Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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?

share|improve this question
    
I'm not sure if I can wrap my head around this stuff well enough to explain it, but I would point you to the end of section 9.6 in P&S. It may be that the problem is asking for the Schwinger-Dyson equation for $T^{\nu}_{\mu}$. –  David Z Nov 1 '11 at 6:55
    
I think for translations $a^\mu$, your conserved current is just $j^{\mu}(x)=a_{\nu}T^{\mu\nu}(x)$. The Ward identity is then just the vanishing of the derivative of the VEV of time ordered products of this current and the field (googling "Ward identity for continous symmetry" should turn something up). –  twistor59 Nov 1 '11 at 8:10

2 Answers 2

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.

share|improve this answer

In quantum field theory, a Ward (Takahashi) identity is an identity between correlation functions that follows from the global or gauged symmetries of the theory, and which remains valid after renormalization. For more setail you may check wikipwdia.

share|improve this answer
6  
While technically correct, this doesn't seem particularly helpful... –  David Z Nov 1 '11 at 5:02
    
Also, it was copied directly from wikipedia.. –  BjornW Nov 1 '11 at 9:49
1  
I guess you love to see your name in print –  Physiks lover Nov 1 '11 at 17:45

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.