Time, Quantum Mechanics and Quantum Fields

Why is time not an operator in Quantum Mechanics even though it is observeable?





One should be careful to distinguish between non relativistic quantum mechanics (Schrödinger equation, etc.) and quantum field theory.

For nonrelativistic quantum mechanics, it is not so surprising that time and space are treated differently, with position being an operator and not time. After all, this is also what happens in Newtonian mechanics: time is absolute, and part of the background, and all other observables are functions of time. This paradigm underlies the formulation of the fundamental problem of Newtonian physics: to determine how a system evolves in time. Time cannot be an observable because an observable is a function of what we consider the system's "state", but the state is considered a function of time in the first place (so time is the independent variable).

Quantum field theory is fully compatible with special relativity, and therefore must treat space and time on equal footing. In nonrelativistic quantum mechanics, position is an observable whereas time is a parameter. That is, position is a function of the state, whereas time is used to label states. In formulating quantum field theory, we therefore have a choice between making spatial coordinates into parameters, or making time into an observable. This choice is discussed by Srednicki:

We can solve our problem, but we must put space and time on an equal footing at the outset. There are two ways to do this. One is to demote position from its stat us as an operator, and render it as an extra label, like time. The other is to promote time to an operator.

Let us discuss the second option first. If time becomes an operator, what do we use as the time parameter in the Schrodinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ$\tau$ of the particle (the time measured by a clock that moves with it) as the time parameter. The coordinate time T$T$ (the time measured by a stationary clock in an inertial frame) is then promoted to an operator. In the Heisenberg picture (where the state of the system is fixed, but the operators are functions of time that obey the classical equations of motion), we would have operators Xμ(τ)$X^{\mu }(\tau )$, where X0=T$X^0=T$. Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so. (The many times are the problem; any monotonic function of τ$\tau$ is just as good a candidate as τ$\tau$ itself for the proper time, and this infinite redundancy of descriptions must be understood and accounted for.)

One of the advantages of considering different formalisms is that they may suggest different directions for generalizations. For example, once we have Xμ(τ)$X^{\mu }(\tau )$, why not consider adding some more parameters? Then we would have, for example, Xμ(σ,τ)$X^{\mu }(\sigma ,\tau )$. Classically, this would give us a continuous family of worldlines, what we might call a worldsheet, and so Xμ(σ,τ)$X^{\mu }(\sigma ,\tau )$ would describe a propagating string. This is indeed the starting point for string theory.

Thus, promoting time to an operator is a viable option, but is complicated in practice. Let us then turn to the other option, demoting position to a label. The first question is, label on what? The answer is, on operators. Thus, consider assigning an operator to each point x$\mathbf{x}$ in space; call these operators ϕ(x)$\varphi (\mathbf{x})$. A set of operators like this is called a quantum field. In the Heisenberg picture, the operators are also time dependent:

ϕ(x,t)=eiHt/ℏϕ(x,0)e−iHt/ℏ$\varphi (\mathbf{x},t)=e^{iHt/\hslash }\varphi (\mathbf{x},0)e^{-iHt/\hslash }$.

Thus, both position and (in the Heisenberg picture) time are now labels on operators; neither is itself the eigenvalue of an operator.

So, now we have two different approaches to relativistic quantum theory, approaches that might, in principle, yield different results. This, however, is not the case: it turns out that any relativistic quantum physics that can be treated in one formalism can also be treated in the other. Which we use is a matter of convenience and taste. And, quantum field theory, the formalism in which position and time are both labels on operators, is much more convenient and efficient for most problems.