A world in which nothing changes would provide no changing clock reading. Whether time could nevertheless exist is a further philosophical question. Physics sharpens the discussion by distinguishing a stationary state, an unchanging measurement result and the absence of every possible temporal comparison.
Stationary measurements can conceal dynamics
Imagine a pointer rotating uniformly around a circle. A detector that measures only the pointer's length records a constant value. A detector sensitive to its direction records a changing angle. The first instrument's stationary output does not establish that the whole system is stationary.
For a unit vector r(t) = (cosωt, sinωt), the squared length is always one, while the first component changes. The distinction is between an invariant of a motion and the motion itself. Specifying the accessible observables is therefore essential.
The same lesson applies to a stable average. A container can have a constant mean density while its constituents move. Constancy of one large-scale quantity gives information about that quantity; it does not remove all microscopic evolution from the model.
A quantum phase needs a comparison
For a time-independent Hamiltonian, an energy eigenstate evolves as |ψ(t)⟩ = exp(−iEt/ħ)|E⟩. Its density operator |ψ(t)⟩⟨ψ(t)| is unchanged because the phase cancels against its conjugate.
All expectation values of fixed observables in that state are therefore time independent. The formal phase factor alone does not give an isolated clock hand that can be read from the state. A physically accessible phase requires a reference or a relative phase between components.
Consider instead equal amplitudes in two energy levels, E₀ and E₁. After a delay t, a measurement projecting onto the initial equal superposition has probability
P(t) = ½[1 + cos(ΔEt/ħ)], ΔE = E₁ − E₀.
It returns to one at T = 2πħ/|ΔE| and reaches zero at T/2. The relevant rate depends on an energy difference. Shifting both energies by the same constant does not alter this probability.
This calculation presumes a preparation and a measurement capable of comparing the components. Writing a phase in a state vector does not supply that apparatus automatically. It identifies what an operational clock construction must preserve and read.
Look beyond a single-time average
A stationary statistical process can also have time-dependent correlations. As an elementary illustration, choose an initial phase φ uniformly around a circle and let x(t) = cos(ωt + φ). At every time the mean is zero and the mean square is one half.
Nevertheless, the two-time correlation is ⟨x(t)x(0)⟩ = ½cosωt. Averaging over the unknown phase removes a preferred starting angle but retains the relationship between observations at different times.
A series of paired measurements can reveal this correlation even though the one-time distribution never changes. Statistical stationarity therefore differs from the absence of every temporal structure. The available question determines which evidence needs to be collected.
What if all comparisons were absent?
Now strengthen the premise: no state changes, no records accumulate, and no correlations distinguish any pair of supposed moments. An operational description would have no experiment within that world that measures an elapsed interval.
One may still introduce a mathematical parameter labelling identical states. Whether that parameter describes something physically additional is a matter for the interpretation of the model. Repetition of an unobservable label is not itself a new measurable process.
This does not prove that every philosophical account of time must identify it with change. It shows where the empirical question ends: a proposed distinction needs an observable consequence, or a clearly stated reason for retaining it at another explanatory level.
Relate generation to accessible change
The book's real-flow construction selects a complex structure on the dynamic sector of an oriented orthogonal flow. A stationary kernel requires separate treatment. This mathematical distinction is useful, but the flow parameter becomes a clock variable only through calibration.
For SMM/DTT, the central question is what completed observations retain of inner generation. A changing inner description may leave a particular outer measurement fixed, just as the rotating pointer leaves its length fixed. A complete account should identify which comparisons can reveal the retained dynamics.
Sources and further reading
- Feynman, Leighton and Sands, The Dependence of Amplitudes on Time, for quantum evolution. The two-level and random-phase calculations here are worked teaching examples.
- Continue with moments and observations and multiple timescales.