Quantum calculations use time in several roles: an evolution parameter, a reading produced by a clock, a duration inferred from changing probabilities, and a label ordering a sequence of operations. Keeping these roles distinct makes the question of inner and outer time more precise.
An evolution parameter is not yet a clock
For a time-independent Hamiltonian H, standard unitary evolution is U(t) = exp(−iHt/ħ). The parameter t labels the transformation. To interpret its numerical value in seconds, a laboratory must connect it to a physical clock and a synchronization procedure.
Take a two-level Hamiltonian with energies 0 and E, where E is positive. Starting from |+⟩ gives |ψ(t)⟩ = [|0⟩ + exp(−iEt/ħ)|1⟩]/√2. A measurement in the +,− basis has P(+) = [1 + cos(Et/ħ)]/2.
This probability oscillates with period 2πħ/E. The state becomes orthogonal to its initial state at t = πħ/E and returns to the same ray after one full period. Those statements follow from the chosen Hamiltonian and preparation. They do not explain why that energy splitting has its measured value.
What a finite quantum clock can distinguish
At times 0 and πħ/E, the two clock states are |+⟩ and |−⟩, which an ideal measurement can distinguish perfectly. At an intermediate time πħ/(2E), the overlap magnitude with the initial state is 1/√2. A single reading cannot perfectly distinguish those two nonorthogonal states.
The reason is elementary. A device that identified the two inputs without error would map them to orthogonal records. A unitary interaction preserves their nonzero inner product, so it cannot make the total outputs orthogonal while retaining exact deterministic discrimination. Additional copies, allowed errors or different operational goals change the task.
The clock also repeats. A phase reading alone cannot tell whether one cycle or many cycles have passed. A cycle counter or another retained record is needed to extend its unambiguous range. Such a counter has its own capacity and stability requirements.
An energy spread controls this example's speed
The initial state's mean energy is E/2 and its energy standard deviation is E/2. Multiplying that spread by the orthogonalization time gives ΔE t = πħ/2. This particular two-level evolution saturates the familiar time-independent pure-state orthogonalization bound.
The bound concerns how fast a state can become distinguishable under a specified unitary dynamics. It is not a statement that a detector always has a time error equal to ħ divided by its energy uncertainty. Arrival-time measurements, control durations and clock estimation are separate operational problems.
Adding a common energy offset C to both levels multiplies the state by exp(−iCt/ħ). The relative phase, the energy spread and the displayed probabilities remain unchanged. In this isolated comparison, the observable frequency is set by the energy difference.
A useful limit on a simple time-operator claim
In a finite-dimensional space of dimension d, no pair of matrices T and H can satisfy [T,H] = iħI. The trace of any matrix commutator is zero, while the trace of the proposed right-hand side is iħd.
This concise obstruction does not rule out every time observable. A finite clock can have labelled readout effects, and more general time measurements can be described through positive operator-valued measures. Infinite-dimensional commutator claims also require careful operator-domain assumptions. The scope of the trace argument should not be extended beyond what it proves.
Identify both temporal roles operationally
SMM/DTT distinguishes inner generative activity from outer completed observations. In a concrete model, one should state which variable orders the generator, which register records completion and how either is calibrated against a physical clock.
Changing the name of t to “inner time” while retaining an unchanged Hamiltonian and unchanged readout does not create a second measurable time. A new physical claim would constrain their relation or predict a difference in accessible records. The book discusses order, duration and clocks and states the calibration requirement when recovering Schrödinger form.
The two-level clock provides a compact benchmark: reproduce its relative-phase rate, its periodic ambiguity and the statistics of repeated readings before claiming a more general temporal reconstruction.
Sources and further reading
- J. Anandan and Y. Aharonov, Geometry of quantum evolution, Physical Review Letters 65, 1697–1700 (1990).
- Continue with duration and order and the principle of an atomic clock.