Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 8.5

Spin 1 and gauge invariance — Schwartz M.D. z-lib.org

Physics — QFT · vector-polarization-sums

Problem

Vector polarization sums. In this problem you can build some intuition for the way in which the numerator of a spin-1 particle propagator represents an outer product of physical polarizations

|\epsilon\rangle\langle\epsilon|

. Calculate the

4 \times 4

matrix outer product

|\epsilon\rangle\langle\epsilon| \equiv \sum_j \epsilon^j_\mu \epsilon^j_\nu

by the following: (a) Sum over the physical polarizations for a massive spin-1 particle in some frame. Re-express your answer in a Lorentz covariant way, in terms of

m

k_\mu k_\nu

and

g_{\mu\nu}

. (b) Show that the numerator of the massive vector propagator (Problem 8.3) is the same as the polarization sum. Why should this be true? (c) Sum over the orthonormal basis of four 4-vectors

\partial_\mu x^\alpha = \delta^\alpha_\mu

with the Minkowski metric

|\epsilon\rangle\langle\epsilon| = \epsilon^0_\mu \epsilon^0_\nu - \sum_{j=1}^3 \epsilon^j_\mu \epsilon^j_\nu

. Express your answer in a Lorentz-covariant way. (d) Sum over the physical polarizations for massless vectors. Express your answer in a Lorentz-covariant way. You may also need the vectors

k_\mu = (E, \vec{k})

and

\bar{k}_\nu = (E, -\vec{k})

. (e) Compare these sums to the numerator of the photon propagator, commenting on the gauge dependence. Do either of these sums correspond to the numerator of one of the

R_\zeta

gauges we derived?

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