Green Function generating functional and Fourier transform spaces

Asked Apr 10 '17 at 13:59

Active Apr 10 '17 at 23:53

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I am given the transform of the generating functional for free Klein-Gordon theory,

$$Z[J]=N\int D\phi \, e^{i\int d^4 J(x)\phi(x)}\tilde{Z}[\phi]$$ where $\phi(x)$ is a scalar field. I'm a little confused on the meaning of the source density function $J(x)$; I know that it is an arbitrary function, but what I see in this transform is that I switch from the space of scalar fields to the one of the source. But then, I would go from a physical space to a non-physical one: what meaning does this have? Are they conjugated quantities just like position and momentum?

I fully understand conventional generating functionals.

edited Apr 10 '17 at 23:53

asked Apr 10 '17 at 13:59

PsycoPulcino

2

Can you indicate in your question that you fully understand conventional generating functions, before moving on to QFT? – Cosmas Zachos Apr 10 '17 at 14:31
well, is it ok now? – PsycoPulcino Apr 10 '17 at 23:54
OK, then just as n derivatives with respect to x of the EGF evaluated at x =0 yield the nth term of the sequence, generalization of x to an infinite dimensional vector J yields the n-point field moment Green's function for the QFT when evaluated at J =0. Yes, fields and sources are generalized fourier conjugates. – Cosmas Zachos Apr 11 '17 at 00:40
Related 249307. – Cosmas Zachos Apr 11 '17 at 00:43
Ok thank you very much :) but i still don't get why a physical quantity is conjugated to a non-physical one – PsycoPulcino Apr 11 '17 at 11:08
It happens all the time to "physical" moments in probability theory, doesn't it? Have you taken statistical mechanics? This is what it runs on. "Physical" is a highly dangerous confusion-generating polemical adjective. – Cosmas Zachos Apr 11 '17 at 13:31

Green Function generating functional and Fourier transform spaces

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