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RQIW 09 1/28 T violation, direction of time and general relativity Joan Vaccaro Griffith University.

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1 RQIW 09 1/28 T violation, direction of time and general relativity Joan Vaccaro Griffith University

2 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 2/28 Arrows of time ▀ Emerge from phenomenological time asymmetric dynamics Cannot be derived from first principles Must be inserted into physical theories by hand pastfuture cosmological arrow big bang expanding universe electromagnetic arrow thermodynamic arrow psychological arrow increasing entropy no memory of the future spontaneous emission memory of the past low entropy excited atom due to asymmetrical boundary conditions

3 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 3/28 matter-antimatter excess of matter balance of matter & antimatter The matter-antimatter arrow - due to 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. - dismissed as not directly affecting the nature of time or everyday life. due to time asymmetric dynamics Time reversal operator Wigner, Group theory (1959), Messiah, Quantum Mechanics (1961) Ch XV unitary operator - depends on spin anti-unitary operator - action is complex conjugation

4 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 4/28 Fundamental question T inversion symmetry violation implies How should one incorporate the two Hamiltonians, and, in one equation of motion? Schrodinger’s equation Backwards evolution is simply backtracking the forwards evolution forwards backwards

5 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 5/28 Physical system: ▀ composed of matter and fields in a manner consistent with the visible portion of the universe ▀ the system is closed in the sense that it does not interact with any other physical system ▀ no external clock and so analysis needs to be unbiased with respect to the direction of time ▀ convenient to differentiate the two directions of time as "forwards" and "backwards” Possible paths through time Forwards and Backwards evolution Evolution of state over time interval  in the forward direction where and = Hamiltonian for forward time evolution. arXiv:0911.4528

6 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 6/28 Evolution of state over time interval  in the backward direction where and = Hamiltonian for backward time evolution. Internal clocks: ▀ assume Hamiltonian of internal clocks is time reversal invariant during normal operation ▀ This gives an operational meaning of the parameter  as a time interval. Constructing paths: ▀ 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 attribute an equal weighting to each [Feynman RMP 20, 367 (1948)]

7 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 7/28 Principle: The total probability amplitude for the system to evolve from one given state to another is proportional to the sum of the probability amplitudes for all possible paths through time. The total amplitude for is proportional to This is true for all states, so which we call the time-symmetric evolution of the system. Time-symmetric evolution over an additional time interval of  is given by

8 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 8/28 Repeating this for N such time intervals yields ▀ is a sum containing different terms ▀ each term has factors of and factors of ▀ is a sum over a set of paths each comprising forwards steps and backwards steps Let

9 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 9/28 Consider the limit   0 ▀ fix total time and set. Take limit as. ▀ we find effective Hamiltonian =0 for clock device no time in conventional sense ▀ Set  to be the smallest physical time interval, Planck time

10 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 10/28 Interference Multiple paths 4 terms interfere Example:

11 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 11/28 Use the Zassenhaus (Baker-Campbell-Hausdorff ) formula for arbitrary operators and and parameter  to get We eventually find that nested sums Simplifying the expression for Using eigenvalue equation for commutator we find

12 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 12/28 where degeneracy eigenvalue trace 1 projection op.

13 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 13/28 Eigenvalues for j th kaon Eigenvalues for M kaons Let fraction Estimating eigenvalues

14 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 14/28 Comparison of with destructive interference Assume constructive interference

15 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 15/28 Destructive interference is much narrower than forward stepsbackward steps total time if Consider:

16 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 16/28 total time if Bi-evolution equation of motion

17 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 17/28 Smoking gun: evidence left in the state Unidirectionality of time Let Hamiltonians and leave distinguishable evidence in state if

18 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 18/28 ▀ and represent evolution in opposite directions of time ▀ in each case corroborating evidence of Hamiltonian is left in the state Repeating......leaves corroborating evidence in the state Interpretation Our experience ▀ Experiments give evidence of exactly one of the Hamiltonians or

19 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 19/28 Compare with universe obeying T invariance In this case Most likely paths for ▀ clocks don’t tick (show t=0 on average) ▀ no physical evidence of direction of time

20 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 20/28 Recall What about zero eigenvalues? Let Then mixed Hamiltonians – not observed we see Hamiltonian of one of these branches

21 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 21/28 Schrödinger’s equation for bi-evolution Consider time increment Rate of change Take limit i.e. ignore

22 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 22/28 Consider Robertson-Walker-Friedman universe General Relativity Metric: Friedman equations: scale parameter closed flat open Square root of last equation: +ve root-ve root

23 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 23/28 ▀ CP and T violation expected to occur in latter part of inflation ▀ Before this period, direction of time is uncertain consider a path with a changing direction of evolution “backwards” evolution is in direction of decreasing t ▀ depends on length of path

24 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 24/28 ▀ CP and T violation expected to occur in latter part of inflation ▀ Before this period, direction of time is uncertain consider a path with a changing direction of evolution “backwards” evolution is in direction of decreasing t ▀ depends on length of path Consider massless balloon containing a gas normal component of tension in membrane pressure of gas balloon expands in both directions of time F F motion of molecule in both directions of time evolution

25 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 25/28 is length of path Unchanging direction of time (conventional GR) Following a path through time...same topology Path length – GR time coordinate Foliation of spacetime space-like slices

26 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 26/28 While is a “good” coordinate for GR, the net time traversed is what clocks measure and what quantum fields depend on. is length of path net time Two time coordinates net time = cosmic time = time since big bang net time path length

27 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 27/28 values of net time (cosmic time) path length net time (cosmic time) CP and T violation from here onwards inflation (present day value)

28 RQIW 09 Arrows Paths through time Interference Unidirectionality General Relativity 28/28 Summary ▀ Feynman path integral method ▀ T violation causes destructive interference of zigzagging paths ▀ empirical evidence determines which branch ▀ early universe – no T violation - direction of time is uncertain ▀ Friedman equations: expansion in both directions – coordinate for GR is path length ▀ radiation and clocks “slow” – cosmic time Q. Is inflation due to uncertain direction of time? Unidirectionality of time Implications for general relativity inflation

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