chapter 7 Path integral quantization of Gauge field and Faddeev-Popov Faddeev Popov ghost

7 1 Problem due to the infinite symmetry 7-1 In the modern field theory, the gauge field is introduced by requiring the invariance of the Lagrangian g g under the g gauge g transformation. We now try to perform the path integral over the gauge field; " (7.1) (7 1)

To do the path integral, one of the key assumptions is to assume the existence of the inverse of the kinetic operator, operator for example in (4.4) (4 4) which does have the inverse. For th F the gauge fi field, ld by b integrating i t ti by b parts t (7.1), (7 1) the th Lagrangian L i becomes

7 1 Problem due to the infinite symmetry 7-1 Hence, we need the inverse of the following operator, " (7.2) (7 2)

It is easy to see, however, (7.2) does have no inverse !! This hi difficulty diffi l arises i from f the h infinite i fi i degrees d off freedom f d generatedd by the gauge transformation. In order to avoid it, we demand the gauge fixing through the path integration as follows follows. To see how the infinity arises from the path integral, integral we will show simple example of the ordinary integration. Consider the integral] g ] which is invariant under the rotation (U(1) symmetry).

7 1 Problem due to the infinite symmetry 7-1 Using the polar coordinate, we find " (7.3) (7 3)

where ‘redundant factor’ due to the rotational symmetry is factored out. This h is what h we want to ddo ffor the h pathh integral. l (7.3) can be written as, " (7.4)

which is equivalent to (7.3). Here, we choose only θ=0 path. I Instead d off ddoing i that, h we consider id generalized li d non-zero θ path, h for f example To apply it to the integration (7.4), the path function is chosen as " (7.5)

7 1 Problem due to the infinite symmetry 7-1 Here, zeros of f are This gives, Thus, (7.4) is now becomes To compensate the additional factor, we define the new function, where " (7.6) (7 6)

In this case, the function f is simple rotation operation;

7 1 Problem due to the infinite symmetry 7-1 Here, (7.6) is rewritten as " (7.7) (7 7)

This equation seem to be the identity. Hence, (7.4) becomes

The function Δ plays a role to isolate the volume factor. We may write Thus, Th "((7.8))

7 2 Gauge fixing 7-2 We use similar method to remove the redundant gauge symmetry. The action is invariant under the g gauge g transformation " (7.9)

or, in i the h infinitesimal i fi i i l limit, li i " (7.10) (7 10)

To remove the infinite degrees of freedoms, we impose " (7.11)

which is called g gauge g fixing g condition. ((similar with ((7.4)) )) For example, the Lorentz gauge is given by

7 2 Gauge fixing 7-2 It is necessary to consider the function Δ which corresponds to (7.8). " (7.12) (7 12)

It includes the delta function (7.11), which is rigorously expressed defined at each space-time space time point over the functional integration of A. A ((7.12)) muse be g gauge g invariant. To see it we write " (7.13)

where h U’ represents t the th additional dditi l gauge transformation. t f ti N Now we use the result that the volume element in group space is not changed by the gauge transformation (gauge invariant measure), measure)

7 2 Gauge fixing 7-2 Thus, putting U’U=U’’, we can rewrite it as " (7.12) (7 12)

which indicates the function Δ is gauge invariant. We introduce " (7.13)

corresponding p g to ((7.6). ) Inserting g it into ((7.1), ), we obtain Then we carry out the gauge transformation

→

, we find " (7.14)

where the integration volume is factored out!!

7 2 Gauge fixing 7-2 It is clear in (7.14) that we can drop the infinite volume factor. We ggeneralize the g gauge g fixing g function in the following g discussion. Instead of the Lorentz gauge , we nay consider " (7.15)

where C(x) is the arbitrary function. (7.14) is modified as " (7.16) (7 16)

The generating functional is independent of C(x) apart from the normalization factors. Thus, we can add to (7.16). Here, α in the parameter. Therefore we can integrate out over the gauge field A(x),which Therefore, A(x) which will be shown in the next section.

7 3 Fadeev 7-3 Fadeev-Popov Popov ghost Remaining task is to deal with the function Δ. ( )), (7.8)),

By definition (see " (7.17)

which hi h is i the h Jacobian bi off the h gauge transformation. f i Using i the h formula for the Gauss integral for fermion field and putting i, we obtain

" (7.18)

Now we have introduced new fields η!! η are Grassmann fields, but behaves like spin-0 bosons. Using this technique, we carry out the functional integral

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost It is easy to find " (7.19) (7 19)

where N is the irrelevant understood d d as

normalization factor.

(7.19) can be

" (7.20)

where the second term of the effective Lagrangian g g is ‘gauge g g ffixingg term’, and the third term ‘Faddeev-Popov ghost term’. The ‘ghost’ η is spin-0 Grassmann field appears in the Feynman diagram to calculate the amplitude. However, the ‘unphysical’ ghost appears only the internal lime of the diagram to maintain the unitarity.

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost In the following discussion, we discuss examples to show how the gghost appears pp in the calculations,, " (7.19) (7 19) ex1: U( ex1 U(11) gauge theory in the Lorentz gauge The h gauge functions f i are given i b by Thus, the generating functional becomes " (7.20)

factored out The ghost term just contributes to the overall normalization, and does not couple with the gauge field. Hence, U(1) gauge theory like QED is free from the ghost field!! (otherwise, you would see the ghost in the text book of the electrodynamics.)

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost The propagator of the gauge boson is calculated with the field strength g tensor and the g gauge g fixing g term. Inverse off these h terms yields i ld " ((7.21))

α =1 1 : Feynman gauge α= 0 : Landau gauge

both covariant!

NB: Most of the text books, the Feynman gauge is adopted to calculate the scattering amplitude. Advantage of Landau choice: As shown in section 5, the inclusion of the interaction changes the propagator as

7-3 7 3 Faddeev Faddeev-Popov Popov ghost 1 p 2 − m 2 + iε

→

1 A( p 2 ) p 2 − B ( p 2 ) + iε

where A and B are obtained by the loop integrals. It is known, known however, however that the use of the Landau gauge always provides 2 A( p ) = 1

which greatly simplifies the calculations. ex2: Non ex2 Non--Abelian gauge theory in the Lorentz gauge In the Yang-Mills theory, we have the gauge functions

(a,b,c : indeces of the SU(n) gauge group)

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost Hence, the ghost part of the Lagrangian becomes

" (7.22)

This gives ‘ghost ghost propagator propagator’ and ‘ghost ghost-gauge gauge field coupling coupling’.

We must include all possible diagrams to calculate the amplitude.

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost In addition to them, we have the gauge field propagator and self coupling p g of the g gauge g fields for the non-Abelian theory. y

(Derive the above Feynman rule by yourself using the functional differentiation.!)

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost ex3 ex 3: Non Non--Abelian gauge theory in the axial gauge We show non-covariant ‘axial g gauge’ g as another example. p It is defined by " (7.23)

Hence, the h gauge fixing fi i term is i expressed d by b (Usually, the space-like vector is chosen as z-component unit vector (0,0,0,1). Hence, the gauge condition implies Az = 0 .) Th gauge functions The f ti are given i b by

The gghost term does not contain the coupling p g to the g gauge g field. Hence, the ghost field is decoupled with the gauge theory!!

7 3 Faddeev 7-3 Faddeev-Popov Popov ghost ex3 ex 3: Non Non--Abelian gauge theory in the axial gauge In the axial g gauge g we can forget g about the g ghost,, but we must ppay y the costs due to the loss of the covariance. Now the effective Lagrangian becomes which casts into This gives the inverse " (7.24) (7 24)