chapter 7 Path integral quantization of Gauge field and Faddeev-Popov ghost 7-1 Problem due to the infinite symmetry In the modern field theory, the gauge field is introduced by requiring the invariance of the Lagggrangian under the gggauge transformation. We now try to perform the path integral over the gauge field; "(7. 1) To do the path integral, one of the key assumptions is to assume the existence of the inverse of the kinetic operator , for example in (4 .4) which does have the inverse. FthFor the gauge fildbittibfield, by integrating by part t(71)thLs (7.1), the Lagrangi an becomes 7-1 Problem due to the infinite symmetry Hence, we need the inverse of the following operator, "(7. 2) It is easy to see, however, (7.2) does have no inverse !! This difficul ty ari ses f rom th e i nfi ni te d egrees of f reed om generated by the gauge transformation. In order to avoid it, we demand the gauge fixing through the path integration as follows. To see how the infinity arises from the path integral , we will show simple example of the ordinary integration. Consider the integg]ral] which is invariant under the rotation (U(1) symmetry). 7-1 Problem due to the infinite symmetry Using the polar coordinate, we find "(7. 3) where ‘redundant factor’ due to the rotational symmetry is factored out. Thhhis is what we want to dfhhdo for the path integra l. (7.3) can be written as, "(7.4) which is equivalent to (7.3). Here, we choose only θ=0 path. Instea d of do ing t hat, we cons ider generali zed non-zero θ pathfh, for 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 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) In this case, the function f is simple rotation operation; 7-1 Problem due to the infinite symmetry Here, (7.6) is rewritten as "(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, "(()7.8) 7-2 Gauge fixing We use similar method to remove the redundant gauge symmetry. The action is invariant under the gggauge transformation "(7.9) or, inthe ifiiinfinitesi mal lim it, "(7.10) To remove the infinite degrees of freedoms, we impose "(7.11) which is called gggauge fixing condition. (similar with (7.4)) For example, the Lorentz gauge is given by 7-2 Gauge fixing It is necessary to consider the function Δ which corresponds to (7.8). "(7.12) It includes the delta function (7.11), which is rigorously expressed defined at each space-time point over the functional integration of A. (7.12) muse be gggauge invariant. To see it we write "(7.13) where U’ represents theadditiona l gauge tftransforma tion. Now we use the result that the volume element in group space is not changed by the gauge transformation (gauge invariant measure), 7-2 Gauge fixing Thus, putting U’U=U’’, we can rewrite it as "(7.12) which indicates the function Δ is gauge invariant. We introduce "(7.13) corresppgonding to (7.6). Inserting it into (7.1), we obtain Then we carry out the gauge transformation → ,wefind "(7.14) where the integration volume is factored out!! 7-2 Gauge fixing It is clear in (7.14) that we can drop the infinite volume factor. We generalize the gggauge fixing function in the following 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) 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 will be shown in the next section. 7-3 Fadeev-Popov ghost Remaining task is to deal with the function Δ. By definition (see (7.8)), "(7.17) which isthe Jacobian of thegaugetransfiformation. Using the 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-Popov ghost It is easy to find "(7. 19) where N is the irrelevant normalization factor. (7.19) can be unddderstood as "(7.20) where the second term of the effective Lagggrangian is ‘gggauge ffgixing 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-Popov ghost In the following discussion, we discuss examples to show how the ghost appears in the calculations, "(7. 19) exex11::U(U(11))gauge theory in the Lorentz gauge The gauge fifunctions are given 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-Popov ghost The propagator of the gauge boson is calculated with the field strength tensor and the gggauge fixing term. Inverse of these terms yildields "(()7.21) α =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 Faddeev-Popov ghost 11 → pmi22− +++εε AppBpi() 22− () 2+ where A and B are obtained by the loop integrals. It is known, however, that the use of the Landau gauge always provides Ap()12 = which greatly simplifies the calculations. 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-Popov ghost Hence, the ghost part of the Lagrangian becomes "(7.22) This gives ‘ghost propagator’ and ‘ghost-gauge field coupling’. We must include all possible diagrams to calculate the amplitude. 7-3 Faddeev-Popov ghost In addition to them, we have the gauge field propagator and self couppgling of the gggauge fields for the non-Abelian theory. (Derive the above Feynman rule by yourself using the functional differentiation.!) 7-3 Faddeev-Popov ghost exex33::NonNon--AbelianAbeliangauge theory in the axial gauge We show non-covariant ‘axial gggauge’ as another example. It is defined by "(7.23) Hence, the gauge fix ing term is expressed by (Usually, the space-like vector is chosen as z-component unit vector (0,0,0,1). Hence, the gauge condition impliesAz = 0 .) The gauge ftifunctions are given by The ghost term does not contain the couppgling to the gggaugefield. Hence, the ghost field is decoupled with the gauge theory!! 7-3 Faddeev-Popov ghost exex33::NonNon--AbelianAbeliangauge theory in the axial gauge In the axial gggaugewecanforget about the gg,host, but we must ppyay the costs due to the loss of the covariance. Now the effective Lagrangian becomes which casts into This gives the inverse "(7. 24).