What makes background gauge field quantization work?

Question

[Again I am unsure as to whether this is appropriate for this site since this is again from standard graduate text-books and not research level. Please do not answer the question if you think that this will eventually get closed. Since otherwise the software doesn't allow me to delete questions once it has answers!]

• A priori the technique of quantization using the background gauge field seems to be a "mere" redefinition of the good old Lagrangian whereby one "artificially" seems to double the number of variables by writing all of the fileds like say the fermions $\psi$, the gauge field $A_\alpha^\mu$, the ghosts $\omega_\alpha$ and $\omega_\alpha ^*$ as a sum of two fields one with a prime (the quantum fluctuation) and one unprimed (the classical background) - like replace every $A_\alpha ^\mu$ by $A_\alpha ^\mu + A_\alpha^{\mu'}$. Till this step nothing seems to distinguish the primed from the unprimed fields.

• Then one defines the a new set of covariant derivatives where the the connection is defined only by the classical background gauge field (the unprimed fields) only.

Then one rewrites the Lgrangian in terms of these new covariant derivatives and these doubled the number of fields. But I would think that the Lagrangian is still exactly the same and nothing conceptually has changed.

Now two statements come in which seem to be somewhat "magical" and thats where it seems that the crux lies,

1. Firstly one says that it is possible to set the background fields such that all the unprimed gauge fields are constant and every other unprimed field is 0.

2. Then one says that the 1-loop contribution to the running of the coupling constant (and also the effective action?) is completely determined by looking at only that part of the orginal action which is quadratic in the primed fields ("quantum fluctuations"?)

I would like to understand why the above two steps are consistent and correct and how to understand them. Also if there is some larger philosophy from which this comes out and if there are generalizations of this.

{This technique and idea seems quite powerful since after one says the above it is almost routine calculation to get the beta-function of a $SU(N_c)$ Yang-Mill's theory with say $N_f$ flavours - and hence asymptotic freedom of QCD! - things which I would think are corner stones of physics! }

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Comments

For me this one is better since it talks about general ideas rather than minute details of a specific step in some derivation. As I see it the basic confusion lies in the distinction that does exist between the primes and unprimed fields above. The primed fields are quantum operators, whereas the unprimed ones are just regular numbers (in addition, both are functions of location). There is no doubling in this split, you just shift the original quantum operator by a number. The rest of the manipulations you mention are meant to choose those numbers conveniently, to simplify perturbation theory.

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Answer 1

One way to see the validity of the background field method (BFM) lies in the proof of the equivalence of the effective action calculated with the BFM to the standard effective action.

Let $\Gamma[v]$ be the effective action (Legendre transform of the connected generating function $W[J]$) where $v=v(J)=\frac{\delta W[J]}{\delta J}$ is the "classical" field generated by the sources $J$. If we modify the classical action by splitting the quantum fields into quantum + background, then the resultant modified effective action $\Gamma[v(j),V]$ now depends on both $v(J)$ and the background fields $V$. You can show (under reasonable assumptions) that $\Gamma[0, V] = \Gamma[V]$.

One of the cleaner and more general discussions of this is in section 3.1 of the 2007 thesis by Grasso: "Higher order contributions to the effective action of N = 2 and 4 supersymmetric Yang-Mills theories from heat kernel techniques in superspace". Here he considers the more general (non-linear) quantum-background splitting that is needed for $N=1$ supersymmetric gauge theories. References to the original literature can also be found in this thesis.

Also worth reading are the introductory papers by Abbott: "Introduction to the Background Field Method" and "The Background Field Method Beyond One Loop". Abbott et al also show that the S-matrix is the same when using the BFM: "The Background Field Method and the S Matrix". This holds irrespective of the gauge fixing, while the equivalence of the effective actions only holds if the same gauge is chosen.

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Note that the quantum-background splitting is not really a doubling of the number of fields, but only just a background dependent change of variables (in the simple case, just a shift) in the path integral.

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Comments

Can you elaborate on how the effective action sees this splitting? I guess this $J$ of yours couples to the gauge fields only. Then I guess the point is that the classical value of the gauge fields (..which I guess you are calling as $v(J)$..) is the same as the background part of the gauge fields which are being set to zero. And does this argument go through for the $J$'s that couple to the other fields since they too are being made to go through this split.

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