Time Operator in Standard Quantum Mechanics?

[CPL.Luke]

is there a time operator (not the time evolution operator) in standard qm?

I'm a bit curious as while it might not matter all that much for non-relativistic qm it seems as if it would be important to have some measure of a the probability that a particle is at a given place and time.

If I had to guess at one I'd say that its something like i (partial with respect to t) in the position basis.

 

[Mentz114]

There's no time operator in the usual sense. Have a look here.

http://math.ucr.edu/home/baez/uncertainty.html.

Some time operators have been proposed, but I think they are non-Hermitian. There's a lot about this in the arXiv. It's worth doing a search ( or consult a textbook ).

I found this thread, also worth a read.

https://www.physicsforums.com/archive/index.php/t-113311.html

 

[CPL.Luke]

yeah I've seen that derivation before, however it seems that while that may be the only implication for non-relativistic stuff, in order to treat the time observable in a relativistic picture you would need a time operator, more so it should be related to the energy operator i partial with respect to t as that would make a nice translation to four momentum and four position.

 

[StatusX]

In non-relativistic QM, time is a parameter while position is an operator. Since we expect the two quantities to be on an equal footing relativistically, there are two things we can do to modify QM before generalizing it to a relativistic setting:

1. Demote position to a parameter. Then operators become functions of both space and time, ie, fields. This leads to quantum field theory, and is the standard approach.

2. Promote time to an operator. You still need a parameter for the wavefunction to evolve over, which can be taken as proper time or left arbitrary, giving the theory a reparametrization gauge symmetry that must be dealt with. Many particles and interactions among them become a little tricky to handle. But generalizing the single parameter to two parameters (ie, making the 1D worldline a 2D worldsheet) gives a nice path to string theory.

 

[CPL.Luke]

so have there been good formulations of relativistic qm with time as an operator?

or have all of these formulations fallen short and become inconsistent or inaccurate?

 

[meopemuk]

In non-relativistic QM, time is a parameter while position is an operator. Since we expect the two quantities to be on an equal footing relativistically, there are two things we can do to modify QM before generalizing it to a relativistic setting:

1. Demote position to a parameter. Then operators become functions of both space and time, ie, fields. This leads to quantum field theory, and is the standard approach.

2. Promote time to an operator. You still need a parameter for the wavefunction to evolve over, which can be taken as proper time or left arbitrary, giving the theory a reparametrization gauge symmetry that must be dealt with. Many particles and interactions among them become a little tricky to handle. But generalizing the single parameter to two parameters (ie, making the 1D worldline a 2D worldsheet) gives a nice path to string theory.

You seem to believe that relativistic invariance implies certain equivalence between time and position, so that they both should be either parameters or operators. However, I don't think there is a convincing proof of such an equivalence. On the contrary, there exists a well-established approach (Wigner, 1939; Dirac, 1949) in which the principle of relativity and quantum mechanics are unified within the theory of unitary representations of the Poincare group. In this approach the relativistic operator of coordinate (Newton-Wigner, 1949) happily coexists with time described as a classical parameter.

Eugene.

 

[StatusX]

Luke,
Like I said, string theory, as it's typically formulated, has time as an operator. Specifically, the (non-interacting, bosonic) theory is a 2D field theory with fields $X^\mu$, including time, $X^0$. As a warmup in textbooks, they usually look at the 1D case first, which reduces to ordinary free field theory. I haven't seen much else of the this approach, but, for example, in his QFT text Srednicki says "promoting time to an operator is a viable option, but is complicated in practice".

Eugene,
I'm not saying it's strictly necessary for them to be on the same footing, but it is required if we'd like our theory to be manifestly lorentz invariant, since lorentz transformations mix the space and time coordinates.

 

[meopemuk]

Eugene,
I'm not saying it's strictly necessary for them to be on the same footing, but it is required if we'd like our theory to be manifestly lorentz invariant, since lorentz transformations mix the space and time coordinates.

Do we really like our theory to be manifestly lorentz covariant with "mixed" space and time coordinates? Perhaps it is sufficient to have "relativistic invariance" in the Wigner-Dirac sense, i.e., to have the Poincare commutation relations satisfied? These two conditions (the "manifest covariance" and the "relativistic invariance") are not equivalent. The "manifest covariance" condition is too restrictive, and I don't think it can be rigorously derived from the principle of relativity or Poincare group properties. There is no harm if this condition is dropped.

Eugene.

 

[CPL.Luke]

however a formuation which manifestly handles relativity would probably be easier to generalize and also a bit more elegant in theory than a formulation which was merely constructed to be covariant.

 

[meopemuk]

however a formuation which manifestly handles relativity would probably be easier to generalize and also a bit more elegant in theory than a formulation which was merely constructed to be covariant.

I agree about the "ease" and "elegance" of the manifestly covariant formulation of relativity. This path was followed during last 100 years of its development. This path led us to the contradiction with quantum mechanics that you mentioned in the beginning: in QM position is an Hermitian operator and time is a parameter, i.e., the manifest covariance is lost. In this situation the "ease" and "elegance" argument does not look so convincing.

Eugene.