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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. Et 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 AX, P exp B(X, P) 2 x p x p Moyal product Moyal bracket A, BMB 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, BMB X P P X Baker bracket A, BBB 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 AB A B˜ V We have turned a left-right module into a bi-module. What we havedone 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 11 xˆ 2 , 2Pˆ pˆ 1 11 pˆ 2 , Then Xˆ , ˆ ˆ, Pˆ i and ˆ xˆ1 11 xˆ 2 , ˆ pˆ 1 11 pˆ 2 . Xˆ , Pˆ ˆ , ˆ Xˆ ,ˆ Pˆ , ˆ 0 Write Lˆ Hˆ 11 Hˆ and ˆ ˆV Then the Liouvilleequation becomes i ˆV Hˆ 11 Hˆ ˆV 0 t The quantum Hamilton-Jacobi equation becomes ˆ 2 SV ˆ 2 RV Hˆ 11 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 expipˆ a( ) expiG a( )expiG 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.