Dirac's 1927 paper 'The physical interpretation of the quantum dynamics' supplied the general statistical interpretation of quantum mechanics through transformation functions, and stated its own limits: relativity mechanics is set aside and time is treated as a c-number parameter; the reciprocal theorem of quantum dynamics holds only when the transformation functions are continuous functions of the characteristic values, i.e. only for continuous spectra; the time-fraction results are stated disregarding difficulties introduced by degeneration; and while the theory shows that the energy actually changes discontinuously from one quantised value to another and computes the fraction of the total time spent in each, it can give no information about the times of the transitions. The first three of these limitations are absorbed by later standard developments (relativistic quantum mechanics; von Neumann's spectral theory together with the rigged Hilbert space; the ensemble reading of the probabilities with degeneracy handled by complete commuting sets). The fourth survives as the individual-events core of the modern measurement problem: experiments observe individual quantum transitions and their temporal structure, quantum trajectory theory reproduces single-event records with postulated jump statistics, collapse models and hidden-variable dynamics supply candidate event-time laws at the cost of new parameters or a different theory structure, and no accepted derivation of occurrence times from the underlying dynamics exists.

Background

By late 1926, quantum mechanics could compute atomic energy levels exactly but lacked a general physical interpretation. Dirac's paper supplied one. It distinguishes q-numbers — the non-commuting dynamical variables of quantum theory — from c-numbers, the ordinary numbers of classical physics, and introduces transformation functions such as (ξ′/α′)(\xi'/\alpha'), whose squared modulus gives the probability that a measurement of a variable ξ on a system known to be in state α' yields the value ξ'. This generalizes Born's probability rule and Heisenberg's resonance treatment into the statistical interpretation of all of quantum mechanics. To handle variables whose possible values form a continuum, Dirac introduces the delta function δ(x)\delta(x), an 'improper' function that vanishes everywhere except at x=0x=0.

Two constructions matter for this problem. First, interpreting a diagonal element of a quantum variable as an average over the system's phase space (η-space) lets the theory compute the fraction of time for which a variable, in a stationary state, lies in a given range — an ergodic-type reading in which probabilities become fractions of time. Second, a 'reciprocal theorem of quantum dynamics' equates two η-space volumes, but only when the transformation functions are continuous functions of the characteristic values, which excludes discrete and mixed spectra. The paper closes by suggesting that the initial state may determine definitely the state at all subsequent times, with probabilities entering only through prior information about η-space.

The limitation that survived is the one about individual events. The theory establishes that a system's energy actually changes discontinuously from one quantised value to another and computes what fraction of the total time it spends in each state, but it can give no information about the times of the transitions. Whether a given excited atom emits its photon now or later — whether the occurrence times of individual transitions obey any dynamical law at all — is left open, and it is exactly this question that later became the individual-events core of what is now called the quantum measurement problem: quantum mechanics predicts probabilities for ensembles and experimenters record individual outcomes (quantum jumps), but no accepted dynamics derives when a particular event occurs.

Problem Statement

The problem, as left open by the source, is: does there exist a dynamics — a principled completion, extension, or interpretation of quantum theory — from whose own dynamical law the actual occurrence times of the individual transitions of a single system between definite states (for example, the instants at which a particular atom's energy changes from one quantised value to another) are obtained, rather than postulated as an independent statistical rule of the kind the 1927 theory was intended to replace (statistics postulated as an independent measurement or jump rule calibrated to reproduce the ensemble probabilities, such as the projection postulate or quantum-trajectory jump prescriptions, do not qualify, nor do completions that change the statistical predictions of quantum theory, introduce entities not determined by the wave function, or add parameters whose values are chosen to fit the ensemble data), such that: (1) ensemble statistics over the individual-event histories reproduce the statistical content that the 1927 transformation theory delivers and that experiment confirms — transition probabilities, the diagonal averages now identified with Born-rule expectations, and the time-fraction results recoverable as ensemble means — to within the precision at which that content is established; and (2) the dynamics is internally consistent: positive probabilities, and consistency, in the limit of large quantum numbers, with the classical correspondence on which the source's interpretation of the diagonal averages rests (§6, p. 637). Alternatively, an impossibility theorem from explicitly stated axioms (the verified ensemble statistics plus a consistent, non-empty set of operational constraints — relations among the preparation, transformation, and measurement statistics of the 1927 transformation theory — satisfied by every dynamics that reproduces those statistics, none of which mentions occurrence times or single-event records, so that the failure of the candidate dynamics is derived from the axioms rather than assumed) shows that no dynamics can derive the occurrence times from its own dynamical law while reproducing that statistics. 'Principled' means that the occurrence times are consequences of the same evolution that yields the ensemble statistics, introducing no entity not determined by the wave function, no parameter whose value is chosen to fit the ensemble data, and no independent rule to generate the event records. An event law obtained by unravelling or conditioning that evolution qualifies only if the occurrence times are fixed by the evolution itself and not by the choice of unravelling — distinct unravellings that share the same ensemble statistics must not assign different occurrence times. The three other limitations stated in the source paper are treated here as absorbed — the non-relativistic restriction, the continuous-spectrum restriction of the reciprocal theorem, and the degeneration caveat — except where they re-enter through this same core: a relativistic account of events in the presence of particle creation, and the role of degeneracy in deciding when two configurations count as the same transition. These two re-entering items are scope caveats rather than required parts of every resolution: each binds only when a proposed dynamics already involves relativistic particle creation or degenerate configurations. The question posed is not the general measurement problem in its whole interpretive extent, but its specifically temporal, single-event core as derived from the 1927 source: the occurrence times of individual transitions.

References

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