[Physics] The Energy-Momentum Tensor and the Ward Identity


I have a question regarding a homework problem for my quantum field theory assignment.

For the purposes of the question, we can just assume the Lagrangian is that of a real scalar field:
$$\mathcal{L}=\frac{1}{2}\left( \partial _\mu \phi\right) ^2-\frac{1}{2}m^2\phi ^2$$.

The first part of the problem was to calculate the energy-momentum tensor corresponding to the translation symmetry $x\rightarrow x+a$. I obtained:
T_\mu ^\nu =(\partial _\mu \phi )(\partial ^\nu \phi )-\delta _\mu ^\nu \mathcal{L}.
So far, so good (at least I hope so).

The second part of the problem states: "Write the corresponding Ward Identity." What does he mean by this? I looked up "Ward Identity" in the index of our text (Peskin and Scroeder), but was unable to find something that seemed relevant to the problem. Where do I even begin?

Best Answer

The Noether current is a conserved quantity in classical field theory. In quantum theory, it is mirrored by the Ward identity. You may possibly find this formula looking into the derivation for Schwinger-Dyson equations.

It is derived in chp 22 of Srednicki's book: http://web.physics.ucsb.edu/~mark/qft.html.

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