Time from an Algebraic Theory of Moments.

B. J. Hiley.

www.bbk.ac.uk/tpru

Time through notion of Dynamical Moments.

Can we get any insights into time through quantum theory?

Compare and contrast classical mechanical time with

quantum mechanical time.

We are led to consider non-locality in time.

Ambiguity in time.

I will develop the appropriate mathematics

Groupoids bi-locality bi-algebra Hopf algebra.

Two time operators Schrödinger timeTransition time.

But there is no time operator!

duronMoment or

Explore relation betweenClassical mechanical time.

Quantum mechanical time.

In CM we have the notion of “flow” :- ;

Determined by Hamilton’s eqns of motion

In QM we have a “flow”

Determined by Schrödinger’s eqn.

Classical Hamilton flow enables us to define mechanical measure of time

Can we use Schrödinger flow to define a quantum measure of time?

Problem.Schrödinger eqn doesn’t tell us what happens

It simply tells us about future potentialities

It is the registration of a ‘mark’ that tells us something has happened.

Mechanical Time.

Peres Quantum Clock.

Attempted to design a QM clock to measure time evolution of a physical process.

Need to include clock mechanism in the Hamiltonian.

The system ‘fuses’ with the clock and changes its behaviour.

Also is operationally meaningless

Conclusion: need a different formalism, even one non-local in time[Peres, Am. J. Phys. 47, (1980) 552-7]

Fröhlich also suggested we should consider the implications of non-local time.

[Fröhlich, p. 312-3 in Quantum Implications, 1987]

We cannot make smaller than time resolution

Thus we need a relation at two distinct times and

Feynman’s Time.

contains information ‘coming’ from the future

contains information coming from the past; y

x

Feynman showed Schrödinger equation

Time-energy uncertainty.

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ΔEΔt ≈ h

The past and future mingle in the ill-defined present.

Ambiguous moments

I want to look at

[Feynman, Rev. Mod. Phys.,20, (1948), 367-387].

Not Instant but Moment.

Replace ‘instant’ by ‘moment’

Development of process is enfoldment-unfoldment

€

′e

€

M1

€

M2

€

′′e

Becoming is not merely a relationship of the present to a past

that has gone. Rather it is a relationship of enfoldments that

actually are in the present moment. Becoming is an actuality.

Bohm:-

[Bohm, Physics and the Ultimate Significance of Time, Griffin, 177-208, 1987]

What we perceive as present is a vivid fringe of memory tinged

with anticipation. [Whitehead, The Concept of Nature, p. 72-3]

Whitehead:-

How do we turn a set of moments into an algebra?

not but

Succession of Moments.

Groupoid

Regard this as a set X of arrows, sources and targets, s and t

P1 is the source s P2 is the target t

is P1 BECOMING P2

Our interpretation

P1

P2

P1Note

1 is a left unity.

3. Inverse

2. is a right unity.

is our BEING.

Since , being is IDEMPOTENT.

P1

Rules of composition.

(i) [kA, kB] = k[A, B] Strength of process.

(ii) [A, B] = - [B, A] Process directed.

(iii) [A, B][B, C] = ± [A, C] Order of succession.

(iv) [A, B] + [C, D] = [A+C, B+D] Order of coexistence.

(v) [A, [B, C]] = [A, B, C] = [[A, B], C]

Notice [A, B][C, D] is NOT defined (yet!) [Multiplication gives a Brandt groupoid]

[Hiley, Ann. de la Fond. Louis de Broglie, 5, 75-103 (1980). Proc. ANPA 23, 104-133 (2001)]

Lou Kauffman’s iterant algebra

[A, B]*[C, D] = [AC, BD] [Kauffman, Physics of Knots (1993)]

Raptis and Zaptrin’s causal sets.

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A B * C D →δBC A D [ Raptis & Zaptrin, gr-qc/9904079 ]

Bob Coecke’s approach through categories.

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If f : A → B and g : B → C, f og : A → C [Abramsky& Coecke q-ph/0402130]

The Algebra of Process.

Feynman Paths.

Interference ‘bare-bones’ Feynman

with

If

[Kauffman, Contp. Math 305, 101-137, 2002]

Classical Groupoids.

Is there anything like this in classical mechanics?

Hamilton-Jacobi

Free symp. requires

Under free symplectomorphisms,

In general Action.

Time-dependent Hamiltonian flows from a groupoid

the 2-form is preserved.

Generating functionThis means

Time Evolution Equation (1).

In the limit as Δt 0, T t we find

Liouville equation

Consider

Change coordinates

Then

What about ?

Time Evolution Equation (2).

Write

Again in the limit as Δt 0, T t we find

If we write

Quantum Hamilton-Jacobi

Quantum potential

Bohm trajectories.

Screen

Slits

Incidentparticles

x

t

x

tBarrier

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∂S∂t

+(∇S)2

2m+V + Q = 0

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Q = −h2

2m∇2R

R

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p = ∇S,

[Bohm & Hiley, The Undivided Universe. 1993]

Barrier

The Quantum potential as an Information Potential.

Nature of quantum potential TOTALLY DIFFERENT from classical potential.

It has no EXTERNAL SOURCE.

The particle and the field are aspects of the process

SELF-ORGANISATION.

The QP is NOT changed by multiplying the field by a constant.

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Q∝∇2R

R[Recall ]

STRENGTH of QP is INDEPENDENT of FIELD INTENSITY.

QP can be large when R is small.Effects DO NOT necessarily fall off with distance.

QP depends on FORM of NOT INTENSITY.

NOT LIKE A MECHANICAL FORCE.

Post-modern organic view.

The Newtonian potential DRIVES the particle.

The QP ORGANISES the FORM of the trajectories.

The QP carries INFORMATION about the particle’s ENVIRONMENT.

e.g., in TWO-SLIT experiment QP depends on:-

(a) slit-widths, distance apart, shape, etc.

(b) Momentum of particle.

QP carries Information about the WHOLE EXPERIMENTAL ARRANGEMENT.

BOHR'S WHOLENESS.

"I advocate the application of the word PHENOMENON exclusively

to refer to the observations obtained under specific circumstances,

including an account of the WHOLE EXPERIMENTAL ARRANGEMENT."

[ Bohr, Atomic Physics and Human Knowledge, Sci. Eds, N.Y. 1961]

The QUANTUM POTENTIAL has an INFORMATION CONTENT.[To inform means literally to FORM FROM WITHIN]

Active Information.

Input

channel

Output

channel

Channel I

Channel II

With particle in channel I, the Quantum Potential, QI, is ACTIVE in that channel,

while the QP in channel II, QII, is PASSIVE.

If interference occurs in the output channel, we need information from BOTH CHANNELS.

INFORMATION IN THE 'EMPTY' CHANNEL BECOMES ACTIVE IN THE OUTPUT CHANNEL.

[It cannot be thrown away.]

Does information ever become inactive?

Inactive information

Input

channel

Output

channel

Irreversible

process

Once an IRREVERSIBLE process has taken place the information becomes INACTIVE

[Shannon information enters here]

There is NO COLLAPSE, but it behaves as if a collapse has taken place.

How do we include the irreversible process?

Close Connection with Deformed Poisson Algebra.

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A∗B = A X, P( ) expih2

s ∂

∂x

r ∂

∂p−

r ∂

∂x

s ∂

∂p

⎡

⎣ ⎢

⎤

⎦ ⎥B(X,P)Moyal product

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A, B{ }MB= A∗B − B∗A = 2A(X, P)sin

h2

s ∂

∂X

r ∂

∂P−

r ∂

∂X

s ∂

∂P

⎡

⎣ ⎢ ⎤

⎦ ⎥B(X, P)

Moyal bracket

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A, B{ }MB≈ h

∂A∂X

∂B∂P

−∂A∂P

∂B∂X

⎡ ⎣ ⎢

⎤ ⎦ ⎥+....

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A,B{ }BB= A(X,P)cos

h2

s ∂

∂X

r ∂

∂P−

r ∂

∂X

s ∂

∂P

⎡

⎣ ⎢ ⎤

⎦ ⎥B(X,P)

Baker bracket

[Baker, Phys. Rev., 109,2198-2206 (1958)]

this becomes the Poisson bracket,To

this becomes the ordinary product,To

[Moyal, Proc. Camb. Phil. Soc. 45, 99-123, 1949].

Time evolution of Moyal Distribution

Again we find two time evolution equations

this becomes the Liouville equation,To

The second equation is

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H , f{ }BB= −

∂S∂t

f + O(h2 )

Writing and expanding in powers of

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∂S∂t

+ H = 0

which becomes

Hamilton-Jacobi eqn.

Liouville eqn.

Cells in Phase space.

P =p

2+p

1

2π =p2 −p1

X=x

2+x

1

2η=x2 −x1

We use cells in phase space New topology.

[Hiley, Reconsideration of Foundations 2, 267-86, Växjö, Sweden, 2003]

Quantum blobs of de Gosson based on symplectic capacity

[de Gosson, Phys. Lett. A317 (2003), 365-9]

x

p

(x1, p1)

(x2, p2)

In general we have

Change coordinates

Now we can use the Wigner transformation

So that

where

Symplectic Camel

Can we live with Ambiguity?

Ambiguous moment.

e.g. Wigner-Moyal

Can we ensure this mathematics containing the symplectic symmetry?

Can we capture mathematically the ambiguity that Bohr emphasizes?

Can we reproduce present physics by averaging over the ?

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{ } = ∂∂X

∂∂Δp

−∂

∂Δp∂

∂X+

∂∂Δx

∂∂P

−∂

∂P∂

∂Δx

€

X,Δp{ } = Δx,P{ } = 1

€

X,P{ } = Δx,Δp{ } = X,Δx{ } = P,Δp{ } = 0

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H (t2) − H (t1),T{ } = H (t2) + H (t1),Δt{ } = 1

This is all classical mechanics.

Generalised Poisson Brackets.

How do we structure the variables

Introduce new Poisson brackets

Define

Then

Suggestion

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x1 , p1[ ] = p2 , x2[ ] = i

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x1 , x2[ ] = x1 , p2[ ] = x2 , p1[ ] = p1, p2[ ] = 0

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X,Δp[ ] = Δx,P[ ] = i

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X,P[ ] = Δx,Δp[ ] = X,Δx[ ] = P,Δp[ ] = 0

€

X, P, Δp = i∂

∂P, Δp = −i

∂

∂X

What about Quantum Mechanics?

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p1 = −i∂

∂x1

and p2 = i∂

∂x2

Use the operators,

From the commutators

Change variables to find

We have formed superoperators

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x1 , x2 , p1 , p2{ } ⇒ ˆ x 1 , ˆ x 2 , ˆ p 1 , ˆ p 2{ }

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AρB = A⊗ ˜ B ( )ρ V

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ρ =ρ11 ρ 12

ρ 21 ρ 22

⎛

⎝ ⎜

⎞

⎠ ⎟ → ρ V =

ρ 11

ρ 12

ρ 21

ρ 22

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

[Prigogine, Being and Becoming, 1980]

A A

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A⊗A

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A⊗A

Formal Doubling.

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D ≠ A⊗ ˜ B ( )In the super-algebra we now have the possibility

Non-unitary transformations possible Decoherence.

What we have done is

We have turned a left-right module into a bi-module.

Essentially a GNS construction.

This can be written as

general transformationWe can formalise all this by considering the

Thermodynamics?

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2 ˆ X = ˆ x 1 ⊗1+1⊗ ˆ x 2 , ˆ η = ˆ x 1 ⊗1−1⊗ ˆ x 2 ,

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2 ˆ P = ˆ p 1 ⊗1+1⊗ ˆ p 2 , ˆ π = ˆ p 1 ⊗1−1⊗ ˆ p 2 .

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ˆ X , ˆ π [ ] = ˆ η , ˆ P [ ] = i

€

and ˆ X , ˆ P [ ] = ˆ η , ˆ π [ ] = ˆ X , ˆ η [ ] = ˆ P , ˆ π [ ] = 0

€

i∂ ˆ ρ V∂t

+ ˆ H ⊗1−1⊗ ˆ H ( ) ˆ ρ V = 0

€

2 ˆ R V2 ∂ ˆ S V

∂t+ ˆ H ⊗1+1⊗ ˆ H ( ) ˆ ρ V = 0

Then

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Write ˆ L = ˆ H ⊗1−1⊗ ˆ H and ˆ ρ → ˆ ρ V

Algebraic Doubling.

Form a bi-algebra.

The quantum Hamilton-Jacobi equation becomes

Then the Liouville equation becomes

Only single time

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ˆ T , ˆ ε [ ] = ˆ τ , ˆ E [ ] = i

€

and ˆ T , ˆ E [ ] = ˆ τ , ˆ ε [ ] = ˆ T , ˆ τ [ ] = ˆ E , ˆ ε [ ] = 0

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ˆ τ , ˆ E [ ] = i

Many time operators?

Two Time Operators.

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T =t1 + t2

2; τ = t2 − t1; E =

E1 + E2

2; ε = E2 − EWe have

Let these exist in the algebra so that

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ˆ T Age operator,

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ˆ τ The duron operator,

Thus we have possibility of TWO time operators.

Thus we have a time operator proportional to time parameter

Prigogine

[Being and Becoming]

Formal Notation.

Only non-vanishing commutators are

Heisenberg equation of motion gives

As well as super-operators we also have time super-operators

Thermal Time Hypothesis.

Generally covariant theory no preferred time.Thermal state picks out a particular time.

Thermal time defines physical time.

Claim:

The von Neumann algebra is intrinsically a dynamical object.

[Connes and Rovelli, Class. Quant. Grav., 11, (1994) 2899-2917]

Gibbs state

The Tomita-Takesaki theorem.

Modular groupwith

For the state

Introduce Swith

Then

Why the Doubling?

We need no longer be confined to one Hilbert space.

Consider temperature expectation values.

Can only construct by doubling the Hilbert space.

Two evolutions

Schrödinger

Bogoliubov

[Umewaza, Collective Phenomena 2 (1975) 55-80]

[Umezawa, Advanced Quantum Field Theory 1993]

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A =12

a + ˜ a ( ) = 2 ˆ X + i ˆ P ( )

€

and A† =12

a† + ˜ a †( ) = 2 ˆ X − i ˆ P ( )

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B =12

a − ˜ a ( ) = −12

ˆ η + i ˆ π ( )

€

and B† =12

a† − ˜ a †( ) = −12

ˆ η − i ˆ π ( )

€

ˆ X =1

2 2A + A†( ) and ˆ P =

i2 2

A − A†( )

A and B are a way of defining ambiguous moments

The Double Boson Algebra.

So that

We need to express in terms of

Then we introduce {A, B, A†, B†} so that

and

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a = ˆ x 1 + iˆ p 1 ˜ a = ˆ x 2 + iˆ p 2a† = ˆ x 1 − iˆ p 1 ˜ a † = ˆ x 2 − iˆ p 2

First we write

Thermal QFT algebra is a Hopf algebra of constructed from a and ã

Deformed Boson Algebra.

Introduce a deformed co-product

when

Then

Introduce

We can write

if

€

a(θ ) =12

A(θ )+ B(θ )( ) = a coshθ − ˜ a † sinhθ

€

˜ a (θ ) =12

A(θ )− B(θ )( ) = ˜ a coshθ − a† sinhθ

and

Bogoliubovtransformations

[Celeghini et al Phys Letts A244, (1998) 455-416]

€

−iδ

δθa(θ ) = G,a(θ )[ ] and − i

δδθ

˜ a (θ ) = G, ˜ a (θ )[ ]

€

where G = −i a† ˜ a † − a ˜ a ( )

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exp iθ ˆ p θ[ ]a(θ ) = exp iθG[ ]a(θ )exp −iθG[ ] = a θ +θ ( )

Bogoliubov transformations and Time.

Let parameterise the time. Introduce conjugate momentum

€

Then for a fixed value of

This is equivalent to the transformation

describes movement between inequivalent Hilbert spaces.

€

0(θ )

€

0(θ +θ )

Hilbert space q

Schrödinger

time

Picture for Time.

This is like a “thermal” time “irreversible” (‘real’) time

Schrödinger time is “implication” time.