Why field redefinitions that leave Lagrangians unchanged are allowed?

Question

This may be a stupid question but I will ask it because I am not fully happy.

In any quantum field theory, we start with a couple of fields $\phi_1,\phi_2,...$, and a Lagrangian involving various combinations of these fields and their derivatives consistent with the symmetries of the theory. Now, I often see that people re-define the fields keeping the Lagrangian unchanged to argue how certain couplings can be made real. For example, see this post.

So it is fine to do such redefinitions, if the Lagrangian is left unchanged. What is the assurance that by doing such redefinitions I am not losing any physics (e.g. possibility of CP violation)?

What does this freedom of redefinitions mean? Is it like "we don't know whether the field $\psi$ destroys an electron or the field $e^{i\alpha}\psi$, and it doesn't matter which one I work with."?

Am I only allowed to do $\phi\to e^{i\alpha}\phi$ type redefinitions?

Answer 1

Field redefinitions come in two types: the first is due to the equations of motion and is typically used in Effective field theory to reduce the number of terms at a certain order; the second type is closely related to the gauge invariance of the system and I think that is what you are most interested in here.

If I perform a field redefinition of the form $\psi\rightarrow e^{i\alpha}\psi$, and the physical equations of motion remain the same then I have a global transformation which leaves my system invariant and via Noether's theorem this means I have a conserved quantity. A field redef is not a redef if it changes the physical outcome of the system, so if you were to perform a redef that got rid of CPT conservation or that prevented CP violation in the electro-weak theory then you wouldn't have actually performed a field redef, instead you have just change the problem all together.

So, in general we use field redefs to make some part of the physics more clear, or to reveal some conserved quantity within the system. There are of course many other uses of field redefs and for the most part they just make the maths easier, especially in EFTs since we can typically get rid of $\Box\phi$ terms in the EFT expansion.

I hope this clears things up for you, if not then just let me know, I'm happy to provide some examples etc.

Answer 2

The quantum field $\phi$ is not a measurable quantity. It's a crutch you use to calculate physical observables like cross sections. So there's nothing that prevents you from redefining your field, as far as physically relevant quantities remain unchanged, and if it helps you to simplify computations.

You can redefine $\phi \to f(\phi)$ for any function $f$, and the S-matrix (the physically relevant quantity) remains unchanged. You need to ensure that $f'(0)=1$ so that the 'new' field is still a canonically normalized free field far away ($t\to \pm \infty$).