/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Unidirectionality of time induced by T violation Joan Vaccaro Centre for Quantum Dynamics Griffith University Australia 1

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe 2 Direction of time  arrows of time (Eddington) thermodynamic (entropy doesn’t decrease) cosmological (expanding universe) electromagnetic (spontaneous emission not absorption) psychological (remember past not future) matter-antimatter, … (CP violation favours matter)  description of dynamical systems is time-symmetric Introduction [The Nature Of The Physical World, 1928] unidirectionality of time problem BUT why do we experience only one direction of time? This is the ? time

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Time reversal operator [Wigner, Group theory (1959] unitary operator anti-unitary operator - action is complex conjugation Typical Schrodinger equation Backwards evolution is simply backtracking the forwards evolution 3 forwards “backwards” mirror symmetry

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe But kaons don’t behave this way Violation of time reversal invariance - a small (0.2%) violation of CP & T invariance in neutral kaon decay - discovered in 1964 by Cronin & Fitch (Nobel Prize 1980) - partially accounts for observed dominance of matter over antimatter gives time asymmetric dynamics What effect does this have on the direction of time? us _ boson, neutral, ½ m p lifetime 10  8 s 4 Conventional answer: nothing!!! forwards “backwards” broken mirror a fundamental time asymmetry

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe where and = Hamiltonian for forward time evolution. Model of the universe: ▀ it is closed in the sense that it does not interact with any other physical system ▀ it has no external clocks and so analysis needs to be unbiased with respect to the direction of time ▀ both versions of the Hamiltonian should appear in the dynamical equation of motion Forwards and Backwards evolution Evolution of state over time interval  in the forward direction Preprint arXiv: Paths through time

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe 6 Evolution of state over time interval  in the backward direction where and = Hamiltonian for backward time evolution. ▀ and are probability amplitudes for the system to evolve from to via two paths in time ▀ we have no basis for favouring one path over the other so assign an equal statistical weighting to each using Feynman’s sum over histories [Feynman Rev. Mod. Phys. 20, 367 (1948)] Constructing paths:

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe which we call time-symmetric evolution. 7 c.f. double slit: The total amplitude for is proportional to This is true for all states, so Time-symmetric evolution over an additional time interval of  is given by t

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Repeating this N times yields ▀ is a sum containing different terms ▀ is a sum over a set of paths each comprising forwards steps and backwards steps Let 8 t

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe The limit   0 ? effective Hamiltonian =0 for conventional clock device no time in conventional sense ▀ Set  to be a small physical time interval, Planck time 9 t ▀ fix total time and set. Take limit as. ▀ we find

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Interference Multiple paths 4 terms interfere Example: 10 t

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe 11 The Zassenhaus (Baker-Campbell-Hausdorff ) formula gives Simplifying Eigenvalue equation for commutator degeneracy trace 1 projection op. eigenvalue

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe where degeneracy eigenvalue trace 1 projection op. 12 Eigenvalues for j th kaon Eigenvalues for M kaons Estimating eigenvalues phenomenological model [Lee, PR 138, B1490 (1965)]. Let fraction total # of particles

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Comparison of with destructive interference constructive interference 13 width

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe total time Bi-evolution equation of motion Only two paths survive if 14 fraction total # of particles # kaons t Destructive interference

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe t only observe evidence of in this branch we observe only one of these terms phenomenological unidirectionality of time only observe evidence of in this branch 15 Direction of time

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Shortest path through time 16 shortest path same time

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe 17 Early universe T violation would be relatively rare, so no interference: = zero eigenvalue of Power method ~ Hamiltonian constraint of the Wheeler-DeWitt eqn. Hence t largest eigenvalue = history

/18 IntroductionPaths through timeInterferenceDirection of timeEarly universe Summary ▀ must use Feynman’s sum over histories to account for both directions ▀ destructive interference leaves only 2 paths ▀ physical evidence shows which path we experience ▀ quantum algorithm for the shortest path to the “future” the unidirectionality of time T invariance T violation t Universe has no reference for direction of time