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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.
I will develop the appropriate mathematics
Groupoids  bi-locality  bi-algebra  Hopf algebra.
Two time operators
Schrödinger time
Transition time.
Moment or
duron
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
We cannot make
is operationally meaningless
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;
contains information ‘coming’ from the future
[Feynman, Rev. Mod. Phys.,20, (1948), 367-387].
 Schrödinger equation
Feynman showed
I want to look at
Time-energy uncertainty.
Et 
The past and future mingle in the ill-defined present.

Ambiguous moments
y
x
Bohm:-
Not Instant but Moment.
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]
Whitehead: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
Development of process is enfoldment-unfoldment

How do we turn a set of moments into an algebra?

M1

M2

e
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
is our BEING.
Since
P1
, being is IDEMPOTENT.
Note
1
is a left unity.
2.
is a right unity.
3. Inverse
P1
P1
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
This means
is preserved.
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
What about
?
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
t
2m
2
2R
p  S, Q  
2m R
Bohm trajectories.
Slits
Incident
particles

Screen
Barrier
x
t
x
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.
[Recall
2 R
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
channel
Output
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
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.
i     
A  B  AX, P exp 

B(X, P)
2 x p x p
Moyal product
Moyal bracket
     

A, BMB  A  B  B  A  2A(X, P)sin 
B(X, P)
2 X P X P 

this becomes the Poisson bracket,
To
A B A B 

 ....
A, BMB  

X P P X 

Baker bracket
     

A, BBB  A(X, P)cos 
B(X, P)
2 X P
X P 
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
H, f BB  
which becomes

S
f  O( 2 )
t
S
H 0
t
Hamilton-Jacobi eqn.
Cells in Phase space.
In general we have
Change coordinates
X
x x
2
2
1
x x
2
1
So that
Now we can use the Wigner transformation
where
p p
P
2
2
1
p p
2
p
(x 2, p2 )
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
X,p  x,P  1
Then

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.

What about Quantum Mechanics?
x1 , x 2 , p1 , p2   xˆ1 , xˆ 2 , pˆ 1 , pˆ 2 
p1  i
Use the operators,

From the commutators

x1
and
p2  i
x1 , p1    p2 , x 2   i

x1 , x 2    x1 , p2   x 2 , p1    p1 , p2   0
 to find
Change variables
X,p  x,P  i

X,P  x,p  X,x  P,p  0
 superoperators
We have formed

X,
P, p  i


, p  i
P
X

x 2
Formal Doubling.
We can formalise all this by considering the general transformation
This can be written as
A A
AB  A  B˜ V
We have turned a left-right module into a bi-module.

What we havedone is
11 
 
11 12 
12 

  
  V 
 21 
 21  22 
 
 22 
A A

A
A
Essentially a GNS construction.
In 
the super-algebra we now have the possibility
D  A  B˜ 
Non-unitary transformations possible  Decoherence.

[Prigogine, Being and Becoming, 1980]
Thermodynamics?
Algebraic Doubling.
Form a bi-algebra.
2 Xˆ  xˆ1 11  xˆ 2 ,
2Pˆ  pˆ 1 11  pˆ 2 ,

Then

Xˆ , ˆ  ˆ, Pˆ  i
and
ˆ  xˆ1 11  xˆ 2 ,

ˆ  pˆ 1 11  pˆ 2 .

Xˆ , Pˆ  ˆ , ˆ   Xˆ ,ˆ  Pˆ , ˆ  0
Write Lˆ  Hˆ 11 Hˆ and 
ˆ  
ˆV

Then the Liouvilleequation becomes

i

ˆV
 Hˆ 11 Hˆ 
ˆV  0
t
The quantum Hamilton-Jacobi equation becomes
ˆ
2 SV
ˆ
2 RV
 Hˆ 11 Hˆ 
ˆV  0

t
Only single time
Two Time Operators.
T
We have
t1  t 2
E  E2
;   t 2  t1; E  1
;   E2  E
2
2
Let these exist in the algebra so that

ˆ
T
,
 ˆ ˆ, Eˆ  i
and
Tˆ , Eˆ  ˆ,ˆ  Tˆ , ˆ Eˆ ,ˆ 0
Thus we have possibility of TWO time operators.


Age operator, Tˆ

The duron operator,

ˆ
ˆ, Eˆ  i
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.
in terms of
We need to express
First we write
a  xˆ1  ipˆ 1
a˜  xˆ 2  ipˆ 2
a †  xˆ1  ipˆ 1
a˜ †  xˆ 2  ipˆ 2
Then we introduce {A, B, A†, B†} so that
1
and
A† 
A
a  a˜   2 Xˆ  iPˆ 

2
1
1
ˆ  and
B
ˆ  i
B† 
a  a˜   
2
2
So that
1 †
a  a˜ †   2 Xˆ  iPˆ 

2
1 †
1
ˆ
a  a˜ †   
ˆ  i


2
2

1
Xˆ 
A  A† 

2 2 
and
i
Pˆ 
A  A† 

2 2
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( )  B( )  a cosh  a˜ † sinh 
2
1
a˜( ) 
A( )  B( )  a˜ cosh  a † sinh 
2
a( ) 
[Celeghini et al Phys Letts A244, (1998) 455-416]

Bogoliubov
transformations
Bogoliubov transformations and Time.
Let  parameterise the time. Introduce conjugate momentum
describes movement between inequivalent Hilbert spaces.
i

a( )  G,a( )

where
and
i

a˜ ( )  G, a˜ ( )

G  i a † a˜ †  aa˜ 
Then for a fixed value of 

expipˆ   a( )  expiG a( )expiG  a  


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.