# Does a symmetry of Lagrangian reserve in each Feynman diagram?

1. Dec 3, 2011

### ndung200790

Does a symmetry of Lagrangian be reserved in each Feynman diagram of perturbative QFT,because even Ward Identity still deduces from U(1) symmetry that we consider each diagram has?.

By the way, does effective action reserve the symmetry that Lagrangian has?.

Thank you very much for your kind helping.

2. Dec 4, 2011

### ndung200790

Must antiparticle and particle be created or annihilated in a couple of antiparticle and particle?If this happens,I think the Feynman diagrams would reserve the unitary symmetries of Lagrangian(this maybe being deduced from functional integral formalism for Green functions).

3. Dec 4, 2011

### tom.stoer

Symmetries in the path integral formalism do exist on the level of the effective action or matrix elements (e.g. Ward identities, Slavnov-Taylor identities). Both quantum corrections (loops) and especially the PI measure have to be taken into account.

It is not always possible to lift a classical symmetry to a quantum symmetry; breakdown of a classical symmetry during quantization is called an anomaly. A famous example, the axial (or chiral) anomaly which is generated by the non-invariance of the PI measure.

In canonical quantization a symmetry is represented via the (closure of the corresponding) operator algebra (derived from the classical algebra in phase space) after regularization.

4. Dec 4, 2011

### ndung200790

What does ''PI measure'' mean?

5. Dec 4, 2011

### ndung200790

And ''the anomaly'' would be the ''intermediate thing'' during quantization process,but it finishes its role after quantization.Is this correct?

6. Dec 4, 2011

### tom.stoer

PI = path integral measure in the space of fields.

I don't understand.

What happens is that the classical equation for current conservation

$$\partial_\mu j^\mu = 0$$

is replaced by something like

$$\partial_\mu j^\mu = \mathcal{O}$$

where both l.h.s. and r.h.s. are to be understood as marix elements and where the r.h.s. contains an operator that does not vanish globally on Hilbert space.

Check e.g. http://en.wikipedia.org/wiki/Chiral_anomaly as a starting point

Last edited: Dec 4, 2011
7. Dec 4, 2011

### ndung200790

Thank Mr Tom.Stoer very much for your teaching.

8. Dec 4, 2011

### ndung200790

Please teach me once more :
So, wouldn't there be anomaly in the case of Ward Identity and Slavnov Identity(meaning the U(1) and SU(3) symmetry would be reserved at quantum level because we consider both loops and PI measure)?

9. Dec 4, 2011

### ndung200790

At the moment,I have heard that in case of Ward Identity,there exist anomaly.So,where is the meaning of Ward Identity at quantum level.

10. Dec 5, 2011

### tom.stoer

Of course it depends on the symmetry you are investigating. There is an anomaly in all UA(1) symmetries in the SM where the axial current is constructed like

$$j^\mu_5 = \bar{\psi}\gamma^\mu\gamma_5\psi$$

That means that the Ward identity for the axial current is violated.