Graduate Position is no more an operator in QFT

[Heidi]

In quantum mechanics there is no operator for time (problem with unbounded energy).
position is no more an operator in field theory. was there still a problem in QM?

 

[mitochan]

Say there are many particles of same kind, position lose its meaning and number density takes its palce.

 

[vanhees71]

In general there is no position operator in relativistic quantum theory, at least not within the only kind of relativistic QT that's successful in describing the real world in terms of the Standard Model, which is local relativistic QFT.

However, for all massive particles you can define a position operator having the usual properties. Since only massive particles have a useful non-relativistic limit, there is no contradiction between having a position operator in non-relativistic quantum theory and local relativistic QFT.

The representations of the Galilei group for massless particles doesn't lead to a physically interpretable quantum theory. See also my comment on this here:

https://www.physicsforums.com/threa...r-the-gravitational-field.997062/post-6433476

 

[Demystifier]

Is there a position operator in QFT? The question does not make sense until one defines what exactly one means by "position operator". There is operator that satisfies some properties one would expect from a decent position operator, but not all. In particular, the Newton-Wigner position operator does not transform as a Lorentz vector.

 

[vanhees71]

I was referring to the usual definition of position operators, as explained in

https://arnold-neumaier.at/physfaq/topics/position.html

It's of course not part of Lorentz four-vector operator, because otherwise time would be an operator too, but that cannot be by construction, because the energy spectrum is bounded from below.

 

[andresB]

Is there a position operator in QFT? The question does not make sense until one defines what exactly one means by "position operator". There is operator that satisfies some properties one would expect from a decent position operator, but not all. In particular, the Newton-Wigner position operator does not transform as a Lorentz vector.

Position in relativistic physics is an interesting thing. It happens that the classical position operator IS a Newton-Wigner operator and, also, does-not transform as a 4-vector (can't give a reference, is still in peer review).

Moreover, I think the Newton-Wigner position function (https://arxiv.org/abs/2004.09723) of Hamiltonian mechanics has the same property, though I'm not sure, I haven't read the article in full details.