Req: A conceptual QFT example

[pellman]

I need someone to point me towards an instance of a relatively simple QFT problem which illustrates what can be calculated from QFT, how it is calculated, and its physical significance. (Note the qualification "relative". I realize there are likely no simple problems in QFT.)

If someone were asking the same of me with regard to QM, I would take them to the standard examples in the texts: The potential well, the step-potential barrier, the harmonic oscillator and the ideal hydrogen model. If I were to single one out it would probably be the step-potential barrier, since it illustrates how a boundary value problem is solved and clearly gives two or more distinct relative probability regions while also exhibiting a non-classical behavior: "tunnelling".

I would also bring up the two-slit experiment since one can discuss all the quantum behaviors and easily visualize the significance of both amplitude and phase--without actually solving the math.

What QFT problem(s) can do this for beginner?

Edit: It should clarify the physical significance of a quantum field and (if possible) show how it is related to the physical entities we call particles.

 

[CompuChip]

First of all, obviously, free field theory (which can be "derived" from a classical spring system), and the simplest interacting theory, $\phi^4$-theory; this is for example how Zee's "Introduction to QFT" starts.

The simplest "physical" example I have seen is that of electron-electron scattering, which is ugly algebra but it is one of the things that in "just" a few pages actually calculates a quantity which experiments can measure. Peskin & Schroeder's "An introduction to QFT" starts with this in chapter 1 (although the actual calculation is only in chapter 5, or so).

Perhaps the most successful example of a QFT that we know is the Standard Model of particle physics, which has in (almost) every aspect turned out one of the greatest successes in theoretical physics of the last 50 years or so, and which we now actually trust enough to build a ~ 7 billion dollar experiment (the LHC) just to look for a particle which has so far only been predicted

 

[Demystifier]

QM is about systems in which the number of particles (of each kind) is fixed, so it cannot describe particle creation and destruction. This restriction of QM is removed by a more general theory, QFT.
Example:
electron-positron annihilation into a pair of photons

 

[CompuChip]

Actually Chapter 1 of Zee's book is freely available in PDF from the publishers website: http://press.princeton.edu/chapters/s7573.pdf

It very clearly introduces the need for QFT and makes the step from QM, I think you will find it useful.

 

[Vanadium 50]

If you are asking the question "why do we need this?", that is, what physical observables require more than just QM, I would suggest a) the Lamb shift and b) the electron's anomalous magnetic moment.