Entropc Dynamcs: an Inference Approach to Tme and Quantum Theory Arel Catcha Department of hyscs Unversty at Albany - SUNY EmQM-13 Venna, 10/013
Queston: Do the laws of hyscs reflect Laws of Nature? Or... Are they rules for processng nformaton about Nature? Our objectve: To derve Quantum Theory as Entropc Dynamcs and dscuss some mplcatons for the theory of tme, quantum measurement, Hlbert spaces, etc.
Entropc Inference: pror posteror constrant q p S[ p, q] d p log p q r dv e S [ p, q] dv Mamze S[p,q] subject to the approprate constrants. MaEnt, Bayes' rule and Large Devatons are specal cases. 3
The subject matter the world the system partcles, felds... other relevant varables 4
The subject matter The goal s to predct the postons of partcles,. artcles have defnte but unknown postons. There est other varables y wth probablty py and entropy S dy p y log p y q y y varables are not hdden varables. 5
Dynamcs: Change happens. p[ y ] p[ y ] M statstcal manfold X Confguraton space non-localty 6
7 Entropc Dynamcs Mamze the jont entropy,, log, ], [ y Q y y d dy Q S J unform M y p Short steps: b a ab
8 The result: ] 1 ep [ 1 S where log y q y p y p dy S Dsplacement: w 1 S a a Epected drft : Fluctuatons: ab b a w w 1 1 O 1/ O!!!
Entropc Tme The foundaton of any noton of tme s dynamcs. Tme s ntroduced to keep track of the accumulaton of many small changes. d, d 1 Introduce the noton of an nstant, t d, t 9
Instants are ordered: the Arrow of Entropc Tme 3 Duraton: the nterval between nstants For large the dynamcs s all n the fluctuatons: w a w b 1 ab m t ab Defne duraton so that moton looks smple: t 10
Entropc dynamcs:, t d, t Fokker-lanck equaton: v t 11
Fokker-lanck equaton: v, t m v t S log 1/, t v b u drft velocty: osmotc velocty: b u S m log m 1/ dffuson current: u m 1
Entropc dynamcs: S M X roblem: ths s just standard dffuson, not QM! 13
The soluton: allow M to be dynamc. S, t M X wave partcle Non-dsspatve dffuson Quantum Mechancs 14
15 Manfold dynamcs? Impose energy conservaton 1 1 ], [ 3 V mu mv d S E 0 1/ 1/ m V m Quantum Hamlton-Jacob equaton:!!!
16 The result: two coupled equatons m t 1 Fokker-lanck/dffuson equaton 0 1/ 1/ m V m Quantum Hamlton-Jacob equaton
Combne and nto 1/ e to get Quantum Mechancs, m V Comple numbers, lnearty, and Hlbert spaces are convenent but not fundamental. 17
Measurement of poston n ED artcles have defnte but unknown postons. Born rule: d d d Measurement of poston s classcal : a mere ascertanng of pre-estng values, a mere amplfcaton from mcro to macro scales. 18
What about other observables? poston detectors potental source comple detector  U ˆ A p Uˆ A a 19
Other observables and the Born rule: If a U ˆ a A then c a U ˆ c Uˆ a A A c and p c a the general Born rule The comple detector measures all operators of the form Aˆ a a 0
Conclusons: Entropc Dynamcs provdes an alternatve to Acton rncples. No need for quantum probabltes or logc. The t n the Laws of hyscs s entropc tme. Quantum measurement s just an eercse n nference. Untary evoluton and wave functon collapse correspond to nfntesmal and dscrete updatng. oston s real. Other observables are not. They are created by the act of measurement. 1
Acknowledgements: Davd T. Johnson Shahd Nawaz Carlo Cafaro Danel Bartolomeo Adom Gffn Carlos Rodrguez Nestor Catcha Marcel Regnatto Kevn Knuth hlp Goyal Keth Earle
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Three ngredents: E. Jaynes E. Nelson J. Barbour entropy dffuson tme Entropc Dynamcs 4
Entropc nference: the pror q the posteror p To update from old belefs to new belefs when new nformaton becomes avalable. the constrants 5
6 6 Amplfcaton from mcro to macro scales The amplfer s prepared n an unstable state. The problem s to nfer from. r A a U a ˆ When r r r Use Bayes rule: the amplfer jumps to state. r r r r
An analogy from physcs: ntal state of moton. fnal state of moton. Force Force s whatever nduces a change of moton: F p t 7
Inference s dynamcs too! old belefs new belefs nformaton Informaton s what nduces the change n ratonal belefs. Informaton s what constrans ratonal belefs. 8
The quantum measurement problem: The sources of all quantum werdness: ndetermnsm superposton What s the source of ndetermnsm? Why don't we see macroscopc entanglement? How do we get defnte outcomes? Are observed values created by measurement? 9
30 Instants are ordered: the Arrow of Entropc Tme Bayes' theorem A tme-reversed evoluton:,, t d t There s no symmetry between pror and posteror. Entropc tme only goes forward. Confguraton space
More on entropc tme A clock follows a classcal trajectory. t 3 t c 3 c c 1 "physcal" space t 1 31
More on entropc tme For the composte system of partcle and clock: c 3 c c 3 c c 1 "physcal" space c 1 3
Entropc tme vs "physcal" tme? We observe correlatons at an nstant. We do not observe the "absolute" order of the nstants. c 3 c c 3 c c 1 "physcal" space c 1 Entropc tme s all we need. There s an arrow of entropc tme. 33