The Problem(s) of Time

Given our ambition to find the most minimal fundamental description of reality, it is natural to ask whether time as well as space could be emergent from the wave function. The Wheeler-deWitt equation of canonical quantum gravity takes the form for some particular form of H in a particular set of variables.

then constructing an effective Hamiltonian describing evolution of the universe with respect to the clock.

Given our discussion thus far, the problem with such a procedure should be clear: what determines the decomposition (15)? In the Schrodinger case we can have data in the form of the spectrum of the Hamiltonian, but in the Wheeler-deWitt case the universe is in a single eigenstate; no other features of the Hamiltonian, including its other energy eigenvalues, can be relevant. This problem has been dubbed the “clock ambiguity” [18].

One potential escape would be to imagine that the fundamental state of the universe is described not by a vector in Hilbert space, but by a density operator acting on it. Then we have an alternative set of data to appeal to: the eigenvalues of that density matrix. These can be used to compute a modular Hamiltonian (given by the negative of the logarithm of the density operator), which in turn can yield an effective notion of time evolution, a proposal known as the “thermal time hypothesis” [19]. Thus it

is conceivable that time as well as space could be emergent, at the cost of positing a fundamental density operator describing the state of the universe.^[21]

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