How to interpret the field function Φ in QFT?

jfy4

Is there a good QFT text that deal primarily in the wave functional approach, versus the fock space approach? I would be interested in reading more about using the wave functional.

A. Neumaier

I haven't seen it in a textbook (though I haven't looked at many). But search for terms such as ''multifingered time'', ''Tomonaga-Schwinger'', ''functional Schrodinger equation'', and you'll find plenty of journal references.

Tomonaga and Schwinger developed this approach to QED at the same time as Feynman developed the path integral approach. All three were shown to be equivalent by Dyson in 1948. But Feynamn's approach turned out to be easier to teach and was aslo computationally less demanding. This is why textbooks usually concentrate on the latter.

In classical general relativity, multifingered time is just an expression of the fact that one can prescribe initial conditions on any maximal spacelike hypersurface, solve the field equations, and look at the result at any ''later'' spacelike hypersurface. See, e.g., Section 21.3 (p.497) in the well-known book Gravitation by Misner, Thorne and Wheeler.

But the same is possible in any Lorentz invariant classical theory, though much less useful in the flat case.

For multifingered time in quantum gravity, see, e.g.,
http://arxiv.org/pdf/1209.0065, Section 2B

jfy4

Thanks Arnold.

mpv_plate

When reading about the basics of QFT I found there is a so called "density operator" which gives the particle density at a given location. It is the combination of annihilation and creation operator in the position space.

Can the density operator be understood as another possible answer? It basically tells where the particles are in the space. When I use the scattering theory it seems I get similar answer: where the particles go (spatially) after they interact (if I understand that correctly). Is the density operator used in the scattering theory?

tom.stoer

Can you please provide a defintion or a reference for this density operator? it seems that I have something in mind which does not allow for such an interpretation

Sonderval

The wave functuional is very clearly explained in the book by Hatfield "Quantum Field Theory of particles and strings" - was quite a revelation to me this summer, when I got hold of it.
If you can read german, you can find a rather simple explanation of what the concpet behind it are on my blog, where I have a series on QFT. The wave functional is explained here:
http://scienceblogs.de/hier-wohnen-d...stehen-nichts/

Demystifier

I think the book by Hatfield "QFT of point particles and strings", mentioned also by Sonderval above, is exactly what you need. In this book many problems are solved in parallel by 3 methods: wave functional, Fock space, and path integrals.

Demystifier

One general comment.

Many people seem to mix three logically different concepts:
1. QFT per se
2. Interacting QFT
3. Relativistic QFT

For example, they often say that "the number of particles is not constant in QFT", but this is true only for interacting QFT.

Or they say that "the particle position operator is not well defined in QFT", but such a claim makes certain sense only in relativistic QFT, because the particle space-position operator turns out to be not Lorentz invariant.

Or sometimes they say that "the number of particles is not well defined in QFT", but it may have at least three different meanings.

- One possible meaning is that the number of particles changes with time, which, as I already said, is true only in interacting QFT.

- Another possible meaning is that the number of particles may be uncertain, i.e., the quantum state does not need to be a particle-number eigenstate. This is a genuine property of QFT per se, valid even for free and/or non-relativistic QFT. But it should be stressed that it does NOT imply that the OPERATOR of the number of particles is not well defined.

- Yet another meaning is that the OPERATOR of the number of particles may not be well defined, but this occurs only in attempts to combine QFT with GENERAL relativity. Namely, the operator of the number of particles may not be invariant under general coordinate transformations, even though it is invariant under Lorentz transformations.

I hope these notes will reduce confusion stemming from fails to distinguish the different concepts above.

jfy4

Thank you Demyistifier, I'll check it out.

A. Neumaier

Hatfield's book is actually called ''Quantum Field Theory of Point Particles and Strings''.

For a commentary on Hatfield's book, see Forrester's Winter 08 Lecture Notes http://aforrester.bol.ucla.edu/educate.php ,
starting with week 4.

mpv_plate

The density operator is described for example here. See the expression (54) on page 6.

It seems to describe how particles are distributed in the space.

tom.stoer

OK, I see. You are right, interpreting this expression in QM it corresponds to a density (like a charge density) in space.

I would never call this density a "density operator" b/c a density operator is already defined in non-rel. QM and its meaning is something totally different

andrien

Where else can it be used.

A. Neumaier

The correct name is (operator-valued) density field.

A. Neumaier

Yes, the expectation of the density field multiplied by the mass is what becomes in the case of macroscopically many particles the thermodynamical mass density, and tells about where the mass is concentrated.