In this chapter we apply the path integral formalism developed in the previous chapter to the quantization of gauge theories. The simple case of an Abelian theory (QED) is treated first, and then we discuss the quantization of non-Abelian gauge theories and obtain the new Feynman rules for ghost lines and vertices in QCD. Ghosts are particles which violate the connection between spin and statistics (in this case they are scalar particles which obey Fermi statistics), and hence they cannot exist in initial or final states but only appear as virtual particles inside of loop diagrams (hence the name "ghost"). We will show that one of their roles in QCD is to maintain unitarity. We conclude the chapter with a discussion of the current theory of the electroweak interactions, referred to as the Standard Model.

The path integral formalism will now be used to quantize gauge field theories. The general results we obtain will first be applied to QED at the end of this section and to QCD in the next section.

We begin the discussion by considering the following generating function for the free electromagnetic field with source

Note that the source term is identical to a current interaction term, giving us the familiar identification of currents as sources of the electromagnetic field.

Recall from our discussion in Sec. 2.2 that quantization of the EM field presented a problem because dAo/dt was not contained in the Lagrangian, and hence A0 was not a dynamical variable. This problem was "solved" by choosing a gauge (the Coulomb gauge) in which A0 could be easily eliminated and the fields quantized. One constraint remained, the Coulomb gauge condition:

We have a similar problem with the generating function (15.1). The action is invariant under the gauge transformation A'^ — Afl + d^A (we assume the source current is conserved; d^ = 0). This means that the field can be separated into two parts:

where Aail are the "gauge" components which leave the action invariant and ADtl are the "dynamical" components upon which the action depends. Since the action is independent of the gauge components, they cannot be determined from the variational principle which gives the field equations and must be integrated out. One of the nice features of the path integral is that it is possible to express the integration over the gauge components as an overall factor which is independent of the dynamical components ADtl. This factor is infinite but has no effect on the dynamics because it is an overall multiplicative factor which can be absorbed into the normalization constant which is divided out when we evaluate propagators, S matrix elements, and other physical observables.

Before the gauge degrees of freedom can be separated from the dynamical degrees of freedom, we must define how the separation is to be made. This is done by imposing a constraint on the fields. The constraint, or gauge condition, defines the dynamical fields; all fields which satisfy the constraint are dynamical. The constraint is chosen so that fields which do not satisfy it differ from those which do by a gauge transformation which leaves the action invariant and therefore these additional fields are redundant. These fields which do not satisfy the constraint are gauge fields which must be integrated out.

An example, taken from Cheng and Li (1984), will illustrate this discussion.

'juppose we have a complex field, ip =

and an action A{ip*ip) which depends on ip*ip transformation only. The action is therefore invariant under the gauge

The path integral which determines the dynamics of this system is d6rdrelA{rQ)

where in the last expression we have integrated out the "gauge dependent" degrees of freedom by integrating over the redundant variable 0 and dropped this factor because it is a constant. While elimination of the redundant degrees of freedom was trivial in this case, it is good to have a systematic method for carrying out the separation in the general case, and one method is to insert unity, written in the following form:

494 QUANTUM CHROMODYNAMICS AND THE STANDARD MODEL into the original integral. This gives

Z = J d<t> J d6rdr6(6-<j>)elA{r3) = J dip Z*

= Z$ J dcj> —> , where the integral over <f> can be separated out because Z^ does not depend on <t> (gauge invariance). This method separates the integral into a "dynamical" part, in which the fields are specified by the "gauge condition" 6 = <j>, and a "gauge" part which includes the redundant dependence on the gauge angle 4>. The gauge group in this example is the group U(\) of multiplications by a complex phase, and the integral over <j> is an integral over all elements of the gauge group.

To prepare the way for application of these ideas to gauge theories, we will generalize the above example. A constraint which is more complicated that 6 — 4> might be chosen to define the dynamical fields. Such a constraint can be written in the form

Then, in place of Eq. (15.6) we have a more general result,

A ~1(r,0) = J d<t>6[F{r,6-<}>)) , where A can be evaluated directly, dF{r,9)

It can also be shown that A is gauge invariant, which for this example means that it is independent of 6:

(r, e + d>') = j d<t>6 [F{r, 6 + <p' - <p)}

The crucial step in this "proof" is the invariance of the measure for the group integration, referred to as a Hurwitz measure, which is expressed mathematically as f dcp = f d(<f> — <j>'). This is trivially true in this example.

Using these results, the path integral for the general constraint (15.8) can be written

where

Z+ = J dOrdr&{r,0)6[F(r,O-<p)} eiA(r^ . (15.13)

This new path integral includes the constraint and yet is still gauge invariant. To prove the latter, use the invariance of the group measure, of A, and of the action

= J d{0' -4> + 4>') rdr A(r, 9' - <j> + 4>')6 [F{r, 9' - <f>)} eiA^

= J d9'rdrA{r,9')S [F(r,6' - 0)] eiA(r^ = Z* . (15.14)

The path integral will now undergo one more transformation before it is in its final form. It is convenient to generalize the gauge fixing condition (15.8) by the more general constraint

and then average over all values of G using the following integral:

where q is a constant referred to as the gauge fixing parameter (to be discussed below). Since G is independent of all the other variables, A will be unchanged by this substitution, and discarding the unimportant constant in Eq. (15.16) the r>ath integral (15.13) can be replaced by

Z+ = J dG j dOrdr A(r, 0)6 [F(r, 6 - <f>) - G] e^^-^0')

This is the form we will employ in our discussion of gauge theories. Note the following features

• The gauge degrees of freedom have been removed from the field integration.

• The new integral includes the gauge constraint in two places: the effective action includes a gauge fixing factor of F2, and the integration measure includes the factor A. In spite of these factors, the overall expression is gauge invariant.

In the example we have been discussing, the initial gauge constraint was <f> = 0, which is equivalent to the function F — 9. In this simple case, A = 1, and the gauge fixing term reduces to a constant,

constant which can be dropped, showing that (15.17) reduces to (15.7).

We now return to our discussion of gauge theories. As discussed above, the gauge condition will be written in the following general form:

where F is some function or operator which depends on the components of AM = \\aA£ (where ^ An are the generators of the gauge transformation). Familiar choices are

In this chapter we will choose the Lorentz gauge because it is manifestly covariant.

The example presented above outlined how the path integral is to be constructed, but because of the greater complexity of the realistic problem, we will review the steps again here. This way of treating the gauge condition was invented by Faddeev and Popov [FP 67] and is referred to as the Faddeev-Popov trick. This trick is not required for the quantization of QED, but is useful in the quantization of QCD. The argument begins by considering the quantity

where dU is an invariant integration over the elements U of the gauge group, defined in general by the transformation

[recall Eq. (13.27)]. For an Abelian gauge group, AM = A^,

and we recover the familiar A^ = All + d^A(x). The invariant group integration (the Hurwitz measure) is defined by the requirement

for any fixed element U' in the group. The idea behind this statement is that the sum (integral) over all elements U of a finite (continuous) group is the same as the sum (integral) over all elements U'U, because multiplication by U' maps the group into itself. For an Abelian gauge transformation we may take

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