If you work in the path integral formalism, you realize that you have to deal with *two* issues: the classical action and its symmetries-and the measure of integration over the field configurations. It's this latter that is sensitive to field configurations that are not classical, i.e. don't satisfy the classical equations of motion. When you perform a variation of the fields, this variation is a classical symmetry if the action is invariant (the Lagrangian becomes at worst a total derivative). To ensure that this variation is a ``quantum symmetry'', you must check that the measure remains invariant, too, so that the partition function remains invariant-that the Jacobian of the transformation is a quantity that is non-zero and doesn't depend on the field configuration. An anomaly means that the measure isn't invariant-so, while you're changing variables in an integral, you end up with a quantity that depends on them, so, in fact, you're ``missing'' field configurations, which is a way of saying your theory isn't unitary. To make it unitary, you must include the Jacobian.

If you work in the path integral formalism, you realize that you have to deal with *two* issues: the classical action and its symmetries-and the measure of integration over the field configurations. It's this latter that is sensitive to field configurations that are not classical, i.e. don't satisfy the classical equations of motion. When you perform a variation of the fields, this variation is a classical symmetry if the action is invariant (the Lagrangian becomes at worst a total derivative). To ensure that this variation is a ``quantum symmetry'', you must check that the measure remains invariant, too, so that the partition function remains invariant-that the Jacobian of the transformation is a quantity that is non-zero and doesn't depend on the field configuration. An anomaly means that the measure isn't invariant-so, while you're changing variables in an integral, you end up with a quantity that depends on them, so, in fact, you're ``missing'' field configurations, which is a way of saying your theory isn't unitary. To make it unitary, you must include the Jacobian.

Stam Nicolis said "...you end up with a quantity that depends on them, so, in fact, you're ``missing'' field configurations, which is a way of saying your theory isn't unitary." This is interesting could you possibly expand on this last part?

I just added a phrase to my answer :) , but let me elaborate. You start with a partition function, an integral over all field configurations of exp(-S). S is the *classical* action.

You've found that phi->phi+deltaphi is a (classical) symmetry, delta S = (delta S/deltaphi)deltaphi=0. This transformation is a change of variables in the partition function: phi+deltaphi=psi. So your partition function over the psi has an extra term, namely the Jacobian, J=|det(deltapsi/deltaphi)|. If this is a nonzero constant, you can take it out of the integral. If it isn't a constant, but depends on the fields, it will enter in the action as Tr(log J) and will contribute to amplitudes. If its contribution is non-local, then the theory has a problem-then you need to introduce additional fields, whose contribution will cancel the contribution of this term and give you a local action.

I see yes I have performed this calculation but I am trying to get a physical intuition. Let me attempt a hand wavey explanation of why quantum mechanics spoils the classical symmetry. In classical physics you have 1 path in the path integral: the path that minimizes the action. If you have a symmetry of the action you have a symmetry of your theory. When things become quantum the fields follow every possible path from a to b with a weighting given by the exponential of the negative action (neglecting normalization). The weighting is invariant to you change in the case of anomalously violated symmetries but the paths aren't. The new set of paths doesn't cover every possible path - some of them are missing. I'm not sure why this is a problem but you said it means your theory isn't unitary? You then calculate the jacobian you acquire from transforming the measure and that gives you the anomaly which is a nonlocal function of the fields (it depends on the topology of your fields). You add to your Lagrangian to keep things local. Is this right?

You always sum over all paths-the issue is with what weight. If your transformation does leave behind a Jacobian and you don't include it, or include it incorrectly, then you'll find unphysical effects, for instance, in field theory, where you do need to regularize the expressions, else you have ambiguities, you'll find that the result will depend on the regulator, i.e. on how you count configurations. Finally the assertion about topology of the field configurations, while appealing, must be qualified by what it means: it assumes certain regularity properties-and, in the path integral one is summing over all possible configurations, so one must show that the configurations that are not as smooth as one likes don't, in fact, contribute, if one wants to appeal to topology. The non-locality means it contributes an infinite number of derivatives; but not all anomalies lead to such terms: their contribution can be a term with a finite number of derivatives, that's a surface term-as is the case of a theta term, for instance.

The bottom line is that you have to deal with the definition of the Jacobian of the transformation: you can view its contribution, either as a change in the measure or as a change in the action, signifying that the weight of a configuration in the amplitude must be modified-you thought it was exp(-S[phi]), in reality it's

exp(-S[phi]+Tr(log(J)) ). If this contribution has an imaginary part, this indicates a vacuum instability. So here's the ``physical effect'' you were looking for.