Abstract
This chapter generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time t. Doing so assigns a conditional state to the rest of the Universe |ψ S(t)〉, referred to as the system. It is shown that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state |ψ S(t)〉 satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator.
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Notes
- 1.
The Schrödinger equation is replaced with a master equation, which induces decoherence.
- 2.
We should emphasize this is a choice of inner product and normalization of the physical states, which may severely reduce the size of the physical Hilbert space. Furthermore, it may not be necessary to preserve the probabilistic interpretation of the system state. For example, the probabilistic interpretation may only be applicable in some limit, and it is the task of the physicist to explain how this limit comes about. To quote DeWitt on this point [4, 15]:
…one learns that time and probability are phenomenological concepts.
And Kiefer’s clarification of DeWitt’s statement [15]:
The reference to probability refers to the ‘Hilbert-space’, problem, which is intimately connected with the ‘problem of time’. If time is absent, the notion of a probability conserved in time does not make much sense; the traditional Hilbert-space structure was designed to implement the probability interpretation, and its fate in a timeless world thus remains open.
- 3.
A proof of this can be found on pages 197–198 of [26].
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Smith, A.R.H. (2019). The Conditional Probability Interpretation of Time: The Case of Interacting Clocks. In: Detectors, Reference Frames, and Time. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-11000-0_8
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