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In Mark Srednicki's QFT book, he talks about the fact that one of the problems with combining quantum mechanics with special relativity is that in QM, position is an operator and time is just a parameter. He then says there would be two natural ways to remedy this, either promote time to an operator or demote position to a parameter. Referring to the first option he says that we can indeed do so if we use the proper time as the parameter in our differential equation and promote the observed time to an operator. He then continues:

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 is just as good a candidate as τ itself for the proper time, and this infinite redundancy of descriptions must be understood and accounted for.

He then goes on to describe the second option, the one that I am familiar with, where we label our quantum field operators by a position label $x$.

My question is two-fold:

  1. What exactly does understanding the infinite redundancy in choices for the time parameter entail, as in how does one end up dealing with that mathematically?
  2. Can it be shown that these two formulations are completely equivalent?

EDIT: Upon further reading, Srednicki asserts in fact that the two formulations are equivalent, so I would like to change to question 2 to a reference request to where I can find such a proof.

Qmechanic
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pmal
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    A reference related to your second question is here: http://physics.stackexchange.com/q/220697/ would you please read through it, in case it's a duplicate of your question. I am looking for a proof. –  Oct 06 '16 at 16:55
  • SM Kravec's answer in that previous question is explaining the same thing Srednicki is talking about. More detail is given in chapter 1 of Polchinsky's string theory book. – octonion Jan 06 '17 at 07:21

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For one thing, such a theory would automatically be a gauge theory, because the freedom to reparameterize the proper time in infinitely many ways would be a gauge freedom. So you'd always need to gauge fix your path integral, even for spin-0 and spin-1/2 theories, and that's always a huge mess.

tparker
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