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The path integral for photons

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The path integral for photonsbased on S-57We will discuss the path integral for photons and the photon propagator more carefully using the Lorentz gauge:Problem: the matrix has zero eigenvalue and cannot be inverted.as in the case of scalar field we Fourier-transform to the momentum space:we shift integration variables so that mixed terms disappear... REVIEW256To see this, note:whereis a projection matrixand so the only allowed eigenvalues are 0 and +1Sinceit has one 0 and three +1 eigenvalues.REVIEW257component does not contribute to the quadratic term becauseand it doesn’t even contribute to the linear term becauseand so there is no reason to integrate over it; we define the path integral as integral over the remaining three basis vector; these are given bywhich is equivalent toLorentz gaugeWe can decompose the gauge field into components aligned along a set of linearly independent four-vectors, one of which is and then thisREVIEW258Within the subspace orthogonal to the projection matrix is simply the identity matrix and the inverse is straightforward; thus we get:going back to the position spacepropagator in the Lorentz gauge (Landau gauge)we can again neglect the term with momenta because the current is conserved and we obtain the propagator in the Feynman gauge:REVIEW259The path integral for nonabelian gauge theorybased on S-71Now we want to evaluate the path integral for nonabelian gauge theory:for U(1) gauge theory, the component of the gauge field parallel to the four-momentum did not appear in the action and so it should not be integrated over; since the U(1) gauge transformation is of the form , excluding the components parallel to removes the gauge redundancy in the path integral.nonabelian gauge transformation is nonlinear:260we have:we have to remove the gauge redundancy in a different way!for an infinitesimal transformation:or, in components:the covariant derivative in the adjoint representation (instead of that we have for the U(1) transformation)261Consider an ordinary integral of the form:the integral over y is redundantwe can simply drop it and define:this is how we dealt with gauge redundancy in the abelian caseor we can get the same result by inserting a delta function:this is what we are going to do for the nonabelian casethe argument of the delta function can be shifted by an arbitrary function of x262if is a unique solution of for fixed x, we can write:then we have:we dropped the abs. valuegeneralizing the result to an integral over n variables:263Now we translate this result to path integral over nonabelian gauge fields:i index now represents x and ax and y y G becomes the gauge fixing function:fixed, arbitrarily chosen function of xfor gauge we use:264let’s evaluate the functional derivative:and we find:Recall, the functional determinant can be written as a path integral over complex Grassmann variables:where:Faddeev-Popov ghosts265the ghost lagrangian can be further written as: we drop the total divergenceghost fields interact with the gauge field; however ghosts do not exist and we will see later (when we discuss the BRST symmetry) that the amplitude to produce them in any scattering process is zero. The only place they appear is in loops! Since they are Grassmann fields, a closed loop of ghost lines in a Feynman diagram comes with a minus sign!Comments:For abelian gauge theory and thus there is no interaction term for ghost fields; we can absorb its path integral into overall normalization.266At this point we have:fixed, arbitrarily chosen function of xThe path integral is independent of ! Thus we can multiply it by arbitrary functional of and perform a path integral over ; the result changes only the overall normalization of .we can multiply by:our final result is:integral over is trivialgauge fixing termnext time we will derive Feynman rules from this

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