QFT Source Function: What is J Physically?

[NanakiXIII]

I just picked up a book on QFT and have already found myself stuck on something that appears to be quite elementary. When disturbing the field, the potential gets an added term with a source function $J$,

$$J(x, t) \phi(x).$$

I'm not quite getting what $J$ physically represents. It 'describes how the field is being disturbed', but how exactly? Also, why is the added term necessarily linear in $\phi$?

 

[The_Duck]

Are you perhaps reading Zee's "Quantum Field Theory in a Nutshell"? I just started on QFT with that; here's how I see it:

Zee talks about a mattress with the displacement of the mattress at a point x in spacetime phi(x). Then imagine that J(x) is a force field that acts on the mattress. Then the potential energy of the mattress in this force field would be -J(x)phi(x). Potential energy appears with a minus sign in the Lagrangian, so we add J(x)phi(x) to get the new Lagrangian.

 

[Haelfix]

Students have sometimes told me that the whole 1st chapter by Zee confuses them, which I can understand as its very loose.

So I always recommend going back to classical field theory, where a similar thing happens without all the 'quantum' confusion.

Quantum field theory is really an example of the correspondance principle at play

 

[haushofer]

First the mathematics: sources J are used in order to calculate n-point functions. That's why these source terms are linear in the field phi.

Diagrammatically you see that these source terms can be seen as "particles traveling from point x to point y" looking in coordinate space. This "traveling" is described by the propagator. That's the reason why peoply say that such a source term "disturbs the vacuum and introduces a source for a particle".

 

[vanhees71]

I don't like Zee's book as a starter precisely for the reason already made: it confuses more than it helps in understanding the quite difficult subject of quantum field theory (qft). It's fun to read if you already know quantum field theory from a more conventional source, where the concepts are worked out carefully.

A good book is, e.g., the textbook by Ryder or Bailin and Love. Also Peskin/Schroeder is quite useful, but sometimes a bit sloppy when it comes to the finer issues of renormalization and the renormalization group.

If you are a bit more familiar with qft from such introductory books, the best text-book source is Weinberg's Quantum Theory of Fields (2 volumes + 1 volume supersymmetric extensions) which derive the qft formalism very carefully and with much physical insight. Weinberg keeps his promise made in the preface to explain "why qft is the way it is". Just my 2 cts.

 

[NanakiXIII]

I'm indeed reading Zee's book and the first chapter so far has indeed been confusing, with the sections on Path Integrals being quite low on information and with his throwing Wick contractions at you seemingly at random in one of the appendices. This particular point really had me a little stuck, though, since Zee simply states it as "obvious".

So is $J$ really just a force field? If it is, it must be a forcefield that only acts perpendicularly to the mattress, right? I got the feeling Zee meant everything to happen perpendicularly to the mattress, but this isn't quite clear to me.

 

[element4]

You might get a better understanding of the $$J\phi$$-term by calculating the equations of motion. There you should see that in this sense, $$J$$ represents a current (external source).

It might get more clear if you coupled your theory to a electromagnetic field,
$$\mathcal L = \frac 12 D_{\mu}\phi D^{\mu}\phi - \frac 14F_{\mu\nu}F^{\mu\nu} + J_{\mu}A^{\mu}$$

where $$D = \partial_{\mu} - ieA_{\mu}$$ and $$F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$. Here I have added a source term linear in the gauge field $$A_{\mu}$$ rather than $$\phi$$, since this is more familiar. Now calculate the equations of motion wrt. $$A_\mu$$:

$$\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0.$$

You should get a term $$\partial_{\nu}F^{\nu}_{\mu}$$, which represents the Maxwells equations, a term depending on $$\phi$$ and a term which is just $$J_{\mu}$$. This means that the gauge field $$A_{\mu}$$ can get disturbed by fluctuations of $$\phi$$ (which vanishes for a charge less field $$e=0$$), and by $$J_{\mu}$$. This is nothing but an (external) source as known from electromagnetism $$J_{\mu} = (\rho, \mathbf J)$$.

Similar terms added to other fields (whether it's scalar, spinor, vector...) , should be understood as generalization of this idea. More practically, a source term is rarely used as something physical in QFT. It's considered as a mathematical trick to calculate correlation functions, as haushofer mentioned.