Universe as a quantum computer: algorithm for the shortest path through time

1. Universe as a quantum computer: algorithm for the shortest path through time Joan VaccaroCentre for Quantum Dynamics Griffith University

2. Lloyd: Universe is a giant quantum computer whose purpose is to calculate its own state. [Programming the Universe (2006)] Feynman: classical path of least action is determined by quantum interference i.e. quantum algorithm for minimizing the classical action [Rev. Mod. Phys. 20, 367 (1948)] sum the probability amplitudes of all possible paths from A to B x B ~ destructive interference leaves path of least action (= classical trajectory in limit) A t action Lagrangian Hamiltonian

3. [Wigner, Group theory (1959), Messiah, Quantum Mechanics (1961) Ch XV] Time reversal operator forwards anti-unitary operator - action is complex conjugation unitary operator backwards Usual Schrodinger equation Backwards evolution is simply backtracking the forwards evolution

4. But kaons don’t behave this way boson,neutral, ½ mp lifetime 10-8s _ u s 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 forwards gives time asymmetric dynamics backwards a fundamental time asymmetry What effect does this have on the computation?

5. Possible paths through time arXiv:0911.4528 • Physical system: • 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 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. • 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 Rev. Mod. Phys. 20, 367 (1948)] 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.

7. c.f. double slit: The total amplitude for is proportional to This is true for all states , so which we call evolution in an ambiguous direction of time. 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 conventional 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 eigenvalue Eigenvalue equation for commutator trace 1 projection op. degeneracy

12. eigenvalue trace 1 projection op. degeneracy where

13. Estimating eigenvalues l phenomenological model [Lee and Wolfenstein, Phys. Rev. 138, B1490 (1965)]. Eigenvalues for j th kaon Eigenvalues for M kaons Let fraction

14. Comparison of with destructive interference constructive interference

15. Destructive interference Consider: forward steps backward steps Condition for destructive interference: is much narrower than That is destructive interference: total time

16. Only two paths survive if fraction total time Bi-evolution equation of motion

17. Smoking gun: evidence from the Hamiltonian only observe evidence of in this branch only observe evidence of in this branch we observe only one of these termsphenomenological unidirectionality of time

18. Shortest path through time shortest path

19. Outlook Does this formalism represent reality? In early hot universe T violation would be rare, so no interference. The evolution is not unitary, it doesn’t conserve energy ! What does this mean ?

20. Consider Robertson-Walker-Friedman universe - matter & radiation are isotropic and homogenous Metric (flat spacetime for brevity): scale parameter Friedman equation: expansion of space Time reversal operation: slope changes sign

21. “backwards” evolution is in direction of decreasing t Space expands due to mass and pressure – these are unchanged by time reversal. Universe expands in both directions of time. scale parameter depends on length of path

22. Foliation of spacetime is the length of path in time space-like slices state of quantum fields at path length

23. Evolution of quantum fields = zero eigenvalue of constant Result: pressure ~ -r quantum field ~ “vacuum energy” exponential rate of expansion r ~ constant generation of radiation energy photogenesis

24. Summary Lack of T violation processes inflation • radiation appears as vacuum energy • constant energy density as space expands exponentially • photogenesis origin of all the matter of the universe Sufficient number of T violation processes • destructive interference eliminates all paths except continuously forwards and continuously backwards • physical evidence shows which path we experience • quantum algorithm for the shortest path to the “future”...