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The Quantum MysteryThe Arrow of Time

A Unified Theory

Solving the Quantum Mystery withthe Arrow of Time

Eddy Keming Chen

Department of Philosophy,Rutgers University, New Brunswick

[email protected]

February 5, 2018


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The First Puzzle

Our best quantum theories require something called the “wavefunction.”

It is a metaphysically scary object:

high-dimensional

complex numbers

full of arbitrary conventions.

The “quantum mystery”: what is the wave function?


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Nature is scary...


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The Second Puzzle

We often see:

Ice melts

Air spreads out

Bananas go bad

We never see:

An ice cube gets bigger

Air contracts to a corner

Bananas become fresher

What explains the arrow of time?


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Explain with physics?

There is no arrow of time in fundamental physics.

The fundamental equations are blind to the past /future distinction.


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Nature is weird!


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The Two Puzzles

They are often thought to be independent phenomena.

But, are they?


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How are the two puzzles related?


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In this talk, I propose a new theory of time’s arrow in a quantumuniverse.

It solves both puzzles.

It replaces the wave function with something new.

Further evidence of the ongoing fruitful connections betweenphilosophy and the sciences.


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Killing Two Birds with One Stone

Figure: Solving two puzzles with one theory.


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Overview

1 The Quantum Mystery

2 The Arrow of Time

3 A Unified Theory


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2 The Arrow of Time

3 A Unified Theory


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The Quantum Mystery

All of our best quantum theories refer to the gadget called the“wave function.”

It explains:

The double-slit experiment.

Stability of atoms.

Galaxy formation.

Applications:

Electronics.

Quantum computing.


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A Wave Function

Figure: Something wavy—the wave function.


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Quantum Mechanics

Physical systems are described by a new mathematical gadget:

The wave function: ψ.

It obeys the Schrodinger equation...

It evolves into “branches”: ψa + ψb + ψc + ...

...except when it’s being “observed”:

The wave function “collapses” into one of the branches: ψa

The outcome is random (probability given by the Born rule).


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Schrodinger’s Cat Thought Experiment

Figure: QM describes all physical systems—big and small.


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Schrodinger’s Cat Described by Quantum Mechanics

Physical systems are described by a new mathematical gadget:

E.g. Schrodinger’s cat at t0: ψKitty(t0)

It obeys the Schrodinger equation...

E.g. Schrodinger’s cat at t1: ψKitty(t1) = ψAlive + ψDead.

...except when it’s being “observed”:

E.g. Schrodinger’s cat at t2: ψKitty(t2) = ψAlive.


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The Measurement Problem

There is something wrong...

If the Schrodinger equation is a fundamental law of nature,then it should govern all physical interactions, includingmeasurements.

Hence, the wave function should never collapse; the wavefunction branches will never be reduced to just one.

QM should predict that there is no definite outcome ofexperiment.

But we have definite outcomes of experiment.


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Solutions

The measurement problem has been solved.

1 Bohm (1952). Particles with definite locations.

2 Ghirardi, Rimini, and Weber (1986). Wave functionspontaneously collapses. [+ some variants]

3 Everett (1956). All branches of the wave function are real. [+some variants]


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Solving the Measurement Problem

These theories clean up the quantum mess:

Logically consistent, very precise.

Nothing special about the use of observations orconsciousness.

But the wave function is still there!


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The Quantum Mystery

Even in the cleaned-up versions of quantum mechanics, we are stillleft with the wave function.

It “governs” local ontologies (e.g. particles and fields).

But what is the wave function?


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The wave function is metaphysically scary...


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The wave function: ψ : R3N → C, where N ≈ 1080.

High-dimensional.

Complex numbers.

Defined with arbitrary conventions.

Quite unlike the local ontology of particles and fields.


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It is a threat to many metaphysical theses:

Nominalism: The fundamental ontology does not containmathematical objects.

Intrinsicalism: The fundamental ontology can be characterizedintrinsically, without using any arbitrary conventions.

3D-ism: The fundamental space is (more or less)3-dimensional.

Humeanism: The fundamental ontology is a Humean mosaic oflocal matters of particular facts, and everything elsesupervenes on that.


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Against Nominalism:

Malament 1982: it is impossible to reformulate quantummechanics without reference to mathematical objects.

Against Intrinsicalism:

Maudlin 2013: we need to use arbitrary conventions to definethe wave function.

In Chen (2017b), I prove that we can specify the wave function incompletely nominalistic and intrinsic relations on physicalspace-time.


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Against 3D-ism:

Albert 1996: because of the wave function, the fundamentalspace is ridiculously high-dimensional—≈ 1080 dimensions.

Against Humeanism:

Teller 1986, Maudlin 2007: the wave function assigns holisticproperties that do not supervene on the Humean mosaic.

In this talk, I will use the data from the arrow of time to build anew quantum theory that is compatible with both theses (and havemany other goodies).


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Responses to the wave function

High-dimensional realism.

Albert 1996, Loewer 1996, Ney 2012, North 2013

Multi-field realism.

Chen 2017a, 2017b, Hubert and Romano 2018

Brute Humeanism.

Darby 2012

The nomological view about the wave function.

Goldstein and Zanghi 2013, Miller 2014, Esfeld 2014, Bhogaland Perry 2015, Esfeld and Deckert 2017, Chen ms.

All have heavy costs; some face devastating objections.


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Realism

We might need to be less confident in metaphysical realism, ifnone of these realist responses is plausible.

A partial victory for the metaphysical anti-realists.


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The Quantum Mystery

Figure: What exactly is the wave function? How to make metaphysicscome out right with QM?


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2 The Arrow of Time

3 A Unified Theory


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The Arrow of Time

I suggest that the arrow of time holds the key to the quantummystery.

This will become clear after this section.


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Back to the second puzzle

There is an arrow of time in our experiences.

Ice cubes will melt in hot waterAir will spread out in a roomFlowers will decay

But there is no arrow of time in fundamental physics.


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Macroscopic Physics

Thermodynamics: a branch of physics that deals with macroscopicphenomena.

Macroscopic properties:

volume

pressure

density

entropy


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Entropy

Lower entropy:

Ice cube in a cup of coffee.

Air concentrated in a small corner.

Fresh potatoes.

Universe: homogeneous, dense, contracted matter.

Higher entropy:

Ice completely melted in a cup of coffee.

Air spread out to the entire room.

Decayed potatoes.

Universe: matter clumped together as galaxies and planets.


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The Second Law

The Second Law of Thermodynamics

Systems evolve from lower to higher entropy (until they reach theentropy maximum).

1 There is an arrow of time in the 2nd Law: future events havehigher entropy than past events.

2 It explains the time-irreversible phenomena—to some extent.

3 It concerns primarily macroscopic phenomena; it is notfundamental physics.


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Let’s Go Deeper!

Can we explain the arrow of time with fundamentalphysics?

The fundamental equations are symmetric in time.

Example: ~F = m~a

A piece of chalk

A billiard ball


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Time Reversal Invariance (TRI)

A theory T satisfies TRI =def for any sequence ofevents, it is possible in T if and only if its time

reversal is possible in T.


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The equations of fundamental physics are time reversal invariant.

Classical Mechanics (e.g. the Newtonian equation ~F = m~a)

Quantum Mechanics (the Schrodinger equation)

General Relativity (Einstein field equations)

Solution: we can add an initial condition.


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The Past Hypothesis:

The universe started in a low-entropy state.


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The Cosmic Arrow of Time


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What is entropy?

Macrostates vs. Microstates. [Picture: gas in a box]

Classically:

Macrostate: a macroscopic description of the physical system(in terms of volume, density, pressure).

Microstate: a microscopic description of the physical system(in terms of positions and velocities of the particles).

Entropy measures how many microstates are compatible with amacrostate. (Boltzmann)


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Translations

A microstate ⇔ a possible world.

A macrostate ⇔ a set of possible worlds (that aremacroscopically indistinguishable).

A phase space⇔ the set of all (nomologically) possible worlds.


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The Phase Space

Figure: The phase space has an interesting geometry. Macrostates aredisproportionate in volume. From Penrose (1989).


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Figure: Why we need the Past Hypothesis. From Penrose (1989).


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The Past Hypothesis (PH): classical

The initial state of the universe has very low entropy.

For the experts: at t0, the microstate of the universe was lying in aspecial macrostate with very small volume on the phase space.

Only one microstate is actual, but there are many to choose from.

Remarkably simple.

Highly informative.

Explains the 2nd Law of Thermodynamics.

It is a law of nature in addition to the equations of CM.


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Classical vs. Quantum

A classical system:

Space of possibilities: phase space.

Microstate: a point in phase space X .

Macrostate: a region of phase space M(X ) that includes X .

Entropy of X : volume of M(X ).

A quantum system:

Space of possibilities: Hilbert space.

Microstate: a wave function, ψ.

Macrostate: a subspace of the Hilbert space Hv that includesψ.

Entropy of ψ: dimension of the subspace Hv .


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A simple Hilbert space


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Subspaces of a Hilbert space


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The Past Hypothesis (PH): standard quantum version

The initial quantum state of the universe is a low-entropy wavefunction Ψ0.

For the experts: at t0, the universe is described by a wave functionchosen at random from the set of wave functions in thelow-dimensional subspace.

Only one wave function is actual, but there are many to choosefrom.

Remarkably simple.

Highly informative.

Explains the 2nd Law of Thermodynamics.

It is a law of nature in addition to the equations of QM.


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PH is not enough...

In the classical case: given PH, there are still infinitely manypossible phase points to choose from.

In the quantum case: given PH, there are still infinitely manypossible wave functions to choose from.

Some of the microstates are “bad.”


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The Statistical Postulate

The Statistical Postulate (SP)

There is a uniform probability distribution over the microstatescompatible with the PH.

SP will ensure that our universe is unlikely to be one of those“bad” microstates.


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The Cosmic Arrow of Time


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A Worry

We now have two sources of “randomness” in quantum mechanics:

Probabilities from the Born rule.

Probabilities from SP.

How are they connected?


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2 The Arrow of Time

3 A Unified Theory


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A Unified Theory of Quantum Mechanics and Time’sArrow

The wave function is the source of the quantum-mechanicalphenomena.

The Past Hypothesis is the source of the arrow of time.

I will propose a unified theory that

gets rid of the wave function;

explains the quantum phenomena;

gets rid of the Statistical Postulate;

explains the arrow of time.


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A Unified Theory of Quantum Mechanics and Time’sArrow

Three steps

1 “Ontic Coarse-graining.”

2 The Initial Projection Hypothesis.

3 The Nomological Thesis.


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Step 1. Coarse-Graining


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Figure: A vector.

Figure: A subspace.


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Two kinds of coarse-graining:

Epistemic: there is a more fine-grained fact but we do nothave epistemic access to it. So we will use the coarse-grainedfact to represent our ignorances. [standard]

Ontic: the coarse-grained state that we use to represent our“ignorances” in fact carves nature at the joint. There is nomore fine-grained fact. [radical]

Let us try an “ontic coarse-graining” on the quantum state.


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Let us describe the universe at t0.

The old theory:

Macrostate: S , the low-entropy subspace(PH).

Microstate: Ψ0, a wave function inside S .

SP: a probability distribution over thesubspace S .

My proposal:

Macrostate: the low-entropy subspace S .


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Some nice mathematical gadgets to describe the subspace S :

Geometric perspective: projection onto S .

Matrix representation: an identity matrix IS restricted to S .

Density matrix: aIS , where a = 1/dimension(S).


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Step 1. Coarse-Graining: A Simple Example

Geometric perspective: projection ontoS=v1-v2 plane.

Matrix representation: an identity matrixIS restricted to S=v1-v2 plane. 1 0 0

0 1 00 0 0

Density matrix: aIS , wherea = 1/dimension(S)=1/2. 1/2 0 0

0 1/2 00 0 0

Figure: Subspace v1-v2.


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For the universe, the initial subspace S is low-dimensional, but itwill be more than 2-dimensional.

My proposal: the subspace represents the initial quantum statecompletely; there is no additional fact about which vectorrepresents the world.

The state of the universe is described by a density matrix and notby a wave function.


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We just performed an “ontic coarse-graining” on the quantumstate.

Remarks about ontic coarse-graining:

It simplifies the ontology.

It eliminates some unknowable facts.

(Claim:) It is at the right balance of simplicity and empiricaladequacy.

The ontic coarse-graining leads to some surprising consequences.


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Step 2. A New Postulate

The Initial Projection Hypothesis (IPH)

The initial quantum state of the universe is W0 = aIS .

(For the experts: at t0, the universe is described by the uniquenormalized projection on the low-dimensional subspace S .)

W0:

encodes the low-entropy information of the PH, giving rise tothe arrow of time.

is governed by a deterministic equation (the von Neumannequation), giving rise to quantum phenomena.

We can reformulate all the realist quantum theories in termsof W0.


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Compare:



The Past Hypothesis (PH): standard quantum version

The initial quantum state of the universe is a wave function Ψ0

that lies in the subspace S .

Same empirical content: the special subspace S .

IPH is no more complex than PH.

IPH is as informative as PH.


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The New Theory U:

1 Local ontology: particles or fields.

2 A density matrix.

3 The von Neumann equation.



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(Questions about empirical equivalence: wave function vs. densitymatrix.)


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Step 3. The Nomological Thesis



Proposal: IPH is a new law of nature.

PH is a law because it is simple and informative.

IPH is as simple and informative as PH.

Therefore, IPH is a candidate law of nature.


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Step 3. The Nomological Thesis



Proposal: IPH is a new law of nature.

W0 is simple to specify.

W0 is unique, i.e. completely specified by a law of nature.

Hence, W0 is nomological.

Hence, there is no need for a probability distribution overinitial states.


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The Old Theory vs. the New Theory

The Old Theory


2 Non-local ontology: wave function.

3 Dynamical Law: the Schrodinger equation.

4 The Past Hypothesis.

5 The Statistical Postulate

The New Theory U


2 Non-local ontology.

3 Dynamical Law: the von Neumann equation.


5 The Statistical Postulate


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The New Theory

The New Theory U

1 Local ontology: particles or fields. [realist ontology]

2 Dynamical Law: the von Neumann equation. [quantumphenomena]

3 The Initial Projection Hypothesis. [arrow of time]


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Consequences for 3D-ism and Humeanism

3D-ism and Humean supervenience are no longer in conflict withquantum mechanics.

The problematic non-local ontology became part of a law inthe best system.

The best system supervenes on the 4-dimensional Humeanmosaic of particles and local fields.


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Other interesting consequences

The New Theory U has many surprising features:

A unique quantum history.

Parallel to the Weyl Curvature Hypothesis.

Unification of:

1 the universe and subsystem;

2 statistical mechanics and quantum mechanics;

3 two kinds of probabilities.


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The Big Picture

Two puzzles:

The quantum mystery.

The arrow of time.

They were thought to be independent puzzles.

Our new theory U provides a unified explanation.


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The Big Picture

We have finally solved the quantum mystery.

The quantum state is more coarse-grained than we thought.The quantum state is not ontological but nomological.

The solution depends on our understanding of the arrow oftime.

Interesting consequences to explore in future work.


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Two Birds with One Stone


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The Big Picture

Why is the world so unified?

Why is Nature so kind to us?


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Thank you! The end.


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Bonus: Empirical Equivalence

Criterion for Empirical Equivalence

Two theories A and B are empirically equivalent if at any time t,they assign the same probability distribution over all possibleexperimental outcomes.

We can prove this rigorously in some cases.

Is it enough? Bell’s jumpy Everettian world.

Subsystem analysis.


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Bonus: Variations of U

U+

1 No local ontology.

2 Ontic density matrix.



U- (the ghostly theory)

1 No local ontology.

2 No non-local ontology.




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Bonus: Ontic Coarse-Graining on Classical Phase Space?

Open questions:1 Mathematically: what does a set of phase points represent on

physical space?

Microstate Dispensability: “take out the microstate and youstill have a world.”Valid in QM: still have a density matrix.Fails in CM: no more particles; no obvious counterpart todensity matrix.

2 Conceptually: any motivation for doing so in classicalmechanics?


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Bonus: W-BM+IPH

i~dW (t)

dt= [H, W ]. (1)

dQi

dt=

~mi

Im∇qi trCkW (q, q′, t)

trCkW (q, q′, t)(q = q′ = Q), (2)

P(Q(t0) ∈ dq) = trCkW (q, q, t0)dq. (3)

∂trCkW (q, q, t)

∂t= −div(trCkW (q, q, t)v), (4)

W (t0) =IS

dimS, (5)


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Bonus: W-EQM+IPH

i~dW (t)

dt= [H, W ]. (6)

m(x , t) = tr(M(x)W (t)), (7)

M(x) =∑i

miδ(Qi − x) (8)

Qiψ(q1, q2, ...qn) = qiψ(q1, q2, ...qn) (9)

W (t0) =IS

dimS, (10)


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Bonus: W-GRW+IPH

i~∂W (t)

∂t= − i

~[H,W (t)]+λ

N∑k=1

∫d3xΛk(x)1/2W (t)Λk(x)1/2−NλW (t),

(11)

Λk(x) =1

(2πσ2)3/2e−

(Qk−x)2

2σ2 (12)

m(x , t) = tr(M(x)W (t)), (13)

W (t0) =IS

dimS, (14)

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