A Ghost Story-Gauge Fixing
The key question of this chapter is to figure out how to quantize a gauge theory. The first instinct is that, can we obtain the correlation function, or the propagator, as in scalar field theory? Unfortunately, the answer is no., at least not so trivial.
This means the gauge theory itself has way to many degeneracies so that its impossible to get a unique propagator alone from the Lagrangian . Geometrically, we have infinitely many gauge orbits in the space of fields. Every orbit represents the same classical solution of the theory, one has to choose a specific orbit then the quantization is consistent. That is,
The proof is just direct computation. Here, we utilize the concept of Lagrange multiplier method, the additional term does not affect classical solution as long as we choose the Lorentz gauge .
Now, we show that after choosing a special gauge fixing condition, the propagator is well-defined, and the presumably quantum theory. However, does this procedure depends on the choice of gauge fixing?
Faddeev-Popov Construction
From the path integral perspective, integrating over the space of gauge fields results in "unregularizable" infinite, since the gauge redundancy is truly uncountable infinite: at every point , we multiply a factor . In other words, suppose there is a formal Haar measure defined on , the reasonable path integral is of the form here is the pushforward measure of and where .
We first see how physicists utilize such ideas in practice, 8, 4.
Proof. Since is the (formal) Haar measure, we have, for all , , hence ◻
Our aim is the replace the integration domain by restriction to the hypersuface . This is doable by the following thoerem.
Proof. By definition, we have , then by gauge invariance, Note that the first integral has no The rest is to prove that It suffices to prove that is the (formal) Radon-Nikodym derivative .1
Fix a gauge field , we parametrize the neighborhood of the orbit through by , is a basis of , and transverse coordinate on the slice. Locally we write with and . Consider the variation of with respect to , say where is the linear map with the components Then we have the identity and hence (up to normalization). We conclude . ◻
Now we conclude that
Now we express the gauge invariant partition function in the form of functional integral on and a geometric interpretation on the slice of gauge orbit . For computation purpose, we wish to make the integral more convenient to evaluate. This means we have to rewrite the Faddeev-Popov determinant and the Dirac delta distribution.
We can, in fact immediately, deduce the following conclusion.
Proof. See the rigorous finite-dimensional version proof of the theorem from Theorem 4.1.4, 5. ◻
No Ghosts in QED
In this section, we will attempt to give a formalism on quantum electrodynamics as a simplified model of quantum gauge theory with ghosts. We follow the derivation of Chapter 12, 4.
Proof. Note that . We have because the integral is independent of the choice of the gauge-fixing condition. Then is independent of , we can consider up to normalization. Thus Expanding with ghost fields and we have the result. ◻
Proof. It suffices to prove that the Faddeev-Popov determinant is independent of the gauge field. Recall the gauge transformation of -gauge field is for some smooth function . Then ◻
The results shows that the Faddeev-Popov determinant is a functional determinant of Laplacian operator. This means the ghost action is a free field theory without coupling and interactions with matter fields. We conclude that QED is a no ghost gauge theory.
Finite-dimensional Justification
In this section, we give a finite-dimensinoal justification of what we done in the Faddeev-Popov procedure. To set up, we consider the following.
A (compact) Lie group action on some (affine) -vector space such that is a trivial principal -bundle, i.e. .
The partition function is defined as in , i.e. and the notations are given as above.
The gauge-fixing condition is the same except now we consider the level set intersects every -orbit transversally -times for some fix . The exact value of depends on the topology of and (5).
Now the integral on the quotient can be rewritten as where is the Dirac delta distribution pull-backed by . Note that for a -cycle with , we have the distributional form defined by We also denote this by for computational convenience.
Proof of the Lemma. Note that the non-degeneracy of is equivalent to intersecting the -orbit through transversally. If the intersection is non-transversal, then and the statement is trivial.
Let be the tangent space to the -orbit through . Consider the annihilator subspace and its basis, say , . Then is a basis of . W.L.O.G. we may assume By orthogonality of ’s and ’s we have, recall that , Wedging with ’s we conclude the result. ◻
Now the following equation is meaningful.
In fact, it should be .↩︎