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I've just started to learn anomalies in quantum field theories. I have a question.

  1. How to show that QED is free from vector current anomaly and what would happen if it were not? In other words, how can we show that $\partial_\mu j^\mu=0$ even at the quantum level?

As I understand, violation of current conservation will cause a violation of Ward identity. A violation of Ward identity is related to violation of unitarity.

  1. How does the unphysical photon polarization states appear in the theory through anomaly? And how do their appearance violate the unitarity of the theory?

  2. Why would the vector current anomaly be a problem in QED but not the chiral current anomaly? Don't we have to get rid of the axial current anomaly in QED?

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    $\begingroup$ The important point is that vector current and chiral current can not be conserved at the quantum level (i.e. by the regulator) simultaneously. You can choose to conserve chiral current, then you have to break the conservation of the vector current, vice versa. $\endgroup$
    – Meng Cheng
    Jul 19 '15 at 15:46
  • $\begingroup$ Ok. Suppose, I choose to conserve vector current but not the chiral current. Then we will have chiral anomaly. I understand. But I read that anomaly in a gauge theory is disturbing. Is that correct? If yes, then chiral anomaly must also be problematic for QED. Right? $\endgroup$
    – SRS
    Jul 19 '15 at 16:09
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    $\begingroup$ It's not, because there is no chiral gauge field in QED, only a classical global symmetry that must be broken at the level of regularization. In contrast, the gauge symmetry has to be preserved exactly in a gauge theory. in QED, the gauge field is only coupled to the vector current, not the chiral current. $\endgroup$
    – Meng Cheng
    Jul 19 '15 at 16:13
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    $\begingroup$ When is then cancellation of anomalies important? $\endgroup$
    – SRS
    Jul 19 '15 at 18:12
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    $\begingroup$ Cancellation of the anomaly associated with the gauge symmetry is of course important. $\endgroup$
    – Meng Cheng
    Jul 19 '15 at 19:31
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1. How can we show that $\partial\cdot j\equiv 0$ at the quantum level?

For example, by showing that the Ward Identity holds. It should be more or less clear that the WI holds if and only if $\partial\cdot j=0$. There are multiple proofs of the validity of the WI; some of them assume that $\partial\cdot j=0$, and some of them use a diagrammatic analysis to show that the WI holds perturbatively (and this is in fact how Ward originally derive the identity, cf. 78.182). It is a very complicated combinatiorial problem (you have to show inductively that an arbitrary diagram is zero when you take $\varepsilon^\mu\to k^\mu$), but it can be done. Once you have proven that the WI holds to all orders in perturbation theory, you can logically conclude that $\partial\cdot j\equiv 0$. For a diagrammatic discussion of the WI, see for example Bjorken & Drell, section 17.9. See also Itzykson and Zuber, section 7-1-3. For scalar QED see Schwartz, section 9.4.

Alternatively, you can also show that $\partial\cdot j=0$ by showing that the path integral measure is invariant (à la Fujikawa) under global phase rotations. This implies that the vector current is not anomalous.

2.a. How does the unphysical photon polarization states appear in the theory through anomaly?

Take your favourite proof that the WI implies that the unphysical states do not contribute to $S$ matrix elements, and reverse it: assume that $\partial \cdot j\neq 0$ to convince yourself that now the unphysical states do contribute to $S$ matrix elements. Alternatively, make up your own modified QED theory using a non-conserved current and check for yourself that scattering amplitudes are not $\xi$ independent.

2.b. And how do their appearance violate the unitarity of the theory?

Morally speaking, because unphysical polarisations have negative norm. If the physical Hilbert space contains negative-norm states, the whole paradigm of probability amplitudes breaks down.

3. Why would the vector current anomaly be a problem in QED but not the chiral current anomaly?

Because in pure QED the axial current is not coupled to a gauge field, and therefore its conservation is not fundamental to the quantum theory. The axial anomaly in pure QED would be nothing but a curiosity of the theory (a nice reminder that classically conserved current need not survive quantisation).

On the other hand, in QED the vector current is coupled to a gauge field, the photon field, and as such its conservation is crucial to the consistency of the theory: without it the WI fails, and therefore we lose unitarity (or covariance, depending on how you formulate the theory).

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