Time in Quantum Mechanics

Time serves several roles in quantum theory: an external parameter labeling evolution, a reading of a physical clock, and a random event time measured by a specified apparatus. These roles are related but cannot be identified without a model. Time as a Parameter explains ordinary dynamical evolution; the question here is how a physical clock can supply that parameter.

A clock is another physical system

The Schrödinger parameter $t$ is not an extra operator acting on every system. A physical clock has states, a Hamiltonian, a readout, and finite accuracy. A detection-time distribution similarly belongs to an experiment: it depends on coupling, efficiency, and the rule for recording arrival or decay.

A Pauli-type argument excludes a self-adjoint time operator canonically conjugate to a semibounded Hamiltonian in the strong, globally exponentiated translation sense. It does not forbid every symmetric time operator, restricted-domain commutator, periodic clock observable, or time POVM. The hypotheses are crucial: a formal commutator on a restricted domain does not automatically generate unrestricted energy translations.

Conditional dynamics in a stationary total state

An ideal Page–Wootters construction uses a clock $C$ and system $S$ with no clock–system interaction and a constraint

$$(H_C+H_S)\lvert\Psi\rangle=0.$$

Choose ideal clock states $\lvert t\rangle=e^{-iH_Ct/\hbar}\lvert0\rangle$ and define the unnormalized conditional system vector $\lvert\psi(t)\rangle={}_C\langle t|\Psi\rangle$. Differentiating gives

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle =-{}_C\langle t|H_C|\Psi\rangle =H_S\lvert\psi(t)\rangle.$$

Thus a globally stationary correlation can encode ordinary evolution relative to a clock reading. The ideal clock may involve generalized states and spectra unavailable to a finite physical clock. Normalization, readout resolution, repeated measurements, and interactions need separate treatment in an actual clock model.

A finite two-level illustration

Take $H_C=\hbar\omega\lvert1\rangle\langle1\rvert$ and $H_S=-\hbar\omega\lvert1\rangle\langle1\rvert$, with

$$\lvert\Psi\rangle= \frac{\lvert0,0\rangle+\lvert1,1\rangle}{\sqrt2}.$$

The total constraint holds. Conditioning on a clock phase state $\lvert t\rangle=(\lvert0\rangle+e^{-i\omega t}\lvert1\rangle)/\sqrt2$ gives a normalized system state $(\lvert0\rangle+e^{i\omega t}\lvert1\rangle)/\sqrt2$.

This clock is periodic: it cannot distinguish $t$ from $t+2\pi/\omega$. Its phase states are nonorthogonal at generic different times. The calculation illustrates relational evolution, but it does not supply an infinitely accurate clock or derive the gravitational problem of time from two qubits.

What the construction does not settle

Conditional evolution shows how correlations can describe change without assigning evolution to the constrained total state. It does not uniquely determine which subsystem is the clock, explain a thermodynamic arrow, or solve every constraint of quantum gravity. A changing clock record must still be described by physically realizable operations.

Quantum Reference Frames extends the relational idea to spatial descriptions. Energy–Time Uncertainty distinguishes clock accuracy from dynamical speed limits.

Exercise

Compute $|\langle t|t'\rangle|^2$ for the two-level clock. When are two readings perfectly distinguishable?

Solution

The overlap squared is $\cos^2[\omega(t'-t)/2]$. It vanishes when $\omega(t'-t)=\pi$ modulo $2\pi$. A single two-level clock distinguishes at most two mutually orthogonal readings, despite its continuously parameterized phase.

References

• Page, D. N., and Wootters, W. K., “Evolution without evolution: Dynamics described by stationary observables,” Physical Review D 27, 2885–2892 (1983), doi:10.1103/PhysRevD.27.2885 — conditional-clock construction.