Time in Quantum Theory

Abstract

I. In general, time is used in quantum theory as an external ('classical') concept. So it is assumed, as in classical physics, to exist as a controller of all motion – either as absolute time or in the form of proper times defined by a classical spacetime metric. In the latter case it is applicable to local quantum systems along their world lines. According to this assumption, time can be read from appropriate classical or quasi-classical 'clocks'. This conception has to be revised only when general relativity, where one regards the spatial metric as a dynamical object, is itself quantized [1] – as required for consistency (see IV). The thereby achieved 'quantization of time' does not necessarily lead to a discretization of time – just as the quantization of free motion does not require a discretization of space. On the other hand, the introduction of a fundamental gravitational constant in addition to Planck's constant and the speed of light leads to a natural Planck time unit, corresponding to 5.40 10 sec. This may signal the need for an entirely novel conceptual framework – to be based on as yet missing empirical evidence. A formal (canonical) quantization of time would also be required in non-relativistic Machian ('relational') dynamical theories [4], which consistently replace the concept of time by some reference motion. If quantum theory is universally valid, all dynamical processes (including those that may serve as clocks or definers of time) must in principle be affected by quantum theory. What does this mean for the notion of time? Historically, the dynamics of quantum systems seemed to consist of individually undetermined stochastic 'quantum jumps' between otherwise 'stationary' states (energy eigenstates) – see [2] for an early review of the formalism and the attempt of an interpretation. Such stochastic events are observed in quantum measurements, in particular. For this reason, von Neumann [3] referred to the time-dependent →Schrodinger equation as a 'second intervention', since Schrodinger had invented it solely to describe consequences of time-dependent external 'perturbations' of a quantum system. Note, however, that atomic clocks are not based on any stochastic quantum events, even though they have to be designed as open systems in order to allow their permanent reading (representing 'measurements' of the clock – see IV). In a consistent →Schrodinger picture, all dynamics is described as a time dependence of the quantum states, while the observables are fixed formal kinematical concepts (see also Sect. 2.2 of [5]). The time dependence according to the Schrodinger equation can be completely understood as an interference phenomenon between different stationary states |m>, which possess individually meaningless phase factors exp(iωmt). Their →superpositions are able to describe time-dependent quantum states |α(t)> in the form |α(t)> := ∫dq ψα(q,t)|q> = Σmcmexp(iωmt)|m> . The wave function ψα(q,t) is here used to define the time-dependent state |α(t)> in abstract →Hilbert space. The Hilbert space basis |q> diagonalizes an appropriate observable Q. The time dependence of a quantum state is in fact meaningful only relative to such a fixed basis, as demonstrated by means of the wave function in the above definition. In non-relativistic quantum mechanics, the time parameter t that appears in the Schrodinger wave function ψ(q,t) is identified with Newton's absolute time. So it is presumed to exist regardless of how or whether it is measured. The letter q represents all variables qi (i=1...I) that span the required configuration space. The special case of a point mass, where q