Srednicki — Quantum Field Theory · Problem 2.9

Lorentz Invariance — Mark Srednicki

Physics — QFT · lorentz-invariance-vector-representation

Problem

Let us write

\Lambda^\rho{}_\tau = \delta^\rho{}_\tau + \frac{i}{2\hbar}\delta\omega_{\mu\nu}(S^{\mu\nu}_{\rm V})^\rho{}_\tau , \tag{2.32}

where

(S^{\mu\nu}_{\rm V})^\rho{}_\tau \equiv \frac{\hbar}{i}(g^{\mu\rho}\delta^\nu_\tau - g^{\nu\rho}\delta^\mu_\tau) \tag{2.33}

are matrices which constitute the vector representation of the Lorentz generators. a) Let

\Lambda = 1 + \delta\omega

in eq. (2.27), and show that

[\partial^\rho\varphi(x), M^{\mu\nu}] = \mathcal{L}^{\mu\nu}\partial^\rho\varphi(x) + (S^{\mu\nu}_{\rm V})^\rho{}_\tau \partial^\tau\varphi(x) . \tag{2.34}

b) Show that the matrices

S^{\mu\nu}_{\rm V}

must have the same commutation relations as the operators

M^{\mu\nu}

. Hint: see the previous problem. c) For a rotation by an angle

\theta

about the

z

axis, we have

\Lambda^\mu{}_\nu = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} . \tag{2.35}

Show that

\Lambda = \exp(-i\theta S^{12}_{\rm V}/\hbar) . \tag{2.36}

d) For a boost by rapidity

\eta

in the

z

direction, we have

\Lambda^\mu{}_\nu = \begin{pmatrix} \cosh\eta & 0 & 0 & \sinh\eta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh\eta & 0 & 0 & \cosh\eta \end{pmatrix} . \tag{2.37}

Show that

\Lambda = \exp(+i\eta S^{30}_{\rm V}/\hbar) . \tag{2.38}