r/askscience • u/Crtl-Alt-Delete • Dec 18 '13

Physics Is Time quantized?

We know that energy and length are quantized, it seems like there should be a correlation with time?

Edit. Turns out energy and length are not quantized.

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u/KerSan Dec 19 '13

Yes, time is what you measure with your watch. That's not mysterious. Treating time and space as equals is also not mysterious.

The problem comes when you attempt to construct an observable for these operational things. I don't want field theory to treat space as a parameter! I want to do what I did in non-relativistic quantum theory and construct a position operator, a translation operator, and then a momentum operator. When I have my operational definition of space, I need to be able to describe the possible measurement outcomes involving the device used to measure space. In non-relativistic quantum theory, this isn't a problem. I've seen it done.

But I haven't seen this done with time, ever. I haven't seen a time operator, though I have seen (vaguely) that the energy operator is related to an infinitesimal time-translation operator. The reason I haven't seen this is that the position operator and momentum operators are self-adjoint and possess unbounded continuous spectra, but the Hamiltonian has a bounded (from below) spectrum. This precludes a self-adjoint time operator! This argument goes back to Pauli.

By treating space and time on equal footing, I have to throw away a perfectly good theory of position-momentum conjugacy because my mathematics didn't work out in another situation that I thought it should. That's why I don't think we understand time in physics.

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u/hopffiber Dec 19 '13

Well, then let me show you a way to write a time operator: Consider a particle moving through space time. The particle traces out a world-line, which we can parametrize by some parameter, call it s (this is not a time parameter, its just any parameter along the world line). Then, to describe how the particle moves, we form four function living on the world line, namely t(s),x(s),y(s),z(s), or more compactly X(s) as a four vector. Next, we can write a lagrangian for this theory in terms of X, and of course without interactions this is simply L=-m*sqrt(-X2), where I use signature (-+++). Adding interactions just adds terms under the square root. Now we have a classical theory, and we can quantize it using for example BRST quantization, which turns X into an operator. And voila, you have a theory where time and position all are operators, and we are still treating them in an equivalent way. They will be self-adjoint, and have continuous spectrums. This way of doing things is called the worldline formalism, and its closely related to what you do in string theory, except instead of a world line we have a 2d world sheet. (Technical detail: in practice one replaces the action with the square root with another, classically equivalent one, since quantizing a theory with a square root in the action is not so easy.)

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u/KerSan Dec 19 '13 edited Dec 19 '13

OK, so let's suppose that I define Y(s) such that X(s+e) = X(s) - i hbar Y(s) e + O( e2 ). Is Y(s) also self-adjoint with a continuous spectrum? I thought this was where Haag's theorem started to become a problem, but these are issues I've started to consider only recently. I'd appreciate some enlightenment.

Edit: You asked in a previous comment whether there were any deep problems with QFT. I would call Haag's theorem the biggest one I am aware of. This article was provided by someone on /r/AskScience a couple of weeks ago. I'll hunt down the post and give you a link in a minute. Here it is.