Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 3.10

Classical field theory — Schwartz M.D. z-lib.org

Physics — QFT · graviton-polarizations

Problem

Graviton polarizations. We will treat the graviton as a symmetric 2-index tensor field. It couples to a current

T_{\mu\nu}

also symmetric in its two indices, which satisfies the conservation law

\partial_\mu T_{\mu\nu} = 0

. (a) Assume the Lagrangian is

\mathcal{L} = -\frac{1}{2}h_{\mu\nu}\square h_{\mu\nu} + \frac{1}{M_{\mathrm{Pl}}}h_{\mu\nu}T_{\mu\nu}

. Solve

h_{\mu\nu}

's equations of motion, and substitute back to find an interaction like

T_{\mu\nu}\frac{1}{k^2}T_{\mu\nu}

. (b) Write out the 10 terms in the interaction

T_{\mu\nu}\frac{1}{k^2}T_{\mu\nu}

explicitly in terms of

T_{00}, T_{01}

, etc. (c) Use current conservation to solve for

T_{\mu 1}

in terms of

T_{\mu 0}

\omega

and

\kappa

. Substitute in to simplify the interaction. How many causally propagating degrees of freedom are there? (d) Add to the interaction another term of the form

c T_{\mu\mu}\frac{1}{k^2}T_{\nu\nu}

. What value of

c

can reduce the number of propagating modes? How many are there now?

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