Energy and Time

1. Nov 19, 2013

Staff: Mentor

My studies lead me to contemplte the implications of the Schrödinger equation and it's close cousins in the classical world. First, I am struck by the fact that the time derivative of any physical quantity must be zero for any system with a zero Hamiltonian.

In quantum mechanics ${\frac{dL}{dt}}=-i[L,H]$ or in classical mechanics ${\frac{dL}{dt}}=-i\{L,H\}$, where $L$ is the average of any observed physical quantity and $H$ is the average of the Hamiltonian. Even more striking is the expression $<ψ|[L,H]|ψ>$ for the average value of any observable $L$ of any state $ψ$. All of these imply that nothing can happen, nothing can be observed, there can be no events in a system with no energy (zero Hamiltonian).

I guess the simplest possible event must be the engagement of the state of an observing apparatus with the state of the system. That too requires a change in the average state (except in the trivial case of observing up in a system prepared in the up state), and a nonzero Hamiltonian to accomplish.

If we accept the definition of time as: "Time is a dimension in which events can be ordered from the past through the present into the future." , then if there is no energy, there can be no events, and there is no way to observe or even to define the flow of time.

I understand that in this universe, we have zero point energy (vacuum energy), so that a closed system with zero Hamiltonian is not really possible. Correct? I also understand the possible circular semantic trap. A system with no energy, no mass, means no system at all. No system, no states, no observable, no events, no time, just nothing.

I have read books and viewed lectures by Leonard Susskind, Steven Hawking, and Brian Greene that tackle the fundamental questions such as the nature of time, but I've never seen this coupling between time and energy mentioned. Even on PF, questions about the nature of time are frequent, especially in connection with the origin of the universe. Yet I don't recall reading the time-energy relationship mentioned in any of the answers.

Modesty forces me to suspect that my conclusions are wrong or that the energy-time connection is so trivially obvious that nobody mentions it. Is that correct?

Please note what I have not mentioned above. The direction of the arrow of time. The mass energy equivalence. The fact that an event is pinned to a location is both time and space. Uncertainty. Nor have I claimed that time does not exist without energy. Existence would be a metaphysical question. I want to focus first on the more narrow question. Namely, for a closed system with zero Hamiltonian time is undefined.

2. Nov 19, 2013

voko

The energy-time connection is ubiquitous in physics. In classical mechanics, conservation of energy is implied by homogeneity of time. In relativity, energy is a component of the four-momentum vector that corresponds to the temporal dimension. In quantum mechanics, time and energy are bound by Heisenberg's uncertainty relation.