Time from an Algebraic Theory of Moments.

1. Time from an Algebraic Theory of Moments. B. J. Hiley. www.bbk.ac.uk/tpru

2. Compare and contrast classical mechanical time with quantum mechanical time. Time through notion of Dynamical Moments. Can we get any insights into time through quantum theory? But there is no time operator! We are led to consider non-locality in time. Ambiguity in time. Moment or duron I will develop the appropriate mathematics Groupoids  bi-locality  bi-algebra  Hopf algebra. Two time operators Schrödinger time Transition time.

3. :- In CM we have the notion of “flow” ; In QM we have a “flow” Mechanical Time. Explore relation between Classical mechanical time. Quantum mechanical time. Determined by Hamilton’s eqns of motion 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.

4. is operationally meaningless Also We cannot make  smaller than time resolution and Thus we need a relation at two distinct times 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. 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]

5. contains information coming from the past; contains information ‘coming’ from the future Feynman showed  Schrödinger equation y I want to look at x Feynman’s Time. [Feynman, Rev. Mod. Phys.,20, (1948), 367-387]. Time-energy uncertainty. The past and future mingle in the ill-defined present. Ambiguous moments

6. 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] Whitehead:- What we perceive as present is a vivid fringe of memory tinged with anticipation.[Whitehead, The Concept of Nature, p. 72-3] not but Not Instant but Moment. Replace ‘instant’ by ‘moment’ Development of process is enfoldment-unfoldment How do we turn a set of moments into an algebra?

7. Note 1 is a left unity. 2. is a right unity. is our BEING. P1 3. Inverse Since , being is IDEMPOTENT. P1 P2 P1 Succession of Moments. Groupoid Regard this as a set X of arrows, sources and targets, s and t P2 is the target t P1is the source s Our interpretation is P1BECOMINGP2

8. 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. [ Raptis & Zaptrin, gr-qc/9904079 ] Bob Coecke’s approach through categories. [Abramsky& Coecke q-ph/0402130]

9. Feynman Paths. If with Interference ‘bare-bones’ Feynman [Kauffman, Contp. Math 305, 101-137, 2002]

10. Under free symplectomorphisms, the 2-form is preserved. Generating function This means Classical Groupoids. Is there anything like this in classical mechanics? Free symp. requires In general Action. Hamilton-Jacobi Time-dependent Hamiltonian flows from a groupoid

11. Consider Change coordinates Then Liouville equation ? What about Time Evolution Equation (1). In the limit as t 0, T t we find

12. Write If we write Time Evolution Equation (2). Again in the limit as t 0, T t we find Quantum Hamilton-Jacobi Quantum potential

13. Slits Incident particles Screen x Barrier t Bohm trajectories. x t Barrier [Bohm & Hiley, The Undivided Universe. 1993]

14. [Recall ] 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. 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.

15. 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]

16. Active Information. 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?

17. Inactive information 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?

18. Moyal product To To this becomes the Poisson bracket, this becomes the ordinary product, Close Connection with Deformed Poisson Algebra. Moyal bracket Baker bracket [Moyal, Proc. Camb. Phil. Soc. 45, 99-123, 1949]. [Baker, Phys. Rev., 109,2198-2206 (1958)]

19. Time evolution of Moyal Distribution To this becomes the Liouville equation, Writing and expanding in powers of Again we find two time evolution equations Liouville eqn. The second equation is which becomes Hamilton-Jacobi eqn.

20. Cells in Phase space. In general we have Change coordinates So that Now we can use the Wigner transformation where 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]

21. ? Can we reproduce present physics by averaging over the 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? e.g. Wigner-Moyal

22. Generalised Poisson Brackets. How do we structure the variables Introduce new Poisson brackets Define Then Suggestion This is all classical mechanics.

23. Use the operators, What about Quantum Mechanics? From the commutators Change variables to find We have formed superoperators

24. We can formalise all this by considering the general transformation A A In the super-algebra we now have the possibility Formal Doubling. This can be written as We have turned a left-right module into a bi-module. What we have done is Essentially a GNS construction. Non-unitary transformations possible  Decoherence. Thermodynamics? [Prigogine, Being and Becoming, 1980]

25. Algebraic Doubling. Form a bi-algebra. Then Then the Liouville equation becomes Only single time The quantum Hamilton-Jacobi equation becomes

26. We have Age operator, The duron operator, Two Time Operators. Let these exist in the algebra so that Thus we have possibility of TWO time operators. Many time operators?

27. we also have time super-operators As well as super-operators Prigogine [Being and Becoming] Formal Notation. Only non-vanishing commutators are Heisenberg equation of motion gives Thus we have a time operator proportional to time parameter

28. Gibbs state with For the state 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. Thermal time defines physical time. Introduce S with The Tomita-Takesaki theorem. Modular group Then Claim: The von Neumann algebra is intrinsically a dynamical object.

29. Why the Doubling? We need no longer be confined to one Hilbert space. Consider temperature expectation values. by doubling the Hilbert space. Can only construct Two evolutions Schrödinger Bogoliubov [Umewaza, Collective Phenomena 2 (1975) 55-80] [Umezawa, Advanced Quantum Field Theory 1993]

30. in terms of We need to express and The Double Boson Algebra. First we write Then we introduce {A, B, A†, B†} so that So that A and B are a way of defining ambiguous moments

31. when and Deformed Boson Algebra. Thermal QFT algebra is a Hopf algebra of constructed from a and ã Introduce a deformed co-product Then Introduce We can write if Bogoliubov transformations [Celeghini et al Phys Letts A244, (1998) 455-416]

32. describes movement between inequivalent Hilbert spaces. Then for a fixed value of Bogoliubov transformations and Time. Let parameterise the time. Introduce conjugate momentum This is equivalent to the transformation

33. Picture for Time. Hilbert space q Schrödinger time This is like a “thermal” time “irreversible” (‘real’) time Schrödinger time is “implication” time.