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```					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?
But there is no time operator!
Compare and contrast classical mechanical time with
quantum mechanical time.

We are led to consider non-locality in time.

Ambiguity in time.
Moment or
I will develop the appropriate mathematics                   duron

Groupoids  bi-locality  bi-algebra  Hopf algebra.

Two time operators        Schrödinger time
Transition time.
Mechanical Time.
Explore relation between
Classical 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.
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

We cannot make  smaller than time resolution

Thus we need a relation at two distinct times                 and

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]
Feynman‟s Time.
contains information coming from the past;                          y
x
contains information „coming‟ from the future

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

Feynman showed                                        Schrödinger equation

I want to look at

Time-energy uncertainty.

Et 

The past and future mingle in the ill-defined present.

Ambiguous moments
Not Instant but Moment.
Bohm:-
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, 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]

Replace „instant‟ by „moment‟                  not          but

e    M1   M2    
e
Development of process is enfoldment-unfoldment

                 
How do we turn a set of moments into an algebra?
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
Our interpretation                                                  P2
is P1 BECOMING P2
P1
is our BEING.                     P1

Since                                   , being is IDEMPOTENT.
Note                                                          P1
1            is a left unity.
2.           is a right unity.
3. Inverse
The Algebra of Process.
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.

A B * C D  BC A D                       [ Raptis & Zaptrin, gr-qc/9904079 ]

Bob Coecke‟s approach through categories.
If f : A  B and g : B C, f g : A C          [Abramsky& Coecke q-ph/0402130]

Feynman Paths.

If

with

Interference „bare-bones‟ Feynman

[Kauffman, Contp. Math 305, 101-137, 2002]
Classical Groupoids.
Is there anything like this in classical mechanics?
Under free symplectomorphisms,
the 2-form               is preserved.

This means                                         Generating function

Free symp. requires

In general                                                 Action.

Hamilton-Jacobi

Time-dependent Hamiltonian flows from a groupoid
Time Evolution Equation (1).

Consider

Change coordinates

Then

In the limit as t  0, T t we find

Liouville equation

Time Evolution Equation (2).

Write

Again in the limit as t  0, T t we find

If we write

Quantum
Hamilton-Jacobi
Quantum potential
S (S)2
      V Q  0      Bohm trajectories.
t    2m         2
2R
p  S, Q  
2m R
Slits

Incident

particles

Screen

Barrier                                    x

x                                 t
Barrier
t          [Bohm & Hiley, The Undivided Universe. 1993]
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.

2R
[Recall    Q     ]
R
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.

Channel I

Input                          Output
channel                         channel

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                                Output
channel                               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.
i    
Moyal product          A  B  AX, P exp             B(X, P)

x
2  p x p
Moyal bracket
     
A, BMB  A  B  B  A  2A(X, P)sin               B(X, P)

                                      X
2  P X P 
To        this becomes the Poisson bracket,
A B A B 
A, BMB              ....
                                        X P P X 
Baker bracket
     
A, BBB  A(X, P)cos 
                                    B(X, P)

X P X P 
2 

To         this becomes the ordinary product,


[Moyal, Proc. Camb. Phil. Soc. 45, 99-123, 1949].
[Baker, Phys. Rev., 109,2198-2206 (1958)]
Time evolution of Moyal Distribution
Again we find two time evolution equations

To       this becomes the Liouville equation,
Liouville eqn.

The second equation is

Writing                     and expanding in powers of
S
H, f BB        f  O( 2 )
t
which becomes
S
H 0                    Hamilton-Jacobi eqn.
t

Cells in Phase space.
In general we have

Change coordinates
x x
X   2   1
x x
2            2   1

So that
Now we can use the Wigner transformation

where             p p                                                 p
P    2    1
p p                                                     (x 2, p2 )
2                 2    1

(x 1, p1 )

x
We use cells in phase space  New topology.
Quantum blobs of de Gosson based on symplectic capacity
Symplectic Camel
[de Gosson, Phys. Lett. A317 (2003), 365-9]
[Hiley, Reconsideration of Foundations 2, 267-86, Växjö, Sweden, 2003]
Can we live with Ambiguity?

Ambiguous moment.

Can we capture mathematically the ambiguity that Bohr emphasizes?

Can we ensure this mathematics containing the symplectic symmetry?

Can we reproduce present physics by averaging over the   ?

e.g. Wigner-Moyal
Generalised Poisson Brackets.

How do we structure the variables

Introduce new Poisson brackets

                
Define                                
X p p X x P P x

Then                   X,p  x,P  1

X,P  x,p  X,x  P,p  0

Suggestion

H (t 2 )  H (t1 ),T   H (t 2 )  H (t1 ),t  1

This is all classical mechanics.

x1 , x 2 , p1 , p2   x1 , x 2 , p1 , p2 
ˆ ˆ ˆ ˆ
                                 
Use the operators,             p1  i               and          p2  i
x1                               x 2

From the commutators
x1 , p1    p2 , x 2   i

x1 , x 2    x1 , p2    x 2 , p1    p1 , p2   0

Change variables to find

                X,p  x,P  i
X,P  x,p  X,x  P,p  0

We have formed superoperators
                                              
X,      P, p  i          , p  i
P           X
Formal Doubling.
We can formalise all this by considering the general transformation
This can be written as                                     A A        A A
AB  A  BV
˜

We have turned a left-right module into a bi-module.
         
done
What we have is                     11 
                 A         A
11 12 
             V    12 

 21  22            21 
 
 22 
Essentially a GNS construction.

In  super-algebra we now have the possibility
the                                             D  A  B
˜

Non-unitary transformations possible  Decoherence.         Thermodynamics?

[Prigogine, Being and Becoming, 1980]
Algebraic Doubling.

Form a bi-algebra.
ˆ ˆ
2 X  x1  11  x 2 ,
ˆ            x1  11  x 2 ,
ˆ ˆ            ˆ

ˆ ˆ
2P  p1  11  p2 ,
ˆ              p1  11  p2 .
ˆ ˆ             ˆ

Then

   X ,  , P i
ˆ ˆ       ˆ ˆ              and      X , P ,   X , P,  0
ˆ ˆ      ˆ ˆ        ˆ ˆ      ˆ ˆ

ˆ ˆ         ˆ
Write L  H 11 H and  V
ˆ   ˆ
                     
Then the Liouville equation becomes

ˆV
               i         H 11 H   0
ˆ       ˆ ˆV
t
The quantum Hamilton-Jacobi equation becomes                     Only single time
ˆ
     ˆ V SV  H 11 H   0
2R 2        ˆ          ˆ ˆV
t
Two Time Operators.
t1  t 2                    E  E2
We have             T              ;   t 2  t1; E  1     ;   E2  E
2                           2

Let these exist in the algebra so that

ˆ   ˆ
T ,   ˆ , E  i
 ˆ                      and         T , E ,  T ,  E, 0
ˆ ˆ      ˆ ˆ       ˆ ˆ       ˆ ˆ

Thus we have possibility of TWO time operators.
              
Age operator, Tˆ


The duron operator,             
ˆ

, E i
ˆ ˆ

Many time operators?
Formal Notation.
As well as super-operators             we also have time super-operators

Only non-vanishing commutators are

Heisenberg equation of motion gives

Prigogine
[Being and Becoming]

Thus we have a time operator proportional to time parameter
Thermal Time Hypothesis.
[Connes and Rovelli, Class. Quant. Grav., 11, (1994) 2899-2917]

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

Gibbs state
Thermal time defines physical time.
Introduce S
with
The Tomita-Takesaki theorem.
with                               Modular group

For the state

Then

Claim:
The von Neumann algebra is intrinsically a dynamical object.
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]
The Double Boson Algebra.
We need to express                        in terms of

First we write
a  x1  ip1
ˆ     ˆ            a  x 2  ip2
˜ ˆ        ˆ
a †  x1  ip1
ˆ     ˆ          a †  x 2  ip2
˜     ˆ      ˆ

Then we introduce {A, B, A†, B†} so that
1 †
A
1
 a  2 X  iP
a ˜         ˆ ˆ       and       A† 
2
a  a †   2 X  iP
˜          ˆ ˆ
2
1             1                                   1 †               1
B
2
a  a    i and
˜
2
ˆ ˆ               B† 
2
a  a †    2  i
˜            ˆ ˆ

   So that                 
1                                   i

ˆ
X
2 2
A † 
A               and       ˆ
P
2 2
A  A† 
and

       A and B are a way of defining ambiguous moments
Deformed Boson Algebra.
Thermal QFT algebra is a Hopf algebra of constructed from a and ã

Introduce a deformed co-product
when

Then

Introduce

We can write
and

if                       1
a( )     A( )  B( )  a cosh  a † sinh
˜
2                                             Bogoliubov
1                                            transformations
a( ) 
˜          A( )  B( )  a cosh  a † sinh
˜
2
[Celeghini et al Phys Letts A244, (1998) 455-416]

Bogoliubov transformations and Time.
Let  parameterise the time. Introduce conjugate momentum

describes movement between inequivalent Hilbert spaces.

                                      
i      a( )  G,a( )     and      i      a( )  G, a( )
˜           ˜
                                     
where       G  i a† a†  aa
˜     ˜

        Then for a fixed value of 

 ip  a( )  exp iG a( )exp iG  a  
exp ˆ

This is equivalent to the transformation

Picture for Time.

Hilbert space q

Schrödinger
time

0()                              0(  )

This is like a “thermal” time “irreversible” („real‟) time

                                
Schrödinger time is “implication” time.

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