Supergravity and Gamma Matrices - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-08-19T11:58:27Z https://physics.stackexchange.com/feeds/question/484055 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/484055 2 Supergravity and Gamma Matrices huntercallum https://physics.stackexchange.com/users/230990 2019-06-03T13:34:25Z 2019-06-03T16:34:41Z <p>On page 101 of Freedman and Van Proeyen's book on Supergravity they find the propagator of the gravitino, however I'm not sure how to work through the steps in (5.30), and hints or answers would be great help.</p> <p>It begins with the equation of motion</p> <p><span class="math-container">$$\gamma^{\mu\nu\rho}\partial_\nu\Psi_\rho=J^\mu\tag{5.18}$$</span></p> <p>and an ansatz solution of the form,</p> <p><span class="math-container">$$\Psi_\mu(x)=-\int d^DyS_{\mu\nu}(x-y)J^\nu(y).\tag{5.26}$$</span></p> <p>Subbing this into the equation of motion gives,</p> <p><span class="math-container">$$i\gamma^{\mu\sigma\rho}p_\sigma S_{\rho\nu}(p)=i{\delta}{^\mu_\nu}-ip_\nu\Omega^\mu(p) \tag{5.28}$$</span></p> <p>in momentum space where <span class="math-container">$\Omega^\mu$</span> is a pure gauge term and contains all the dependence on <span class="math-container">$p_\mu$</span> only. They then go on to produce an ansatz for this equation of the form,</p> <p><span class="math-container">$$\require{cancel}iS_{\rho\nu}(p)=A(p^2)\eta_\rho\nu\cancel{p}+B(p^2)\gamma_\rho\cancel{p}\gamma_\nu\tag{5.29}$$</span></p> <p>which they then sub into the LHS of <span class="math-container">$(5.28)$</span> to obtain the following,</p> <p><span class="math-container">\begin{align} i\gamma^{\mu\sigma\rho}p_\sigma S_{\rho\nu}(p)&amp;=A\gamma^{\mu\sigma}_\nu\cancel{p}p_\sigma+(D-2)B\gamma^{\mu\sigma}\cancel{p}\gamma_\nu p_\sigma\\ &amp;=A(p^\mu\gamma^\sigma_\nu-p^\sigma\gamma^\mu_\nu)p_\sigma+(D-2)B(-p^\mu\gamma^\sigma+p^\sigma\gamma^\mu)\gamma_\nu p_\sigma+...\\ &amp;=[A-(D-2)B](p^\mu\gamma^\sigma_\nu-p^\sigma\gamma^\mu_\nu)p_\sigma+(D-2)Bp^2\delta^\mu_\nu+... \end{align}\tag{5.30}</span></p> <p>where the ... represents terms that are proportional to the vector <span class="math-container">$p_\nu$</span>. I'm just unsure of how to go from the first to second and then to the third line of the above aligned equations.</p>