Time as Parameter in QM vs Coordinate in QFT

[waterfall]

According to Peterdonis in an old thread

"In non-relativistic mechanics, where time is a parameter, not a coordinate, the single time parameter is "shared" by all the particles, i.e., the multi-particle wavefunction evolves in "time" the same way the single-particle one does. But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact, but you also can't just have a "shared" time coordinate among all of them because they may be in relative motion."

According to Matterwave in https://www.physicsforums.com/showthread.php?t=573589 msg #11:

The distinction between coordinate and parameter is very often not made because essentially they are the same thing. I suppose if you view the non-relativistic space-time structure as a fiber bundle structure where the 3-D slices of constant times are the fibers and the 1-D base manifold is time, and you look explicitly at each fiber, then you would call the space coordinates "coordinates" on these fibers, and the time coordinate a "parameter" which specifies which fiber you are on. But of course you can very simply just consider the entire fiber bundle and now you simply have 4 coordinates. This is especially true since this fiber bundle is isomorphic to R^4, so it's a trivial fiber bundle (as far as I know, somebody correct me if I'm wrong here).

I don't see much merit in making this distinction. But, I have not really studied Bohmian mechanics, so I don't know if it's useful there.

Peterdonis said Time is a parameter in non-relativistic QM while as a coordinate in relativistic QM/QFT. But Matterwave said parameter and coordinate has same meaning. So who is right and why is that?

 

[PeterDonis]

Some of this may be just choice of terminology. However, I think there is an aspect of non-relativistic QM that Matterwave's comment does not address.

If we were talking about classical mechanics, then I would have no problem with viewing the entire spacetime as a single manifold, R^4, with one time and three spatial coordinates. However, in QM (we're talking non-relativistic QM now), we can't do that because the physics is no longer deterministic: there is not a single unique future for a given 3-D spatial "slice" in the spacetime. So you can't just view the spacetime as a single R^4 manifold, because the physics doesn't tell you *which* R^4 manifold it will be; it only gives you a probability distribution over different possible R^4 manifolds, given a set of initial data on a 3-D spatial slice. In other words, in plain English, you can't predict the results of quantum experiments definitely in advance; you can only assign probabilities. So your mathematical framework has to take that into account.

In standard non-relativistic QM, the way this is handled is to describe states of the system we are interested in by labeling each state with its own unique set of "coordinates" (such as x, y, z, if the "system" is just a single particle moving in 3-D space and not interacting with anything else--but there can be lots of other kinds of systems requiring different kinds of coordinates). Then we express the various probabilities for one state changing to other possible states by bringing in this parameter called "time", which is fundamentally distinct from the coordinates.

Once you try to include relativity, the single time parameter no longer works, as I said in that old thread, because there is no single universal time; the "rate of time flow" varies for systems in relative motion. Dealing with *that* ends up leading to quantum field theory, which changes the whole framework again.

 

[Bill_K]

One way to include relativity is, instead of defining the state Ψ as a function of a coordinate t, more generally define it as a function of a spacelike hypersurface σ. Then instead of writing the Schrodinger equation as an evolution of Ψ simultaneously everywhere, write it as a local evolution. That is, let σ' be a spacelike hypersurface which differs from σ by an infinitesimal amount δσ in the neighborhood of a spacetime point x. Then H Ψ = iħ δΨ/δσ, where H is the Hamiltonian density at x.

 

[waterfall]

So there is no more objections from the group like Fredrik or Matterwave on the statements that time is a parameter in non-relativistic QM and a coordinate in relativistic QM/QFT such that I could state this in a lecture to high school students or laymen (or include it in physics textbook) and make this impress into their minds forever? Or are there any objections? Please tell it now before it's too late. Thanks.

 

[Bill_K]

PeterDonis gave three reasons and I disagree with all three of them. Nevertheless I do agree with the original assertion - that time in Schrodinger quantum mechanics is a parameter while in quantum field theory it is a coordinate. For the reason I stated - that in nonrelativistic quantum mechanics it must advance uniformly, while in the relativistic theory it must be permitted to advance by different amounts in different locations.