The quantum time problem

Traditionally the quantum time problem is the question Is time a parameter or an observable in quantum mechanics? The essence of a parameter is that it is a label describing the state of a given system. On the other hand, the essence of a quantum observable is that it is a property of a quantum system that can be subject to measurement, the outcome of whose is described by the laws of quantum mechanics. However, the quantum time problem is more complicated than deciding whether time is a parameter or an observable. The problem is multi-faceted, with the different aspects almost conceptually distinct that no single idea may be applied to them all at once. In this post, I will attempt to spell out the problem and hope to clarify some issues regarding the problem. Some of my opinions here are tentative until they shall have found their way in refereed publications.

Let me right away point out that a quantum observable may provide a parameter labeling a quantum state. Recall that the state of a quantum system is abstractly represented by the wave-function . But this state can be represented or described in several ways, in fact infinitely many ways, such as the position representation and the momentum representation . Of course, and are different functions but they describe the same state, the same amount of information is encoded in them. In these representations, and take on the role of a parameter, merely a convenient label for the state of the system not the observable position and momentum. However, the parameters and are not at all independent of the position and momentum observables because they are in fact the range of possible outcomes of any measurement on them.

But it is not necessary that a parameter labeling the state arise as values of an underlying observable of the system. The label may be external. A quintessential example of such a parameter is the time parameter that appears in the Schroedinger equation. The time there is supposed to be universal and marks the evolution of a quantum object. This time is measured by some clock external to the system and hence not a property or an observable of the system. As such it is obviously nonsensical to ask What time does the system have? The quantum time problem then asks if time has other aspects beyond the parametric aspect it has in labeling the evolution of a system, aspects such as it can be ascribed as inherent to a system that can be subject to measurement.

A stronger version of the problem can be stated as follows Can the wave-function be parametrized by a time variable which constitutes the measurable values of an intrinsic time observable of the system in the same way as the wave-function is parametrized by the possible values of the position and momentum observables? In particular, Does a time operator exist?

Hints into non-parametric nature of time

There seems to be a non-parametric aspect of time as demonstrated by radioactivity. A radioactive particle is characterized by its lifetime, which is the most likely time that it will decay. It is then not unreasonable to suppose that decay time is an inherent property of a radioactive particle that can be subject to measurement. Now the time at which a radioactive particle decays is inherently random, and the measured time of decays of an ensemble of similar particles is distributed. These characteristics of radioactive decay time seem aligned to the well-known nature of quantum measurements. The question now is whether the outcome of the measurement of decay time falls within the ambit of quantum measurement theory.

But decay time may not be the only non-parametric time aspect of a quantum particle. For a quantum particle in certain configurations, we may ask its arrival time, tunneling time or its escape time. For the arrival time, we imagine the quantum particle prepared in some initial state and wait for it to arrive at a given location in the configuration space. For the tunneling time, we imagine the quantum particle entering the barrier and ask how long it takes before it emerges from the other side of the barrier. For the escape time, we imagine the quantum particle to be trapped in a potential well and ask how long before it escapes the well.

It is not unreasonable to expect that these different times are functions of the state of the quantum particle. As such they do not refer to the external parametric time in the Schroedinger equation, but each may be taken as a “property” of the system arising from the system’s specific configuration. One would then be justified in attempting to measure these times and in interpreting the results as reflective of the system itself not of the external clock used to measure them. In fact, such measurements had already been made.

I was careful to write “property” above to indicate that those times may be functions of the state of the system but not necessarily intrinsic observables of the system. Temperature comes to mind. Temperature is a property, say, of a gas, but it is, in fact, nothing but the average kinetic energy of the particles constituting the gas. That is, at a more fundamental description, temperature does not exist as an intrinsic property of the system of particles that is the gas. It is a distinct possibility that decay, arrival, tunneling, and escape times are in the same category as the temperature. They are functions of the state and are measurable, but not intrinsic observables of the quantum particle in the same way as momentum, position and kinetic energy are. Nevertheless, they point to a non-parametric nature of time.

Quantum Observables and the quantum time problem in weak and strong forms

Quantum mechanics is primarily about measurements made on a given system, a set of rules that relates the outcomes of measurements and the pre-measurement configuration or initial state of the system. A quantum system, according to quantum mechanics, is a Hilbert space; and the observables of the system, properties of the system that can be subject to measurements, are represented by a set of positive operator valued measure acting on the underlying system Hilbert space. Equivalently a quantum observable provides a resolution of the identity of the system Hilbert space.

To claim that a property is an observable is to assert the existence of a set of positive operators, either projection valued or non-projection valued, that can describe the distribution of the outcome of measurements of the property. Thus position, momentum, and energy are legitimate quantum observables because outcomes of their measurements are completely described by a resolution of the identity. So the quantum time problem can be stated as follows: Is the result of any measurement time involving a quantum object, such as time of arrival and tunneling time of a quantum particle, describable in terms of a resolution of the identity? This statement of the problem can be taken as the weak form of the quantum time problem.

A stronger version of the problem can be stated as follows. Let us assume that a time measurement distribution is indeed derivable from a resolution of the identity. The question is Does an operator in the system Hilbert space exist such that it is the first moment of the resolution? Furthermore, is this first-moment operator canonically conjugate with the system Hamiltonian? Operators with such properties are called time operators. The strong form of the problem can be restated as follows: Does every time measurement distribution derivable from a time operator? Or does time measurement distribution has an underlying ideal distribution generated from a time operator?

The weak and strong forms of the quantum time problem are not equivalent. An affirmative answer to the strong form implies an affirmative answer on the weak form. However, an affirmative answer on the weak form does not necessarily imply an affirmative answer on the strong form of the problem.

Perspectives on the quantum time problem

There are four prevailing perspectives on the quantum time problem.

The first is the view that time is a mere parameter marking the evolution of a system. It is not a property of a system in the same way as spin and mass are properties of a system. As such time is not a dynamical observable of a given system and not subject to the laws of quantum measurement theory. In particular, no time operator exists in the same way that a position or a momentum operator exist.

The second is the diametrically opposite view that since time is a legitimate measurable quantity it must be a dynamical observable as such it must be represented by an operator in the system Hilbert space and the distribution of measurement is dictated by the resolution of the identity provided by the operator. It is not only that time is an operator but that it has the additional property of being conjugate with the system Hamiltonian, i.e. it evolves in step with the parametric time. That is time observables are time operators.

The third is at the middle of the first two. Time is neither a mere parameter nor an operator conjugate with the Hamiltonian, but it is nevertheless a legitimate measurable quantity reflecting some properties of the system under consideration. In this view, it is possible that the aspect of time under consideration is represented by an operator but not necessarily conjugate to the Hamiltonian. Or that time measurement must always be taken in context with other systems, such as a measuring instrument. It also accommodates the possibility that time measurements are not described by resolutions of the identity, but just functions of the states and other observables of the system.

The fourth is the perspective that time is multifaceted; that is, it has several aspects, e.g. time of arrival, tunneling time, so that it is not expected that the same treatment applies to them all. This view accommodates the possibility that time can be a parameter labeling the system, a dynamical observable, and a measurable quantity but not necessarily represented by a time operator. Here the time in the Schroedinger equation is parametric and measured by a classical clock. And some time observables may indeed be time operators but others just functions of state variables and parameters external to the system.

Conclusion

To date there is still no consensus on the quantum time problem. Researchers are still debating on what exactly the problem is and how exactly it should be solved. The problem is exacerbated by the fact that the problem has several distinct aspects that it is very easy to compound the problem by confusing the different problems. It is possible that some of the present disagreements are actually non-existent because they are in the first place addressing very different aspects of the quantum time problem.

But it is more likely that the disagreements are fundamental in nature (for example, Should the quantum time of arrival problem be solved within the standard formulation of quantum mechanics or within Bohmian mechanics?) that an experimental confirmation of one approach to an aspect of the quantum time problem can lead to a fundamental shift in our understanding of quantum mechanics itself.