will evaluate the wavefunction

65

using the path integral for the region after the two slits. We again utilize (111). Notice

that the initial wavefunction ψ(x0, 0) in that equation starts at time zero, so we will

need to adjust the time at the double slit wall so that it is set to zero. We can do

this by substituting the variable t in the Kernel (111) with t− tM . This can be done

because time has the unique property that any point can be arbitrarily set to t = 0,

and by doing this swap we effectively set the time at the middle wall equal to zero

(starts at tM and evolves from there). So the integral we need to evaluate will have

the following form:

〈x|ψ(t)〉 = N

ˆdxM exp

[im(x(t)− xM)2

2~(t− tM)

]ψ(xM , tM) =

= N

ˆdxM exp

[im(x(t)− xM)2

2~(t− tM)

]exp

[−1

2

mx2M

mσ2x + i~tM

][C1(xM) + C2(xM)]

(116)

where all constants, as is the custom, have been absorbed into N .

This expression above cannot be evaluated explicitly. However, because of the

properties of the step functions (Mauro 2003), it can be rewritten as

〈x|ψ(t)〉 = N(ˆ xA+δ

xA−δdxM exp

[im(x(t)− xM)2

2~(t− tM)− 1

2

mx2M

mσ2x + i~tM

]+

+

ˆ −xA+δ

−xA−δdxM exp

[im(x(t)− xM)2

2~(t− tM)− 1

2

mx2M

mσ2x + i~tM

])(117)

You can consider each of the two integrals to represent an independent wavefunction,

so that you can rewrite the whole expression as

〈x|ψ(t)〉 = N [ψ1(x) + ψ2(x)]

You can think of ψ1 as being the wavefunction coming out of ∆1 and ψ2 as the wave-

function associated with ∆2.

The equation (117) shows us that the phases of the two wavefunctions ψ1 and ψ2

are out of phase. You can see this because the integration is over two different inter-

66

vals, so the two phases will turn out different. This is significant, because wavefunc-

tions out of phase is characteristic of interference. When two wavefunctions with

different phases overlap, they form the characteristic interference pattern (Shankar

1988; Mauro 2003).

If we want the probability of finding a particle at the rightmost wall (the inter-

ference wall), we need to know the wavefunction 〈x|ψ(t)〉 at the rightmost wall. If

it takes time tR to reach this wall, that wavefunction can be simply represented as

〈x|ψ(tR)〉 = ψ(x, tR) (equivalent to taking (117) and plugging in t = tR). Then,

according to the postulates of section 2.5, we just have to apply the Born Rule to find

the probability density as a function of x:

ρ(x, t = tR) = ψ(x, tR)∗ψ(x, tR) = 〈ψ(tR)|x〉〈x|ψ(tR)〉 =

= ψ1(x, tR)∗ψ1(x, tR)+ψ2(x, tR)∗ψ2(x, tR)+ψ1(x, tR)∗ψ2(x, tR)+ψ2(x, tR)∗ψ1(x, tR)

The last two terms are not identically zero (Mauro 2003). If we plot ρ vs x, we will

get the typical interference pattern we expected, as seen in Figure 2b.

This, of course, comes as no surprise. We are still dealing with quantum theory.

Following the same steps, though, will we get the same results with KvNM? We will

follow a similar series of steps to evaluate classical KvN theory.

For the KvNM case, we will use a double gaussian function because we are mak-

ing considerations with both the position and momentum (KvNM is a phase space

theory). A double gaussian (a gaussian in x and y coordinates; see Figure 3b) takes

on the mathematical form:

f(u, v) =1

√πσuσv

e−(u−µu)2/2σu2

e−(v−µv)2/2σv2 (118)

We are interested in the interference pattern that lies along the x direction. Taking

the KvN wavefunction ψ(x, px, t) = R(x, px, t)eiG(x,px,t) (like in section 5.1) and

plugging in (118) for R at t = 0 we will have

67

ψ(x, px, t = 0) =1

√πσxσp

exp

[− x2

2σx2− p2

x

2σp2+ iG(x, px, t = 0)

](119)

Please note here that σp refers to the momentum in the x direction. Following the Path

Integral Formulism, in order to know the know the wavefunction ψ at some later time

t we will need to utilize the Kernel. For this situation, the travelling classical particle

is free of any potential V (x, y), so we will need the free particle wavefunction. As

V (x, y) = 0, there will be no derivatives of the potential. We can take the results (75)

and (76) to get a straightforward and concise Kernel, which will make the calculation

we are about to do simple.

We can take (75) and rewrite it in the following way (dropping the constant 2π,

as it can be normalized away, and replacing q with the desired variable x):

δ(∆t

m(pj −m

xj+1 − xj∆t

))

= δ(pj∆t

m− xj+1 + xj

)In the continuum limit, we can just simply write this as:

= δ(p0t

m+ x0 − x

)

where x0 is the initial position of the particle, p0x is the initial momentum, and m is

the mass as before. This comes straight from elementary kinematics, as since there

is no potential, the derivative of the potential must be 0, so there is no acceleration:

Fx = −∂V∂x

= max ⇒ ax = 0

x(t) = x0 + v0xt+1

2axt

2 ⇒ x(t) = x0 + v0xt = x0 +p0xt

m

Using the momentum relation px = mvx on the last step. In other words, the Dirac

delta functional is enforcing classical position evolution, x(t).

68

The same can be done with (76) to get:

1

∆tδ(pj+1 − pj

∆t+ V ′(qj)

)= δ(pj+1 − pj

)

In the continuum limit remembering the direction is in x:

= δ(px − p0x

)This makes sense, considering that there are no forces present (because potential

is zero), and hence the momentum should be constant. This Dirac delta functional

enforces classical momentum conservation.

The Kernel K can therefore be succinctly represented as

K = δ(px − p0x

)δ(x− p0xt

m− x0

)(120)

And plugging this Kernel into the original equation (68) shows us that indeed it pro-

vides us with the correct result that we desire, forcing classical evolution of the KvN

wavefunction:

〈x, px|ψ(t)〉 =

ˆdx0 dp0xδ

(px − p0x

)δ(x− p0xt

m− x0

)ψ(x0, p0x, 0) =

= ψ(x− pxt

m, px, 0) = ψ(x, px, t),

utilizing the property (19) . This statement checks out, as we know that 〈x, p|ψ(t)〉 =

ψ(x, p, t).

Returning to (119), we will evaluate the same integral utilizing the Kernel. By

equation (68), the evolution of the wavefunction from its original state when t = 0 is

given by

〈x, px|ψ(t)〉 =1

√πσxσp

ˆdx0 dp0xδ

(px − p0x

)δ(x− p0xt

m− x0

)×

69

× exp

[− x2

0

2σx2− p2

0x

2σp2+ iG(x0, p0x, 0)

]=

=1

√πσxσp

exp

[− 1

2σ2x

(x− pxt

m

)2

− p2x

2σp2+ iG

(x− pxt

m, px

)](121)

Note that the above is strictly in terms of the position and momentum in the x direc-

tion. The above equation is true at least until it hits the middle wall with the slits,

because up until that point the wavefunction is that of a free particle. Using geomet-

ric considerations, we can figure out the wavefunction at the double slit wall. Since

there is no acceleration in the direction of travel (as −∂V/∂y = 0), we know from

kinematics:

vy =p0y

m=

∆y

∆t=yf − y0

tf − t0

Under the assumption that y(0) = 0 we can write:

yMt

=p0y

m⇒ tM =

yMm

p0y

, (122)

where tM is the time to the double slit wall. We can substitute this value into (121)

to know what is the classical wavefunction at that point in time.

Then, the wavefunction right after the two slits can be written as

ψ(x, px, tM + dt) = Nψ(x, px, tM)[C1(x) + C2(x)] (123)

Utilizing, as before, the functions (114) and (115) we had defined in the quantum

picture above. This is done in order to filter out the effects of the wavefunction that

actually goes through the slits. We do not care for the portion of the wavefunction

that does not go through the double slit. They will produce no effects. The above

equations make certain that only the wavefunction that goes through slits ∆1 and ∆2

will be studied. The N in (123) is a renormalization constant, similar to before, so

that´dx dpxψ

∗ψ = 1 (the renormalization condition for the KvN wavefunction.)

Using our Kernel for the free particle again, we will evaluate the wavefunction

70

using the path integral for the region after the two slits. We again utilize (68). Notice

that the initial wavefunction ψ(q0, p0, 0) in that equation starts at time zero, so we

will need to adjust the time at the double slit wall so that it is set to zero. We can do

this by substituting the variable t in the Kernel (120) with t− tM . This can be done

because time has the unique property that any point can be arbitrarily set to t = 0, and

by doing this swap we effectively set the time at the middle wall equal to zero (starts

at tM and evolves from there). So the integral we need to evaluate has the following

form:

〈x, px|ψ(t− tM)〉 =

ˆdxM dpMxδ

(px − pMx

)δ(x− pMx(t− tM)

m− xM

)×

×Nψ(xM , pMx, tM)[C1(xM) + C2(xM)]

Plugging everything in:

〈x, px|ψ(t− tM)〉 = N

ˆdx0 dp0xδ

(px − p0x

)δ(x− p0x(t− tM)

m− x0

)×

× exp

[− 1

2σ2x

(x0 −

p0xtMm

)2

− p20x

2σp2+ iG

(x0 −

p0xtMm

, p0x

)][C1(x0) + C2(x0)]

And then using (19) to evaluate:

〈x, px|ψ(t− tM)〉 = Nexp[− 1

2σ2x

(x− px(t− tM)

m− pxtM

m

)2

− p2x

2σp2+

+iG(x−px(t− tM)

m−pxtM

m, px

)][C1

(x−px(t− tM)

m

)+C2

(x−px(t− tM)

m

)]=

= N exp

[− 1

2σ2x

(x− pxt

m

)2

− p2x

2σp2+ iG

(x− pxt

m, px

)]×

×[C1

(x− px(t− tM)

m

)+ C2

(x− px(t− tM)

m

)](124)

71

Similar to (122), under geometric considerations one can show that the time to the

back wall is tR = myR/p0y, and so we can substitute this and (122) into the above

expression to get the expression for the wavefunction at the back wall (wall where we

should see the interference pattern emerge in QM).

We want to know the probability distribution for finding a particle at coordinate

x at this back wall (Mauro 2003). We have no need of the momentum information,

as the interference pattern is a shape that appears in space. So what we can do is get

the probability distribution in terms of x is the following:

P (x) =

ˆ ∞−∞

dpx|ψ(x, px, tR−tM)|2 =

ˆ ∞−∞

dpx 〈ψ(tR − tM)|x, px〉〈x, px|ψ(tR − tM)〉

Plugging in (124) for 〈x, px|ψ(tR − tM)〉 we get:

P (x) = N

ˆ ∞−∞

dpx exp

[− 1

2σ2x

(x− pxtR

m

)2

− p2x

2σp2

]2

exp

[−iG

(x− pxtR

m, px

)]×

× exp

[iG(x− pxtR

m, px

)][C1

(x− px(tR − tM)

m

)+ C2

(x− px(tR − tM)

m

)]2

=

= N

ˆ ∞−∞

dpx exp

[− 1

2σ2x

(x− pxtR

m

)2

− p2x

2σp2

]2

×

×[C1

(x− px(tR − tM)

m

)+ C2

(x− px(tR − tM)

m

)]2

We will use a very special property of the θ-Heavyside step function at this point. We

know that the following important property is true (Mauro 2003):

(C1 + C2)2 = C1 + C2

Therefore:

P (x) = N

ˆ ∞−∞

dpx exp

[− 1

2σ2x

(x− pxtR

m

)2

− p2x

2σp2

]2

×

72

×[C1

(x− px(tR − tM)

m

)+ C2

(x− px(tR − tM)

m

)]This last equation is very important, because it shows that no interference pattern

will emerge from the classical KvN wavefunction. If you look at the above equation

closely, you can see that the probability distribution of x is a superposition of two

probability distributions:

P (x) = N

ˆ ∞−∞

dpx exp

[− 1

2σ2x

(x− pxtR

m

)2

− p2x

2σp2

]2

C1

(x− px(tR − tM)

m

)+

+N

ˆ ∞−∞

dpx exp

[− 1

2σ2x

(x− pxtR

m

)2

− p2x

2σp2

]2

C2

(x− px(tR − tM)

m

)The first integral in the above expression is the probability at x if only the first slit

∆1 was opened and the second slit ∆2 was closed, and the second integral in the

above expression is the probability at x if only the second slit ∆2was opened and

the first slit ∆1 was closed. This can be seen from how C1 filters out any portion

of the wavefunction that does not go through ∆1 and C2 filters out any portion of

the wavefunction that does not go through ∆2. This means that the total probability

distribution is the sum of the probability distribution of each slit, which is exactly

what you would expect for particle instead of wave behavior (no interference effects).

Notice also that the phase of the wavefunction has completely disappeared from

the above equation. You will see this is important, because for interference effects

we will want the phase to play an active role. Waves must be out of phase in order

to produce interference effects, not have the phase cancel each other out like in the

above equations.

If you graph the above function, you will see that indeed no interference fringes

were produced (see Figure 2a). Koopman wavefunctions therefore successfully pre-

dict the behavior of classical particles. Despite their complex nature in a Hilbert

space, they do no produce any interference effects. All the interference effects com-

pletely disappear. It shows how wavefunctions can describe classical physics.

73

6 Miscellaneous KvN Topics of Interest

6.1 Operational Dynamic Modeling

The history of physics is full of failed models, models that did not stand the test of

time. Physics is often carried out in a trial and error fashion, a process that can be

repetitive, messy, and slow. This raises the question if discovery and model building

in physics can be optimized, for the sake of speeding up the progress of discovery.

One such optimization model is called Operation Dynamical Modeling (ODM)

and utilizes KvNM as an essential ingredient (Bondar et al. 2012). The two main

pillars of modern physics is QM and CM. 22 Since we have seen that both QM and

CM must obey the Ehrenfest Theorems, any particular model must conform to either

QM or CM to the best of our knowledge, and if it falls outside either pillar, the model is

quite likely false. If repeated measurements influence the outcome of the experiment,

one knows that there must be a noncommutative algebra involved (i.e., there must

be a nonzero commutator involved, like in QM.) If repeated measurements do not

generally alter the outcome of the experiment in the fashion of QM, you can assume

a commutator of 0 between all variables involved.

Since the Ehrenfest Theorems deal with dynamical weighted averages (expecta-

tion values), this is a statistical test of different models. If a model does not obey

the Ehrenfest Theorems, it is quite likely to be false, and progress can be saved by

either fixing the model so that it is inline with the fundamental Ehrenfest Theorems

or scrapping the model.

Not only does ODM allow quantum and classical physics, it also explains the

nexus of the two. Semi-classical systems are incorporated into ODM, as they must

also obey the Ehrenfest Theorems. By construction, one could create a commutator

that encompasses both the classical and quantum worlds, such as [q, p] = i~κ where

κ is a constant that lies between 0 (classical world) and 1 (purely quantum world).

By taking the limit of κ goes to 0, one should be able to recover either CM.22Note, General Relativity and Special Relativity are here considered to be part of CM.

74

This melding, in fact, results in the derivation of the semi-classical Wigner quasiprob-

ability distribution. This function is known to accurately describe the interface be-

tween the quantum world and the classical world. As the Schrodinger equation is

generally written in terms of configuration space, the Wigner quasiprobability dis-

tribution is important because it recasts QM into a phase space theory. The Wigner

function has a few odd properties, including the fact that it can take on negative prob-

abilities, but thanks to ODM some light has been shed to the reasons why this might

be the case (as we will discuss later.)

We will begin with Stone’s Theorem again like we did in section 3, i.e. i ∂∂t|ψ〉 =

H |ψ〉, and try to create an operator HQC which will encompass both the classical and

quantum worlds using our newly constructed commutator [q, p] = i~κ, 0 ≤ κ ≤ 1.

We must have a unified quantum-classical algebra which gives us the Koopman-

von Neumann Algebra (43) when κ = 0 and a quantum algebra for κ = 1. We

desire:

limκ→0

qQC = qc

limκ→0

pQC = pc

limκ→0

HQC = ~K

In order to reflect the KvN Algebra, Bondar et al. 2013 proposed the following unified

Classical-Quantum Algebra:

[qQC , θQC ] = [pQC , λQC ] = i (125)

All other commutators will equal 0. For the unified dynamics, we first begin with the

Ehrenfest Theorems (equations (36) and (37) for reference) as before:

d

dt〈q〉 =

d

dt〈ψ(t)| qQC |ψ(t)〉 =

1

m〈p〉 =

1

m〈ψ(t)| pQC |ψ(t)〉

d

dt〈p〉 =

d

dt〈ψ(t)| pQC |ψ(t)〉 = 〈−V ′(q)〉 = 〈ψ(t)| − V ′(qQC) |ψ(t)〉

75

Following the same series of steps as in section 3, we will obtain the differential

equations:

κm∂H

∂pQC+m

~∂H

∂θQC= pQC

κ∂H

∂qQC− 1

~∂H

∂λQC= V ′(qQC)

The general solution H to the two above differential equation is simply:

H(qQC , pQC , θQC , λQC) =1

κ

[ p2QC

2m+ V (qQC)

]+ F (pQC − ~κθQC , qQC + ~κλQC)

where the function F is a function undetectable to experiment because one can show

that for any observable A, you have [F , A] = 0 (Bondar et al. 2012).

The quantum operators can be constructed from classical ones, so Bondar et al.

2012 propose that

qQC = qc − ~κλc2

pQC = pc + ~κθc2

θQC = θc

λQC = λc

It is clear that the above expressions have a nice, smooth classical limit as κ → 0.

All the variables operators become classical operators in that limit.

To make the limit smooth as you go from the quantum Hamiltonian H to ~K as

κ goes from 1 to 0, Bondar et al. 2012 dervied:

HQC =1

κ[p2QC

2m+ V (qQC)]− 1

2mκ(pQC − ~κθQC)2 − 1

κV (qQC + ~κλQC) (126)

Equivalently, you can write by plugging in the quantum-classical operators:

76

HQC =~mpcθc +

1

κV (qc −

~κ2λc)−

1

κV (qc +

~κ2λc)

The above expression can be shown to lead to the phase space formulation of QM, the

Wigner quasiprobability distribution. This derivation will be left for the next section.

The Wigner distribution is significant because it is known to accurately model the

‘nexus’ between quantum and classical worlds (Bondar et al. 2012; Bondar et al.

2013). It also has a nice classical limit (Bondar et al. 2012; Bondar et al. 2013).

Independently deriving this distribution indicates that the Ehrenfest Theorems apply

at all times, including in the regime of quantum/classical boundary. ODM, therefore,

is a very successful melding of the two worlds.

ODM suggests that any physical theory will follow KvNM, QM, or the above

melding of the two. KvNM is fundamentally important because any novel classical

theory proposed must be consistent with it. We will see certain applications of ODM

and its successes in the pages that follow. Without KvNM, there would be no unifying

paradigm based in the Ehrenfest Theorems. ODM is a methodology that will pave

for more efficient discovery in Physics.

6.2 Wigner Quasiprobability Distribution

The Wigner Quasiprobability Distribution connects our understanding of the quan-

tum wavefunction to phase space. It is an important area of study developed by Eu-

gen Wigner in 1932 and independently formulated by another eminent mathematical

physicist, Jose Enrique Moyal. The Wigner Distribution has a variety of applications,

from quantum optics to quantum computing (Bondar et al. 2013). It has one signifi-

cantly odd feature though. This distribution allows for negative probabilities, which

is axiomatically impossible by mainstream probability theories (Bondar et al. 2013).

ODM sheds some light on this odd feature. The Wigner Distribution can be derived

from the unified dynamics of ODM, and we will proceed to do so below.

It is very simple to derive the Wigner quasiprobability distribution from ODM’s

77

−〈q, λ| 1κV (qc +

~κ2λc) |ψ(t)κ〉

Utilizing the previous results for Theorem I (see footnote 13), we can see:

i~d

dt〈q, λ|ψ(t)κ〉 =

~m〈q, λ| pcθc |ψ(t)κ〉+

1

κV (q−~κ

2λ) 〈q, λ|ψ(t)κ〉−

1

κV (q+

~κ2λ) 〈q, λ|ψ(t)κ〉

Finally utilizing the results from Theorem IV above:

i~d

dt〈q, λ|ψ(t)κ〉 = −i2 ~

m

∂2

∂λ∂q〈q, λ|ψ(t)κ〉+

1

κV (q − ~κ

2λ) 〈q, λ|ψ(t)κ〉+

−1

κV (q +

~κ2λ) 〈q, λ|ψ(t)κ〉

Let us move all the terms to the left hand side of the equation, so that the left hand

side equals 0 pm the right hand side. Introducing two new variables, u = q− ~κλ/2

and v = q + ~κλ/2, we will rewrite the whole equation so it has a prettier form. To

eliminate the double partial derivative, we can take the following steps:

u− v = q − ~κλ2− q − ~κλ

2= −~κλ

u+ v = q − ~κλ2

+ q +~κλ

2= 2q

Therefore the partial derivative can be rewritten:

∂2

∂λ∂q=

∂2

∂(u−v−~κ)∂(u+v2

)= −2~κ

∂2

∂(u− v)∂(u+ v)= −2~κ

[ ∂2

∂u2− ∂2

∂v2

]

Recasting the entire equation in terms of u and v:

[i~κ

∂

∂t− (~κ)2

2m

( ∂2

∂u2− ∂2

∂v2

)− V (u) + V (v)

]〈q, λ|ψ(t)κ〉 = 0 (131)

Here, 〈q, λ|ψ(t)κ〉 is equal to the density matrix ρκ(u, v, t) up to a constant of pro-

79

portionality C, i.e., ρκ(u, v, t) = C 〈q, λ|ψ(t)κ〉.

We can transition to q, p representation to derive the Wigner distribution from the

above using Theorem II with A = (q, λ):

〈q, p|ψ(t)κ〉 = 〈q, p| I |ψ(t)κ〉 =

ˆdλ 〈q, p|q, λ〉〈q, λ|ψ(t)κ〉

Note that we do not integrate over q due to the commutator relationships [q, q] =

[qc, pc] = 0, whereas [pc, λc] = i. Since [q, q] = [qc, pc] = 0 it makes no sense

to integrate over q, as might be initially implied by looking at the expressions akin

to (31). Since the other commutator is nonzero, we will retain integration over λ,

however. We know that ρκ(u, v, t) = C 〈q, λ|ψ(t)κ〉, so we will plug that in.

The question now is to figure out what 〈q, p|q, λ〉 is. We will utilize Theorem IV

and the commutator relationships in order to deduce this. Since [q, q] = [qc, pc] = 0,

solving this expression is equivalent to solving 〈p|λ〉, as the other terms commute

(i.e., the commutator of their operators is equal to 0) and are therefore irrelevant to

this problem. By Theorem IV, we set A = p, B = λ, and κ = 1. We know that this

expression is therefore

〈q, p|q, λ〉 = 〈p|λ〉 =1√2πeipλ

Putting it all together:

〈q, p|ψ(t)κ〉 =

ˆdλ

C√2πρκ(q −

~κλ2, q +

~κλ2, t)eipλ (132)

The above expression shows us that the wavefunction 〈q, p|ψ(t)κ〉 is equal to the

Wigner quasi-probability distribution up to a constant of proportionality C. The

Wigner distribution is very accurate when it comes to describing the boundary of

Quantum and Classical (Bondar et al. 2012; Bondar et al. 2013), so this is no trivial

derivation.

What ODM reveals is that the Wigner “distribution” is therefore a type of wave-

function for a quantum particle at a point in classical phase space. The strange prop-

80

erty of having negative probabilities can by explained in the ODM framework as

being due to the fact that the Wigner “distribution” is not a distribution at all! It is a

wavefunction. Wavefunctions, unlike probability distributions, can take on negative

values (Bondar et al. 2013). The KvN wavefunction can have negative or imaginary

weights to it, whereas traditional Louiville theory gives only positive, real weights to

the same trajectories (Bondar et al. 2013).

It is worth to highlight one more interesting result of the above analysis. Although

not directly physically observable, the operators θ and λ can be indirectly observed

through the Wigner wavefunction. They are a critical ingredient as the so-called

‘Bopp Operators’.

6.3 Time Dependent Harmonic Oscillator

The Harominc Oscillator is one of the most thoroughly studied models in all of

physics. Its application arises in many situations. We are especially interested in

the Time Dependent Harmonic Oscillator, where factors like the mass or frequency

might change with time. Because of the common mathematical basis for QM and

KvNM, it should come to no surprise that common sets of tools can be used to in-

vestigate both. The Harmonic Oscillator is no exception. The operatorial tools to

investigate quantum oscillators can be used to explore classical (KvN) oscillators

and vise versa.

For a classical or quantum Time Dependent Harmonic Oscillator we need a quadratic

potential of the following form:

V (q, t) = k(t)q2

2

The above potential is modelled off of the classical spring potential. An ideal block-

mass and spring system can undergo harmonic oscillation. The spring potential is

given by V = (1/2)kq2 with spring constant k and displacement q. Here, we add an

extra layer of complexity by making the spring constant time dependent. This is our

81

model potential for all Time Dependent Harmonic Oscillation, whether quantum or

classical.

In the quantum case, we would plug in the above expression for the potential term

into the Schrodinger equation to solve for the Quantum Oscillator. In the classical

case, we plug in the derivative of the above expression into the Koopman Generator

(45):

∂

∂qV (q, t) =

∂

∂qk(t)

q2

2= k(t)q = V ′(q)

i∂

∂t|ψ(t)〉 =

[ pθm− V ′(q)λ

]|ψ(t)〉 =

[ pθm− k(t)λq

]|ψ(t)〉 .

Similar to what we did before (50) in section 3, we can attack both sides with the bra

vector 〈q, p| to get:

〈q, p| i ∂∂t|ψ(t)〉 = 〈q, p|

[ pθm− k(t)λq

]|ψ(t)〉 ,

i∂

∂t〈q, p|ψ(t)〉 = 〈q, p| pθ

m|ψ(t)〉 − 〈q, p| k(t)λq |ψ(t)〉 ,

i∂

∂tψ(q, p, t) =

[ pθm− k(t)λq

]ψ(q, p, t) (133)

where we colloquially say that q = q, p = p, λ = −i ∂∂p

, and θ = −i ∂∂q

, just as we

derived in section 3. Notice as for the Path Integral (section 4.2), we choose to set

the constant C to zero in the Koopman Generator for simplicity’s sake.

One way to solve for the equations of motion of a Harmonic Oscillator is to use

Ermakov-Lewis invariants (Lewis Jr. 1967). If you have a Hamiltonian of the form

H =1

2[p2 + Ω2(t)q2],

where Ω(t) is an arbitrary function with time dependence, you can use the invariant

to write a transformation where the time dependent term Ω(t) completely disappears.

82

This makes solving the problem much simpler. Like it sounds, the Ermakov-Lewis

invariant I is an operator that does not change (stays constant in time). The invariant

I obeys the following rule for the quantum system (Lewis Jr. 1967; Lewis Jr. and

Riesenfeld 1968):

dI

dt=∂I

∂t+

1

i~[I , H] = 0. (134)

H is the generator of motion, usually the quantum Hamiltonian, but in the classical

case it will be the Koopman Generator. Based on the Koopman Algebra (43), Ramos-

Prieto et al. 2018 deduced the following expression for I:

I = α0q2 + α1θ

2 + α2p2 + α3λ

2 + α4qp+ α5θλ

where all the αj are time dependent. This was a scientific guess basically, as there is

no simple way to derive this fact. Plugging the above expression into (134), we get:

∂α0

∂tq2 +

∂α1

∂tθ2 +

∂α2

∂tp2 +

∂α3

∂tλ2 +

∂α4

∂tqp+

∂α5

∂tθλ+

1

i~[I , K] = 0 (135)

Evaluating [I , K], we first can do (13):

[I , K] =[I ,pθ

m− k(t)λq

]=[I ,pθ

m

]−[I , k(t)λq

](136)

Taking the first term and applying (11) and (13):

[I ,pθ

m

]= −

[ pθm, I]

= −[ pθm, α0q

2 + α1θ2 + α2p

2 + α3λ2 + α4qp+ α5θλ

]=

= −[ pθm, α0q

2]−[ pθm, α1θ

2]−[ pθm, α2p

2]−[ pθm, α3λ

2]−[ pθm, α4qp

]−[ pθm, α5θλ

]=

= − p

m

[θ, α0q

2]−[ pm, α0q

2]θ− p

m

[θ, α1θ

2]−[ pm, α1θ

2]θ− p

m

[θ, α2p

2]−[ pm, α2p

2]θ+

83

− p

m

[θ, α3λ

2]−[ pm, α3λ

2]θ− p

m

[θ, α4qp

]−[ pm, α4qp

]θ− p

m

[θ, α5θλ

]−[ pm, α5θλ

]θ

(137)

Utilizing our nifty identities (10) and (12) and Koopman Algebra again the following

can be said about the above commutators:

[θ, α0q

2]

= α0q[θ, q]

+[θ, α0q

]q = −2iα0q

[θ, α1θ

2]

= α1θ[θ, θ]

+[θ, θ]α1θ = 0

[θ, α2p

2]

= α2p[θ, p]

+[θ, p]α2p = 0

[θ, α3λ

2]

= α3λ[θ, λ]

+[θ, λ]α3λ = 0

[θ, α4qp

]= α4q

[θ, p]

+[θ, q]α4p = −iα4p

[θ, α5θλ

]= α5θ

[θ, λ]

+[θ, θ]α5λ = 0

[ pm, α0q

2]

= α0q[p/m, q

]+[p/m, q

]α0q = 0

[ pm, α1θ

2]

= α1θ[ pm, θ]

+[ pm, θ]α1θ = 0

[ pm, α2p

2]

=α2

mp[p, p]

+[p, p]α2

mp = 0

[ pm, α3λ

2]

=α3

mλ[p, λ]

+[p, λ]α3

mλ =

2iα3

mλ

84

[ pm, α4qp

]=α4

mq[p, p]

+[p, q]α4

mp = 0

[ pm, α5θλ

]=α5

mθ[p, λ]

+[p, θ]α5

mλ =

iα5

mθ

Ergo, the above expression (137) reduces to:

[I ,pθ

m

]= 2iα0

pq

m+ iα4

p2

m− 2iα3

λθ

m− iα5

θ2

m(138)

Now, we do the same for [I , k(t)λq]:

[I , k(t)λq

]= −

[k(t)λq, I

]= −

[k(t)λq, α0q

2+α1θ2+α2p

2+α3λ2+α4qp+α5θλ

]=

= −[k(t)λq, α0q

2]−[k(t)λq, α1θ

2]−[k(t)λq, α2p

2]+

−[k(t)λq, α3λ

2]−[k(t)λq, α4qp

]−[k(t)λq, α5θλ

]=

= −k(t)λ[q, α0q

2]−[k(t)λ, α0q

2]q − k(t)λ

[q, α1θ

2]+

−[k(t)λ, α1θ

2]q − k(t)λ

[q, α2p

2]−[k(t)λ, α2p

2]q+

−k(t)λ[q, α3λ

2]−[k(t)λ, α3λ

2]q − k(t)λ

[q, α4qp

]+

−[k(t)λ, α4qp

]q − k(t)λ

[q, α5θλ

]−[k(t)λ, α5θλ

]q

We can perform a similar series of steps to achieve:

[q, α0q

2]

= 0

[k(t)λ, α0q

2]

= 0

85

[q, α1θ

2]

= α1θ[q, θ]

+[q, θ]α1θ = 2iα1θ

[k(t)λ, α1θ

2]

= 0

[q, α2p

2]

= 0

[k(t)λ, α2p

2]

= k(t)α2p[λ, p]

+[λ, p]k(t)α2p = −2ik(t)α2p

[q, α3λ

2]

= 0

[k(t)λ, α3λ

2]

= 0

[q, α4qp

]= 0

[k(t)λ, α4qp

]= k(t)α4q

[λ, p]

+[λ, q]k(t)α4p = −ik(t)α4q

[q, α5θλ

]= α5θ

[q, λ]

+[q, θ]α5λ = iα5λ

[k(t)λ, α5θλ

]= 0

Therefore, putting it all together:

[I , k(t)λq

]= −2iα1k(t)λθ + 2ik(t)α2pq + ik(t)α4q

2 − ik(t)α5λ2 (139)

86

Utilizing equations (136), (138), (139) with (135), we can compute the invariant

to be:

∂α0

∂tq2+

∂α1

∂tθ2+

∂α2

∂tp2+

∂α3

∂tλ2+

∂α4

∂tqp+

∂α5

∂tθλ+

1

i~

(2iα0

pq

m+iα4

p2

m−2iα3

λθ

m−iα5

θ2

m+

+ 2iα1k(t)λθ − 2ik(t)α2pq − ik(t)α4q2 + ik(t)α5λ

2)

= 0 (140)

Combining like terms:

(∂α0

∂t− k(t)α4

~

)q2 +

(∂α1

∂t− α5

~m

)θ2 +

(∂α2

∂t+α4

~m

)p2 +

(∂α3

∂t+k(t)α5

~

)λ2+

+(∂α4

∂t+

2α0

~m− 2k(t)α2

~

)qp+

(∂α5

∂t− 2α3

~m+

2k(t)α1

~

)θλ = 0 (141)

It must equal zero, but each of the operators we know is nonzero, so the only way

for this equation to be true is if all the coefficients above equal zero. Setting all the

coefficients equal to zero and solving the system of equations will lead you to the

dynamics of the (here classical) Harmonic Oscillator.

For instance, if we solve for α1 we will get the equation:

∂2α1

∂t2+ 2k(t)α1 =

1

2α1

(∂α1

∂t

)2

+C

α1

And if we define an arbitrary function called ρ(t) so that α1 = ρ2(t)2

, we will have

d2ρ

dt2+ k(t)ρ = C/ρ3

which is an equation we call the Ermakov equation (Lewis Jr. 1967; Lewis Jr. and

Riesenfeld 1968). Using the Ermakov equation, all other coefficients can be rewritten

so that the invariant has the following form:

I =1

2

[ qρ

+ (dρ

dtq − ρp)2 +

λ2

ρ2+ (

dρ

dtλ+ ρθ)2

].

87

Notice that the above classical Ermakov-Lewis invariant cannot be directly observed.

This is because operators like θ and λ are not directly tied to physically observable

quantities (Ramos-Prieto et al. 2018). Nevertheless, it serves as an important tool in

solving Harmonic Oscillation problems.

We can define following the unitary transformation

T = exp

[iln(ρ)

2(qp+ pq)

]exp

[− i

2ρ

dρ

dtq2

],

which for the quantum case transforms the time dependent problem described above

to a time independent problem (Lewis Jr. 1967; Lewis Jr. and Riesenfeld 1968) by

the following:

1

2[p2 + q2] = T IT †.

In the classical case, we can define the following two unitary transformations (Ramos-

Prieto et al. 2018):

T1 = exp

[i

ρ

dρ

dtqλ

],

T2 = exp

[iln(ρ)

2(qθ + θq)

]exp

[−i ln(ρ)

2(pλ+ λp)

].

We can define the new wavefunction Ψ(q, p, t) = T2T1ψ(q, p, t) and we can rewrite

our equation of interest (133):

i∂

∂tΨ(q, p, t) =

1

ρ2

[ pθm− λq

]Ψ(q, p, t). (142)

where the values of the operators are exactly the same as in (133). The original

wavefunction equation got transformed into a wavefunction equation without the time

dependent potential term in front of the position operator. This can be much more

easily solved to deduce the statistical motion of the classical particles.

Note that this method was originally used in the operatorial form for the Schrodinger

equation (Lewis Jr. 1967). We are able to successfully implement this procedure for

88

KvNM as well. If you graph the phase space diagrams for a selected k(t), you can

see that it is indeed the graph of an oscillator (Ramos-Prieto et al. 2018). KvNM

is successful at bridging methods that involve quantum-like operators with classical

physics.

6.4 Aharonov–Bohm Effect

The Aharonov-Bohm Effect is a surprising phenomena that emerges from the quan-

tum wavefunction interacting with the magnetic potential. In classical electromag-

netic theory, one can take the electric ~E and magnetic ~B fields and decompose them

into various derivatives of electric and magnetic potentials:

~E = −~∇φ,

~B = ~∇× ~A,

where φ is the electric scalar potential and ~A is the magnetic vector potential. Both

of these potentials have the unique property that they make ~E and ~B gauge invariant,

in other words, that a ‘shift’ χ will not produce different magnetic or electric fields.

In other words, the new potentials ~A′ and φ′ defined by

~A′ = ~A+ ~∇χ,

and

φ′ = φ− ∂χ

∂t,

will produce the same magnetic and electric fields as ~A and φ. These potentials have

historically been seen as useful mathematical constructs with little physical bearing

on reality in themselves. The Aharonov-Bohm effect soon put this assumption to the

test.

The Aharonov-Bohm effect is a a quantum interference, much like that in Young’s

Double Slit Experiment, produced by the magnetic potential changing the phases of

89

the quantum wavefunctions. When different wavefunctions with difference phases

meet, they interfere with themselves. The surprising thing is that the interference

pattern does not depend on the magnetic field ~B but directly on the magnetic potential

~A. In other words, if the magnetic field ~B = ~∇ × ~A is zero but the potential is

nonzero, an effect will still be observed. This suggests that the magnetic potential

~A is more fundamental than the magnetic field ~B, which have always been seen as

being on equal footing in classical theory.

Aharonov and Bohm described the theory behind the phenomena in 1959, but

the idea was not experimentally verified for the magnetic potential until after their

deaths. The experiment used a solenoid, which is the model we will use here for the

magnetic potential. Recall that a solenoid is a cylindrical device made of winding

wire which produces a magnetic field inside the solenoid and zero magnetic field

outside the solenoid.

The generic setup of the experiment for either type of potential is depicted in Fig-

ure 4. A coherent beam of electrons would be split into two branches, travel through

an electric or magnetic potential where ~E = ~0 or ~B = ~0, and the two branches would

then be recombined. Each branch’s wavefunction took a slightly different path, so

when the two branches are recombined an interference pattern should emerge be-

cause of a phase difference acumulated along each path. This is a semiclassical way

to visualize the experiment (recall that in QM proper, the particle actually takes all

possible paths between source and sink in the allotted time.)

To understand mathematically how the phase shift happens, first recall the polar

form of the wavefunction (3) and what the Action represents (51). We can write that:

ψ = R exp

[i

~

ˆ t

0

T − V dt]. (143)

The above wavefunction can be written for a region with no potential energy (free

particle case) as:

ψ0 = R exp

[i

~

ˆ t

0

Tdt

].

90

And so (143) can be rewritten in the generic case as:

ψ = ψ0 exp

[− i~

ˆ t

0

V dt

].

For a wavefunction going through an electric potential, from CM we know we can

write that the potential energy is V = eφ, where e is the charge of the particle’s

wavefunction and φ is the scalar potential. And so the generic wavefunction can be

written as the following for the case with electric potential:

ψ = ψ0 exp

[−ie

~

ˆ t

0

φdt

]. (144)

The above formulas can be used to describe what should happen in the experiment in

the presence of the electric potential.

Refer to Figure 4. Starting off, the beam of coherent electrons has wavefunction

ψ0 in the absence of an external potential. It is split into two, and both branches travel

through a region of potential φ, so each wavefunction gains the form of (144). The

branching could be represented as ψ01 and ψ0

2 upon reaching the area with electric

potential at time t1. After some evolution they leave the potential at time t2, and they

are then recombined. The first branch travelled along potential φ1 and the second

branch travelled along φ2, in this semiclassical picture. You can write the following

superposition for the final wavefunction:

ψf = ψ1 + ψ2 = ψ01 exp

[− i~C1

]+ ψ0

2 exp

[− i~C2

],

where we define the phases

C1 = e

ˆ t2

t1

φ1dt,

and

C2 = e

ˆ t2

t1

φ2dt.

91

Since C1 and C2 are different, we have gained a phase shift of (C1 −C2)/~ 24 There

is absolutely no classical force ~F = e ~E exerted on the electrons as there is no elec-

tric field, but a measurable effect should still be produced on the electron’s behavior

(Aharonov and Bohm 1959).

A similar analysis can be done with the magnetic vector potential. Following

Aharonov and Bohm 1959, we can generalize (144) to include the magnetic potential.

In relativity, you can write the electromagnetic potential as the four vector Aµ =

(φ/c ~A) = (φ/c Ax Ay Az) where as before, φ is the electric potential and ~A is

the magnetic potential. The position four vector can be represented is xµ = (ct ~x) =

(ct x y z). Rewriting the position four vector as the differential dxµ, we can generalize

the potential integral to be:

e

~

˛Aµdxµ =

e

~

˛ (φdt−

~A

c· d~x). (145)

The path of integration has to be closed in spacetime (Aharonov and Bohm 1959).

Let us assume that the variable t is a constant in order to understand how interference

could arise strictly from the magnetic potential. The above integral suggests that we

will expect the following phase shift to hold true:

(C1 − C2)/~ = − e

c~

˛~A · d~x

We can set up a similar situation as to before.

Starting off, the beam of electrons has wavefunction ψ0 in the absence of an ex-

ternal magnetic potential. It is split into two, and both branches travel through a

region of potential ~A. Each wavefunction gains the form of (144) except we re-

place exp[−(ie/~)

´φdt]

with exp[−(ie/~c)

´~A · d~x

]as described above. In other

words:24This comes straight from the Euler formula. Recall that:

ei(θ+C) = cos(θ + C) + isin(θ + C)

. Here, the C is the phase. If the phase of two waves are different, we should expect interference.

92

ψ = ψ0 exp

[− ie~c

ˆ~A · d~x

]. (146)

The branching could be represented as ψ01 and ψ0

2 upon reaching the area with mag-

netic potential at time t1. After some evolution they leave the potential at time t2, and

they are then recombined. You can write the following superposition:

ψf = ψ1 + ψ2 = ψ01 exp

[− i~C1

]+ ψ0

2 exp

[− i~C2

],

where we define

C1 =e

c

ˆ ~x(t2)

~x(t1)

~A · d~x′,

and

C2 = −ec

ˆ ~x(t2)

~x(t1)

~A · d~x′,

where we assume equal path length for each branch. If the path length is indeed kept

the same, then you will have phase difference (C1 − C2)/~ = (e/c~)´~A · d~x in the

end (Aharonov and Bohm 1959). Note that this is true even if the magnetic force

~F = q~v × ~B is zero because ~B = ~0. This is a purely quantum effect.

The electric Aharonov-Bohm effect described above has been theorized, but no

experimental data exists for it as of yet. We will focus on the magnetic potential there-

fore. No further space will be dedicated to the electric potential, but all considerations

from here on out will be framed in the context of the magnetic potential.

One may ask the obvious question, in the descriptions above, if all possible paths

are taken in the Path Integral, would not some paths go through the magnetic field,

explaining the outcome of the experiment? Since the Feynman Path Integral defines

a set starting point and endpoint in a set time period, it is actually possible to show

that you can set up a barrier (described in Aharonov and Bohm 1959) so that no

wavefunction directly interacts with the inside of the solenoid in the magnetic case.

93

The changes we see, therefore, must be due to the nonzero magnetic potential instead

of the magnetic field itself.

KvNM, having the mathematical form of quantum theory, might be one of the

best tools to show that the Aharonov-Bohm effect does not exist in CM (Mauro 2003).

This is a purely quantum phenomenon. Interference is not something expected in the

domain of CM, but is expected under QM. This can be traced to the behavior of the

generator of motion, the quantum Hamiltonian, having the following form under QM

in the presence of the magnetic potential for a free particle:

H =1

2m

(p− e

c~A)2

In the presence of the magnetic potential, the free particle Hamiltonian has the p

operator replaced with p − (e/c) ~A. This is described as a minimal coupling rule

(Gozzi and Mauro 2002). If KvNM describes CM precisely, we should expect that

the generator of motion, the Koopman Generator (45), will not be affected by the

magnetic potential even if we treat it mathematically in the exact same ways as we

treat the quantum Hamiltonian. In the following pages, we will put this to the test.

First, imagine two cylinders, one within the other (following Gozzi and Mauro

2002; Mauro 2003). Let them be infinite in height. The smaller cylinder on the

inside will represent the solenoid which houses a magnetic field. The outside of the

solenoid (area inside the larger cylinder but outside of the smaller cylinder) will have

no magnetic field, but there will be a magnetic potential. Since we are dealing with

cylinders, it would be useful to rewrite the generators of motion in terms of cylindrical

coordinates. The conversion of cartesian to cylindrical coordinates is given by:

x = r cos(θ),

y = r sin(θ),

and

z = z, (147)

94

where r is the distance from the z axis and θ is the angle in the xy plane.

With this notation, the quantum Hamiltonian for a free particle can be written:

H = − ~2

2m~∇2 = − ~2

2m

( ∂2

∂x2+∂2

∂y2+∂2

∂z2

)= − ~2

2m

( ∂2

∂r2+

1

r

∂

∂r+

1

r2

∂2

∂θ2+∂2

∂z2

).

This is now in cylindrical coordinates. The eigenvalue problem we want to solve is:

Hψ(r, θ, z) = Eψ(r, θ, z).

where E are the energy eigenvalues. The above expression is often termed the time

independent Schrodinger equation, as it uses the quantum Hamiltonian to extract en-

ergy values with no explicit time derivative.

Since the operators ∂/∂θ and ∂/∂z commute with the above form of H , we can

choose solutions of the form (Gozzi and Mauro 2002):

ψ(r, θ, z) =1

2πexp

[ip0zz

~

]exp[inθ]R(r), (148)

where p0z is a constant and n is an integer. Plugging in this wavefunction into the

time independent Schrodinger equation will give you solutions in the form of Bessel

Functions:

d2R(r)

dr2+

1

r

dR(r)

dr+(2mE

~2− p0

z2

~2− n2

r2

)R(r) = 0.

Bessel Functions are a class of functions that govern cylindrical harmonics.

If you turn on the magnetic potential, we will now have to solve

HAψ(r, θ, z)A = EAψ(r, θ, z)A.

In order to achieve this, we have the following minimal coupling rule in cylindrical

95

coordinates:

pθ = −i~ ∂∂θ⇒ pθ −

e

c

Φ

2π= −i~ ∂

∂θ− e

c

Φ

2π(149)

The quantum Hamiltonian therefore becomes:

HA = − ~2

2m

( ∂2

∂r2+

1

r

∂

∂r+

1

r2

( ∂∂θ− ie

chΦ)2

+∂2

∂z2

),

where Φ is the flux (constant) through an area perpendicular to the axis of the smaller

cylinder. This area will be larger than the smaller cylinder’s cross-section. Notice

that pθ is not present in the wavefunction, so the form of (148) will remain unaltered.

The energy values in the presence of the magnetic potential, as you will see, will be

different.

As before, substituting (148) into the magnetic potential version of the time inde-

pendent Schrodinger equation will give you the following Bessel function

d2R(r)

dr2+

1

r

dR(r)

dr+(2mE

~2− p0

z2

~2− (n− α)2

r2

)R(r) = 0,

where α = eΦBch

. If you compare the two Bessel functions, you can immediately see

that the presence of the magnetic potential shifts the function’s behavior. For the

quantum case, the behavior of the system is impacted by the presence of the vector

potential.

Now we will do the same procedure for L as the generator of motion (33). For

the free particle in 3 dimensions, we know that

L = −ipxm

∂

∂x− ipy

m

∂

∂y− ipz

m

∂

∂z,

based off of (33). Just as in the quantum case, we want this to be in cylindrical coor-

dinates. We will therefore need to convert the cartesian momenta px, py, and pz into

pr, pθ, and pz. The latter are the canonical momenta, which can be calculated directly

from the Lagrangian L.25 As before, any cartesian coordinates can be converted to25Do not confuse the Lagrangian L with the operator L. These are two different mathematical

entities.

96

cylindrical coordinates via (147), which we will make use of.

To start off, we write down the Lagrangian of motion. Recall from section 4.1,

the Lagrangian is just the kinetic energy minus the potential energy, L = T − V .

Since the potential is zero, the Lagrangian is just equal to the total kinetic energy, so

that

L = T =1

2m(x2 + y2 + z2), (150)

where we use the standard dot notation, so that any variable γ has a time derivative

γ = dγ/dt. We can take time derivatives of (147) to get:

x = r cos(θ)⇒ x = r cos(θ)− r sin(θ)θ,

y = r sin(θ)⇒ y = r sin(θ) + r cos(θ)θ,

with the z coordinate unaffected. Plugging these terms into (150), we get our La-

grangian in cylindrical coordinates:

L =1

2mr2 +

1

2mr2θ2 +

1

2mz2

Recalling from Lagrangian Dynamics that the canonical momentum associated with

coordinate σ is given by

pσ =∂L

∂qσ

Therefore applying the above formula for coordinates r, θ, and z, we have:

pr =∂L

∂r= mr

pθ =∂L

∂θ= mr2θ

pz =∂L

∂z= mz

97

If we apply the same rule to the cartesian Lagrangian (150) we get:

px =∂L

∂x= mx,

py =∂L

∂y= my,

with the z coordinate momentum unaltered. Using all the above information, we

can rewrite the cartesian momenta in terms of the cylindrical momenta, using the

momenta expressions above:

px = mx = m[r cos(θ)− r sin(θ)θ] = pr cos(θ)− 1

rpθ sin(θ),

py = m[r sin(θ) + r cos(θ)θ] = pr sin(θ) +1

rpθ cos(θ).

We therefore can rewrite the generator of motion L using the above information in

terms of cylindrical coordinates:

L = −ipxm

∂

∂x−ipy

m

∂

∂y−ipzm

∂

∂z= −ipr

m

∂

∂r−i pθmr2

∂

∂θ−ipzm

∂

∂z−i p

2θ

mr3

∂

∂pr. (151)

For the case with no magnetic potential, we will be solving the analogue to the time

independent Schrodinger equation,

Lψ = Eψ,

where ψ is the KvN classical wavefunction and E is the eigenvalue of the above

expression. Note that E is certainly not the same as the Energy like in the Schrodinger

equation. We observe that L commutes with−i ∂∂θ

,−i ∂∂z

, pθ, and pz. We can therefore

deduce that the wavefunction will have the form

ψ =1

2πR(r, pr)δ(pθ − p0

θ)δ(pz − p0z) exp[inθ] exp

[iλ0zz]. (152)

98

If we plug this wavefunction into (151), we will get (Gozzi and Mauro 2002):

(− i

mpr∂

∂r+p0θn

mr2− ip0

θ2

mr3

∂

∂pr+λ0zp

0z

m− E

)R(r, pr) = 0. (153)

We have a similar expression for the magnetic potential case:

LAψA = EAψA

Just as in the quantum case, we will utilize the minimal coupling rule (149). The

only difference is that the representation of the operator pθ is no longer going to be

a derivative with respect to θ, and instead we will have pθ = pθ for the KvNM case

(Gozzi and Mauro 2002). Making the substitution

pθ ⇒ pθ −e

c

Φ

2π, (154)

we will get the following two expressions:

L = −iprm

∂

∂r− i 1

mr2

(pθ −

e

c

Φ

2π

) ∂∂θ− ipz

m

∂

∂z− i 1

mr3

(pθ −

e

c

Φ

2π

)2 ∂

∂pr, (155)

ψ =1

2πR(r, pr)δ(pθ − p0

θ −eΦ

2πc)δ(pz − p0

z) exp[inθ] exp[iλ0zz]. (156)

Plugging (156) into (155) we will get:

(− i

mpr∂

∂r+p0θn

mr2− ip0

θ2

mr3

∂

∂pr+λ0zp

0z

m− EA

)R(r, pr) = 0 (157)

If you compare (157) (case with magnetic potential) to (153) (case without magnetic

potential) you will notice they happen to be the same equation. This shows us that

the same solutions will exist for classical mechanics regardless of the presence or

absence of the gauge potential. As demonstrated, the gauge potential will however

99

affect quantum systems (Gozzi and Mauro 2002; Mauro 2003).

On a philosophical level, the above analysis raises the question what do the gauge

potentials physically mean? Historically, it was simply seen as a mathematical tool

or convenience in the study of Electromagnetism. The electromagnetic field and its

potentials were fully equivalent descriptions of reality, but the potentials were only

seen as a mathematical construct with useful properties. This might be a historic

example of physicists not taking the mathematics of their theory seriously enough.

For example, Albert Einstein famously invented the theories of special relativity by

acknowledging that the speed of light c is the same in all noninertial reference frames.

This is a fact tacitly present in Maxwell’s equations, but physicists at the time did

not treat the mathematics seriously enough and thought the equations must only be

true relative to a hypothetical ‘fluid’ filling all of space and time, the aether, an idea

Einstein showed to be wrong. Aharonov and Bohm 1959 say in their paper “a further

interpretation of the potentials is needed in the quantum mechanics”, so perhaps this

is an area for further scientific investigation and discovery in the future.

7 Other Research

Wavefunctions in a complex Hilbert phase space can exactly model CM in the form of

statistical physics. Both QM and CM can be articulated in a common mathematical

language utilizing Hermitian operators to recover expectation values of observables.

In the pages above we covered a host of different applications of the KvN wave-

function inside classical physics. We hardly exhausted the possible applications of

KvNM, however. There are many areas of research we have not even touched upon.

Some recent examples of the widespread applicability of the KvNM formulism:

1. Why the usage of finite-dimensional Hilbert Spaces in QM and KvNM is often

ill-advised

A recent trends in QM research has been to try utilize QM with finite dimen-

sions instead of the regular infinite Hilbert space. This is done to aid in clas-

100

sical computer modeling of quantum systems and an interest in the possibility

of rooting QM in a finite instead of an infinite space. For instance, approxima-

tions of the quantum Hamiltonian tend to be finite matrices (Bondar, Cabrera,

and Rabitz 2013). Bondar, Cabrera, and Rabitz 2013 demonstrate that this will

unavoidably violate the Ehrenfest Theorems. This is a serious shortcoming of

finite dimensional treatments of QM, but there are a small number of ways that

the issues might be circumvented (Bondar, Cabrera, and Rabitz 2013).

2. KvNM has been used to study aspects of String Theory and the Gravitational

Principle of Equivalence

The KvN Path Integral is utilized in Carta and Mauro 2002 to study the prop-

erties of Classical Yang-Mills Theories (a subset of String Theories.) The Path

Integral provides geometrical tools like exterior derivatives, forms, etc. The

Classical Path Integral might shed light on the geometrical aspects of these

string theories (Carta and Mauro 2002). Sen, Dhasmana, and Silagadze 2020

investigates the validity of Einstein’s principle of mass equivalence for both

classical KvNM and quantum systems. Fisher information is discussed and

how it relates to insights from KvNM and QM. It is demonstrated that the weak

principle of equivalence holds in QM just as it does in KvNM (Sen, Dhasmana,

and Silagadze 2020).

3. ODM can be used to describe relativistic particles

The ODM paradigm is upheld in relativistic settings. The following two Ehrenfest-

like theorems were postulated (Cabrera et al. 2019) based on relativistic prin-

ciples

d

dt〈xk〉 = 〈cγ0γk〉.

d

dt〈pk〉 = 〈ce∂kAνγ0γν〉.

If the quantum commutator is chosen, you can follow a series of steps similar

101

to those in section 3 to derive the famous Dirac equation, which governs rel-

ativistic quantum particles (Cabrera et al. 2019). If the Koopman Algebra is

chosen and you constrain the system from making antiparticles, the same se-

ries of steps as deriving the KvNM will give you the classical Spohn equation

(Cabrera et al. 2019). This is quite a surprising result - ODM can study spin

1/2 relativistic particles.

4. ODM has been used to study Time’s Arrow and stochastic behavior from de-

terminism

A lot of processes are reversible in physics, but our intuitions of the world are

that it is irreversible and things tend towards decay. How does irreversibility

arise out of the reversible? McCaul and Bondar 2021 utilizes the KvN Path

Integral to formulate a functional model for how irreversibility arises out of

the reversible laws we have.

5. Reformulation of Electromagnetism

Rajagopal and Ghose 2016 provides a reformulation of electromagnetic theory

that draws inspiration from the KvN theory.

6. KvN wavefunctions were utilized to understand the interaction of classical with

quantum systems

The problem of modeling quantum and classical system interaction is an old

one. Utilizing the KvN classical wavefunction, however, had led to new models

of how the interaction occurs. Not only does it describe the classical system’s

effect on the quantum system, but uniquely it models how the quantum system

will impact the classical system in return (Bondar, Gay-Balmaz, and Tronci

2019). It also overcomes difficulty in some other models. For instance, the

quantum density matrix will always be positive-definite in this model (Bondar,

Gay-Balmaz, and Tronci 2019).

Among other areas of inquiry. Doubtlessly, many more areas of research that will

benefit from KvNM and a paradigm like ODM.102

It is important to have many “tools” in science when investigating novel phenom-

ena. Feynman famously said that the reason he had solved problems others could

not, is that his toolbox had a different set of tools than everyone else. KvNM is yet

an another novel tool for you, the reader, to have in your toolkit. KvNM has and will

continue to shed light on new and exciting phenomena in science.

103

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tials in the Quantum Theory”. In: Physical Review 115, p. 485. doi: https :

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Appendices

A Useful Mathematics for Hilbert Space Physics

Note: A lot of the following useful mathematics was found in the Appendix of Bon-

dar et al. 2012.

Theorem I: If you have a function of operator A, i.e. f(A), then the following

eigenvalue expressions are true:

f(A) |A〉 = f(A) |A〉

〈A| f(A) = 〈A| f(A)

This should purposefully resemble the prior eigenvalue problem you are familiar

with, A |A〉 = A |A〉. This theorem is true for multivariable operators, for instance,

A(q, p) |q, p〉 = A(q, p) |q, p〉

.

Theorem II (Resolution of Identity): The following theorem says that

I =

ˆdA |A〉〈A|

where I is the identity operator, functionally equivalent to multiplying an object by

1. Since it is just equal to 1, you can introduce the Resolution of Identity in a variety

of calculations. As always A is an arbitrary placeholder variable, and you can make

it any variable of interest.

Theorem III (Cauchy-Schwartz Inequality): It simply states that the following is al-

107

ways true for complex vectors in Hilbert Space:

〈C|C〉〈D|D〉 ≥ | 〈C|D〉 |2

where C and D are generic placeholder variables.

Theorem IV : If the commutator relationship between two generic operators A and

B is [A, B] = iκ, then the following statements hold true:

〈A| B |ψ〉 = −iκ ∂

∂A〈A|ψ〉

〈B| A |ψ〉 = iκ∂

∂B〈B|ψ〉

Recall that 〈B|ψ〉 = ψ(B) and 〈A|ψ〉 = ψ(A).

Theorem V : If the commutator relationship between two generic operators A and

B is [A, B] = iκ, then the following statements hold true:

〈A|B〉 =1√2πκ

eiAB/κ (158)

This is not difficult to demonstrate. First, take Theorem IV and realize that |ψ〉 is a

general vector that can be substituted for a more specific vector. This is a common

practice in QM. Let us say that |ψ〉 = |B〉. Then by Theorem IV we have:

〈A| B |B〉 = B 〈A|B〉 = −iκ ∂

∂A〈A|B〉

You can solve this common differential equation for the function 〈A|B〉 to get:

〈A|B〉 = CeiAB/κ

where C is a constant of integration. C can be found in the following way: Start with

equation (18)

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〈A|A′〉 = δ(A− A′)

and use Theorem II with A = B 26 to get:

〈A|A′〉 = δ(A− A′) = 〈A| I |A′〉 =

ˆdB 〈A|B〉〈B|A′〉

Since we know what 〈A|B〉 is, we plug it in to get:

〈A|A′〉 = δ(A− A′) =

ˆdB[CeiAB/κ][Ce−iA

′B/κ] = C2

ˆdBei(A−A

′)B/κ

From equation (22) of section 2, we can easily see that

ˆdBei(A−A

′)B/κ = 2πκδ(A− A′)

Ergo,

δ(A− A′) = C22πκδ(A− A′)

C22πκ = 1

C2 =1

2πκ

C =1√2πκ

We now know C and have successfully demonstrated equation (158).

Theorem VI: As strange as it sounds, the mathematics of Hilbert Space defines deriva-

tives with respect to operators. If you have [Ak, B] = Constant, then a function of

operators f(A1, A2, ..., An) will obey the following rule:

[f(A1, A2, ..., An), B] =n∑k=1

[Ak, B]∂f(A1, A2, ..., An)

∂Ak

26In other words, sinceA is just a generic placeholder for Theorem II, we will use I =´dB |B〉〈B|

109

Notice the partial derivative with respect to an operator. The next theorem shows how

to change this odd operator derivative to regular derivatives that we know and love.

Theorem VII The following is the relationship between derivatives with respect to

operators and the regular derivatives we are most generally familiar with:

∂f(A1, A2, ..., An)

∂Ak=

1

(2π)n

ˆ n∏l=1

dεl dµl exp[in∑q=1

µq(Aq − εq)]∂f(ε1, ε2, ..., εn)

∂εk

where on the left you can see the derivative with respect to an operator and on the

right you can see the derivative with respect to a scalar variable.

Theorem VIII: An integral of a Gaussian function has the following form:

ˆ ∞−∞

e−ax2+bx+cdx =

√π

aeb2

4a+c

where a, b, and c can be real or imaginary constants.

B The Principle of Causality and Stone’s Theorem

A unitary operator U(t2, t1) always has the following properties:

1. Group product: U(t3, t2)U(t2, t1) = U(t3, t1)

2. Reversibility: U(t2, t1)† = U(t2, t1)−1 = U(t1, t2)

3. Identity: U(t1, t1) = 1

The job of the unitary operator U(t2, t1) is to bring you from t1 to t2. The group

product encapsulates the concept of cause and effect, as you move from cause to

effect through time. Causality is central to the philosophy of science, and we should

not be surprised that it pops up inside the mathematics of physics.

The three properties of the unitary operator above imply Stone’s Theorem:

110

If the three properties hold true, then the following relationship is also true:

i∂

∂tU(t2, t1) = GU(t2, t1)

where G is a Hermitian operator. If we assume that the wavefunction ket evolves via

the unitary operator

|ψ(t2)〉 = U(t2, t1) |ψ(t1)〉 ,

then we recover the Schrodinger-like equation we spoke of in earlier sections:

i∂

∂tU(t2, t1) |ψ(t1)〉 = GU(t2, t1) |ψ(t1)〉 ⇒ i

∂

∂t|ψ(t2)〉 = G |ψ(t2)〉

This is the reason why Stone’s Theorem becomes an important ingredient in section

3.

C Exponated Operators and the Propogator

Although it might seem odd at first, exponated operators are allowed within Quantum

Mechanics. The Maclaurin series for the f(x) = ex is given by:

ex =∞∑n=0

xn

n!= 1 +

1

1!x+

1

2!x2 +

1

3!x3 +

1

4!x4 + · · ·

Let us say that you have a generic operator A, like a square matrix. As long as An is

defined for any real number n, then you can say:

eA =∞∑n=0

An

n!= 1 +

1

1!A+

1

2!A2 +

1

3!A3 +

1

4!A4 + · · ·

The Maclaurin series defines exponated operators as an infinite sum.

If we have an expression such as eAt where A is time-independent, we can also

derive a derivative rule for it:

111

eAt =∞∑n=0

(At)n

n!= 1 +

1

1!At+

1

2!(At)2 +

1

3!(At)3 +

1

4!(At)4 + · · ·

d

dteAt =

d

dt1 +

d

dt

1

1!At+

d

dt

1

2!(At)2 +

d

dt

1

3!(At)3 + · · · =

=1

1!A+

1

2!A2(2t)+

1

3!A3(3t2)+

1

4!A4(4t3)+··· = A

[1+

1

2!A(2t)+

1

3!A2(3t2)+

1

4!A3(4t3)+···

]=

= A[1+

1

2 · 1A(2t)+

1

3 · 2 · 1A2(3t2)+···

]= A

[1+

1

1!At+

1

2!(At)2+

1

3!(At)3+···

]= AeAt

∴d

dteAt = AeAt (159)

The above relation is true even if a real or complex constant is placed in front of the

exponated term At. You can redo the exercise to convince yourself of this. We will

use this important fact in order to demonstrate the important relationship (52) is fully

equivalent to the Schrodinger equation:

|ψ(t)〉 = e−i~ tH |ψ(0)〉

i~∂

∂t|ψ(t)〉 = i~

∂

∂te−

i~ tH |ψ(0)〉 = i~

−i~He−

i~ tH |ψ(0)〉 = H |ψ(t)〉

∴ i~∂

∂t|ψ(t)〉 = H |ψ(t)〉

A similar derivation can be done for (66), to demonstrate that it is fully equivalent to

Stone’s Theorem (Appendix B).

112

Figure 1: This is the set up of the double slit experiment. Middle wall is at y = yMand rightmost wall is at y = yR. Adapted from Mauro 2002

Figure 2: a. Left: Graph of KvN probability distribution. b. Right: Graph of QMprobability distribution. Adapted from Mauro 2002.

Figure 3: a. Left: Example graph of a single variable gaussian curve. b. Right:Example graph of a double variable gaussian curve. Adapted from Wolfram Alpha2021.

114

Figure 4: Aharanov-Bohm Experiment set up. Reproduced from Aharonov andBohm 1959.

115

introduction to koopman-von neumann mechanics an …

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