defeated by quantum field theory yet again

tom.stoer

as I said in post #5: we can, but you don't find it in many text books (unfortunately)
b/c it's too complicated

Sonderval

"what do particle fields represent"
As I told you - for the photon, the field A is the vector potential.

"why don't we just use good old kets and bras for QFT?"
because - again as I told you - the state in QFT is a wave functional (a wave function of functions), making dealing with it awkward. You can do it - I explained in my post before how for a vacuum and 1-particle state. See the book by Hatfield "QFT oof point particles and strings" (I seem to recommend that a lot...)

"why can't we just solve for a bunch of eigen fields from the sources and then just superimpose them"
Because in the end what we are most interested in are scattering processes and for those that would be difficult to do.

tom.stoer

Sorry to say that but scattering processes are only one sector of QFT; non-perturbative effects like, chiral symmetry breaking, quark condensates, color confinement, QCD bound states, their masses, their form factors etc. are likewise interesting - and there you will a lot of work based on the canonical formalism not using path integrals.

Sonderval

tom is of course right - I was just thinking about path integrals because this is where we started.
Stupid side question, though: Things like QCD bound states can be calculated with lattice gauge theory, which in essence is nothing but numerically solving a path integral, isn't it?

tom.stoer

Not a stupid question. Yes, you are right, lattice gauge theory is based on path integrals; but people use canonical formulations as well, especially to address the more funmdamental questions like e.g. the mechanism behind confinement

HomogenousCow

What do non force mediator fields represent? like the electron field in The QED lagrangian

Sonderval

They represent the electron field (you probably guessed *that*...)

The most intuitive way to understand them may be by their relation to the charge density or current. See for example this page, section 3.7:
http://www.quantumfieldtheory.info/Chap03.pdf
(Be warned though that Bob Klauber has an idosyncratic way of looking at many things which is not standard - nevertheless this text helped me a lot in understanding parts of QFT)

@tom
Thanks - I was just confused because you seemed to imply that path integral always implies perturation theory.

Sonderval

They represent the electron field (you probably guessed *that*...)

The most intuitive way to understand them may be by their relation to the charge density or current. See for example this page, section 3.7:
http://www.quantumfieldtheory.info/Chap03.pdf
(Be warned though that Bob Klauber has an idosyncratic way of looking at many things which is not standard - nevertheless this text helped me a lot in understanding parts of QFT)

@tom
Thanks - I was just confused because you seemed to imply that path integral always implies perturation theory.

A. Neumaier

The path integral in qwuantum mechanics is (unlike that in quantum field theory) perfectly well-defined mathematically.

The Feynman integral book by Johnson and Lapidus
http://tocs.ulb.tu-darmstadt.de/110841727.pdf
tries to do everything in QM with the path integral!

The book ''Feynman integral calculus'' by Smirnov
http://www.amazon.com/gp/search?inde...rds=3540306102
is a textbook, and has problems and solutions.

HomogenousCow

I followed Sondervals link and found it to be a better text, until the third chapter that is
I find the section very confusing, I accepted that the solutions to the Klein Gordon equation are operators ( even though I had no idea how they would operate on states), he then goes on in length about the a and b operators ( and their conjugates), I felt fine with that, a killed off a particle and b killed off an antiparticle, but then he dosen't actually explain why we had to solve for them in the first place, he does not actually make use of the original solutions at any point.
Futhermore how do I actually get results that I used to get in quantum mechanics, for example for non interacting particles how do I calculate the position probability density for a state etc etc???

A. Neumaier

The position representation is very nonrelativistic, as it singles out a time coordinate.
Almost nobody works in the relativisitc domain in a position representation.

Relativistic multiparticle theory is almost always done as field theory, with the free case treated as a warm-up. So one starts with what is familiar from the nonrelativistic regime, to build up motivation for the relativistic field treatment. Afterwards forget the Klein Gordon equation - it is useless and superseded.

Indeed, if you look at Weinberg's QFT book, he builds the fields from scratch, using unitary irreducible representations of the Poincare group as a start rather than free classical fields. This also has the advantage that it works for any spin.

meopemuk

Hi HomogeneousCow,

your questions are very important and relevant. You can find some of them asked and answered in http://arxiv.org/abs/physics/0504062

Cheers.
Eugene.