T violation, direction of time and general relativity

1. T violation, direction of time and general relativity Joan VaccaroGriffith University

2. cosmological arrow electromagnetic arrow thermodynamic arrow psychological arrow Arrows of time • Emerge from phenomenological time asymmetric dynamics Cannot be derived from first principles Must be inserted into physical theories by hand past future expanding universe big bang spontaneous emission excited atom increasing entropy low entropy memory of the past no memory of the future due to asymmetrical boundary conditions

3. matter-antimatter balance of matter & antimatter excess of matter 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 Wigner, Group theory (1959), Messiah, Quantum Mechanics (1961) Ch XV Time reversal operator anti-unitary operator - action is complex conjugation unitary operator - depends on spin

4. Schrodinger’s equation Backwards evolution is simply backtracking the forwards evolution forwards backwards Fundamental question T inversion symmetry violation implies How should one incorporate the two Hamiltonians, and , in one equation of motion?

5. Possible paths through time arXiv:0911.4528 • 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” Forwards and Backwards evolution Evolution of state over time interval t in the forward direction where and = Hamiltonian for forward time evolution.

6. Evolution of state over time interval t 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 tas 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 otherso attribute an equal weighting to each [Feynman RMP 20, 367 (1948)]

7. 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-symmetricevolution of the system. Time-symmetric evolution over an additional time interval of t is given by

8. Repeating this for N such time intervals yields Let • 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

9. Consider the limit t 0 • fix total time and set . Take limit as . • we find effective Hamiltonian=0 for clock device no time in conventional sense • Set t to be the smallest physical time interval, Planck time

10. Interference Multiple paths Example: 4 terms interfere

11. Simplifying the expression for Use the Zassenhaus (Baker-Campbell-Hausdorff ) formula for arbitrary operators and and parameter d to get We eventually find that nested sums Using eigenvalue equation for commutator we find

12. eigenvalue trace 1 projection op. degeneracy where

13. Estimating eigenvalues l Eigenvalues for j th kaon Eigenvalues for M kaons Let fraction

14. Comparison of with destructive interference constructive interference Assume

15. Destructive interference Consider: forward steps backward steps is much narrower than if total time

16. if total time Bi-evolution equation of motion

17. Unidirectionality of time Smoking gun: evidence left in the state Let Hamiltonians and leave distinguishable evidence in state if

18. Repeating... ...leaves corroborating evidence in the state Interpretation • and representevolution in opposite directions of time • in each case corroborating evidence of Hamiltonian is left in the state Our experience • Experiments give evidence of exactly one of the Hamiltonians or

19. 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. What about zero eigenvalues? Recall Let we see Hamiltonian of one of these branches Then mixed Hamiltonians – not observed

21. Schrödinger’s equation for bi-evolution Consider time increment Rate of change Take limit ignore i.e.

22. General Relativity Consider Robertson-Walker-Friedman universe Metric: scale parameter closedflatopen Friedman equations: Square root of last equation: -ve root +ve root

23. 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. 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 Consider massless balloon containing a gas normal component of tension in membrane F motion of molecule in both directions of time evolution F pressure of gas “backwards” evolution is in direction of decreasing t • depends on length of path balloon expands in both directions of time

25. Path length – GR time coordinate is length of path Foliation of spacetime Unchanging direction of time(conventional GR) Following a path through time space-like slices ...same topology

26. Two time coordinates is length of path While is a “good” coordinate for GR, the net time traversed is what clocks measure and what quantum fields depend on. net time net time path length net time = cosmic time = time since big bang

27. CP and T violation from here onwards values of net time (cosmic time) path length inflation (present day value) CP and T violation from here onwards net time (cosmic time)

28. Summary Unidirectionality of time • Feynman path integral method • T violation causes destructive interference of zigzagging paths • empirical evidence determines which branch Implications for general relativity • 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? inflation