How Does QFT Describe or Predict the Position of a Particle?

juanrga

"Labels on operators" is not the same than time and space itself being operators... In QFT space and time are not operators. vanhees71 is right

bhobba

Sorry for any confusion but the alternative the quote was referring to that was more difficult to work with is one in which time is an operator.

I managed to find a copy of Srednicki on the web:
http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf

The relevant quote is from page 25:

'We can solve our problem, but we must put space and time on an equal footing at the outset. There are two ways to do this. One is to demote position from its status as an operator, and render it as an extra label, like time. The other is to promote time to an operator. Let us discuss the second option first. If time becomes an operator, what do we use as the time parameter in the Schrodinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ of the particle (the time measured by a clock that moves with it) as the time parameter. The coordinate time T (the time measured by a stationary clock in an inertial frame) is then promoted to an operator. In the Heisenberg picture (where the state of the system is fixed, but the operators are functions of time that obey the classical equations of motion), we would have operators Xµ(τ), where X0 = T. 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.)'

Am I missing something?

Thanks
Bill

juanrga

In QM time is evolution parameter and well-known theorems explain why cannot be a operator. Moreover, in RQM position cannot be an observable (its operator is non-Hermitian). By this and other inconsistencies RQM was abandoned...

In QFT, the inconsistencies of RQM are partially solved (really ignored) by downgrading x to unobservable parameter, there is not x operator and no localization problems. Wavefunctions are reinterpreted as field operators and the whole theory is formulated in energy-momentum space.

In SHP theory x is maintained as an operator and time t is introduced as another operator, but this t is not the time of QM and this x is not the operator of RQM.

Lacking an adequate time, another concept of time tau is introduced as an evolution parameter. SHP theory has two times, tau plays the role of QM time (and is not an operator) and t is an operator associated to x^0.

SHP is not QFT, the Hamiltonian of SHP is not QFT, but a quadratic Hamiltonian and in general tau is not proper time as Srednicki says, because in general a Hamiltonian using time t as operator cannot be on-shell. The whole theory is very complex, redundant (multiple times), and full of inconsistencies.

bhobba

Without going through the math, or seeing a paper that does it for me I am in a bit of a pickle. But thinking about it in a superficial way I tend to side with what you are saying. It is however a bit of a concern when a standard textbook doesn't get it quite right -it leaves students like me in a rather awkward position.

Does anyone know a paper or other source that discusses this stuff?

I do however understand the issues of RQM and why you need QFT - tons of books explain the negative probabilities and other problems.

BTW thanks for going to the trouble of explaining whats going on.

Thanks
Bill

jimgraber

What is SHP theory, please?
Thanks

kith

Maybe you can answer one of my long-term questions. ;) A laser beam has clearly some kind of localization. How is this accounted for in quantum optics? How can position not be an observable, when I can clearly "observe" the spatial extent of a laser beam or detect photons in a small detector volume?

I don't say it is wrong, I just find it very hard to get the notion of photons together with what I know from non-relativistic QM. As far as I can tell, the experimental methods for detecting electrons and photons are quite similar. So it is kind of odd that a simple wavefunction interpretation is possible in one case and is not in the other.

juanrga

Stuckelberg, Horwitz, & Piron theory.

sheaf

Bialnycki-Birula gives a good overview of the problems of defining position operators for (and single particle wavefunctions for) photons.

kith

Thanks sheaf! This article explains quite a few things, but I don't know yet if it will answer one of my big questions.

atyy

Maybe relevant is a derivation of the LSZ formula for scattering that uses wave packets, which are very approximately localized.

http://www.phys.psu.edu/~collins/563/LSZ.pdf: "I will show a simple construction of suitable coordinate-space wave function ..."

the_pulp

Ok, the arguments for not considering time and position as operators sound convincing, but in the 4th post of this thread someone wrote a link to a paper which describes what it seems to be QFT from a point of view where time and position are things measurables in labs (what it shows that could be measurable is the probability to find certain amount of particles in certain positions during certain time intervals). This paper also sounds very convincing but yet, opposite to the idea that time and position is not an operator.
So who is right, the one who is here saying that position plays no role (dont remember your name, sorry), that paper in the 4th post, both? What am I missing?

strangerep

Do you mean Hendrik Van Hees' reference to the section on position on Arnold Neumaier's FAQ? Arnold mentions that, although a position operator is problematic for the photon, one can (apparently) define satisfactory POVMs (Positive Operator-Valued Measures) instead:

I haven't yet had time to chase down these references, but there seems to be good reason to consider POVMs to be more important in general quantum theory than one might think, compared to the usual Hermitian operators that one usually insists on for observables. In the finite-dim case, in turns out that every POVM arises from a set of generalized coherent states, which in turn arise from quite general considerations of dynamical groups.

If a covariant POVM for photon position is indeed satisfactory, then perhaps the whole puzzle about nonexistence of a photon position operator is moot.

(Kith: this might also be relevant to what you were asking.)

the_pulp

No, sorry, I was meaning this link:

http://xxx.lanl.gov/abs/0904.2287

This sounded very convincing in relation to measure the probability of find certain particles in certain positions during certain times.

So, after reading this paper I dont get why I should keep on saying that QFT and position measurements are not compatible. What am I missing?

Thanks for your help!

Avodyne

The formalism in which both space and time are operators is explained in string-theory books, and is summarized here:

http://www.physics.thetangentbundle....article/action

martinbn

The link doesn't explain that, it only talks about the classical action, there isn't even quantization yet. And the question was how it is in QFT, no?

Demystifier

True, but see
http://xxx.lanl.gov/abs/hep-th/0702060 [Found.Phys.39:1109-1138,2009]
especially Sec. 5.2.

juanrga

In the first place this link is not about anything discussed here.

In the second place string theory explains nothing, that is the reason which is named TON (Theory Of Nothing) these days...