In their book Introduction to Quantum Field Theory, Peskin and Schroeder talk about a trick to form the generators for the Lorentz group from the commutators of the gamma matrices, using their anti-commutation relations. Using the anti-commutation relations
$$\{ \gamma^\mu, \gamma^\nu\} = 2 g^{\mu \nu} \times \mathbf{1}_{n \times n}$$
the generators of the Lorentz group are formed as
$$S^{\mu\nu} = \frac {i}{4} \big[ \gamma^\mu, \gamma^\nu \big].$$
This can be seen as the general case of a set of basis vectors (here the 16 matrices corresponding to the multiples of the gamma matrices) which form a Clifford algebra of $\mathcal{R}(p,q)$, and whose commutators form generators of a Lie group (Lorentz group here) which conserves the quadratic form of the Clifford algebra itself.
Is there a way to formalize this idea? I want to know if we take any arbitrary metric $g^{\mu\nu}$ on some space $V$, will the generators defined as $S^{\mu\nu}$ generate a Lie group whose elements are transformations on $V$ that conserve the inner product corresponding to the metric?
Best Answer
It's pretty annoying that P&S just give you $$S^{\mu \nu} = \frac{i}{4} [\gamma^{\mu},\gamma^{\nu}]$$ from thin air, here is a way to derive it similar to Bjorken-Drell's derivation (who start from the Dirac equation) but from the Clifford algebra directly, assuming that products of the gamma matrices form a basis. Given a Clifford algebra of $\gamma^{\mu}$'s satisfying \begin{align} \{ \gamma^{\mu} , \gamma^{\mu} \} = 2 \eta^{\mu \nu} I \end{align} we note that for an invertible transformation $S$ we have \begin{align} 2 \eta^{\mu \nu} I &= 2 \eta^{\mu \nu} S^{-1} S \\ &= S^{-1}(2 \eta^{\mu \nu}) S \\ &= S^{-1}\{ \gamma^{\mu} , \gamma^{\mu} \} S \\ &= \{ S^{-1} \gamma^{\mu} S, S^{-1}\gamma^{\mu} S \} \\ &= \{ \gamma'^{\mu} , \gamma'^{\mu} \} \end{align} showing us that the Clifford algebra of matrices $$\gamma'^{\mu} = S^{-1} \gamma^{\mu} S$$ also satisfies the Clifford algebra, hence any set of matrices satisfying the Clifford algebra can be obtained from a given set $\gamma^{\mu}$ using a non-singular transformation $S$. Since the anti-commutation relations involve the metric $\eta_{\mu \nu}$, and we know the metric is left invariant under Lorentz transformations $$\eta^{\mu \nu} = \Lambda^{\mu} \, _{\rho} \Lambda^{\nu} \, _{\sigma} \eta^{\rho \sigma} $$ this immediately implies \begin{align} 2 \eta^{\mu \nu} I &= \Lambda^{\mu} \, _{\rho} \Lambda^{\nu} \, _{\sigma} 2 \eta^{\rho \sigma} I \\ &= \Lambda^{\mu} \, _{\rho} \Lambda^{\nu} \, _{\sigma} \{ \gamma^{\rho} , \gamma^{\sigma} \} \\ &= \{ \Lambda^{\mu} \, _{\rho} \gamma^{\rho} , \Lambda^{\nu} \, _{\sigma} \gamma^{\sigma} \} \\ &= \{ \gamma'^{\mu} , \gamma'^{\mu} \} \end{align} which shows that the Lorentz transformation of a gamma matrix also satisfies the Clifford algebra, and so is itself a gamma matrix, and hence can be expressed in terms of some non-singular transformation $S$ \begin{align} \gamma'^{\mu} &= \Lambda^{\mu} \, _{\nu} \gamma^{\nu} \\ &= S^{-1} \gamma^{\mu} S \end{align} where $S$ is to be determined. Since the operators $S$ represent performing a Lorentz transformation on $\gamma^{\mu}$, and Lorentz transformations on fields expand as $I - \frac{i}{2}\omega_{\mu \nu} M^{\mu \nu}$, we expand $\Lambda$ and $S$ as \begin{align} \Lambda^{\mu} \, _{\nu} &= \delta ^{\mu} \, _{\nu} + \omega^{\mu} \, _{\nu} \\ S &= I - \frac{i}{2} \omega_{\mu \nu} \Sigma^{\mu \nu} \end{align} where $\Sigma^{\mu \nu}$ must be anti-symmetric and constructed from a basis of gamma matrices, hence from \begin{align} \gamma'^a &= \Lambda^a \, _{\mu} \gamma^{\mu} \\ &= (\delta^a \, _{\mu} + \omega^a \, _{\mu})\gamma^{\mu} \\ &= \gamma^a + \omega^a \, _{\mu} \gamma^{\mu} \\ &= \gamma^a + \omega_{b \mu} \eta^{a b} \gamma^{\mu} \\ &= \gamma^a + \omega_{b \mu} \eta^{a [b} \gamma^{\mu]} \\ &= \gamma^a + \frac{1}{2} \omega_{b \mu} (\eta^{a b} \gamma^{\mu} - \eta^{a \mu} \gamma^b) \\ &= \gamma^a + \frac{1}{2} \omega_{\nu} (\eta^{a \mu} \gamma^{\nu} - \eta^{a \nu} \gamma^{\mu}) \\ &= S^{-1} \gamma^a S \\ &= (I - \frac{i}{2} \omega_{\mu \nu} \Sigma^{\mu \nu}) \gamma^a (I + \frac{i}{2} \omega_{\mu \nu} \Sigma^{\mu \nu}) \\ &= \gamma^a - \frac{i}{2} \omega_{\mu \nu} [\gamma^a, \Sigma^{\mu \nu}] \end{align} we have the relation (which can be interpreted as saying that $\gamma^a$ transforms as a vector under spinor representations of Lorentz transformations, as e.g. in Tong's QFT notes) \begin{align} i (\eta^{a \mu} \gamma^{\nu} - \eta^{a \nu} \gamma^{\mu}) = [\gamma^a, \Sigma^{\mu \nu}] \end{align} and we know $\Sigma^{\mu \nu}$, since it is anti-symmetric, must involve a product's of $\gamma$ matrices (because of the 16-dimensional basis formed from Clifford algebra elements), only two by the left-hand side, and from \begin{align} \gamma^{\mu} \gamma^{\nu} &= - \gamma^{\nu} \gamma^{\mu} , \ \ \ \mu \neq \nu, \\ \gamma^{\mu} \gamma^{\mu} &= \gamma^{\nu} \gamma^{\mu} , \ \ \ \mu = \nu, \end{align} we expect that \begin{align} \Sigma^{\mu \nu} &= c [\gamma^{\mu},\gamma^{\nu}] \\ &= c (\gamma^{\mu} \gamma^{\nu} - \gamma^{\nu} \gamma^{\mu}) \\ &= 2 c ( \gamma^{\mu} \gamma^{\nu} - \eta^{\mu \nu}) \end{align} for some $c$ which we constrain by the (vector) relation above \begin{align} i (\eta^{a \mu} \gamma^{\nu} - \eta^{a \nu} \gamma^{\mu}) &= [\gamma^a, \Sigma^{\mu \nu}] \\ &= c [\gamma^a, 2( \gamma^{\mu} \gamma^{\nu} - \eta^{\mu \nu})] \\ &= 2 c [\gamma^a, \gamma^{\mu} \gamma^{\nu}] \\ &= 2 c ( \gamma^{\mu} [\gamma^a,\gamma^{\nu}] + [\gamma^a, \gamma^{\mu}] \gamma^{\nu}) \\ &= 2 c [ \gamma^{\mu} 2( \gamma^a \gamma^{\nu} - \eta^{a \nu}) + 2( \gamma^a \gamma^{\mu} - \eta^{a \mu}) \gamma^{\nu}] \\ &= 4 c [ \gamma^{\mu} ( \gamma^a \gamma^{\nu} - \eta^{a \nu}) + ( \gamma^{\mu} \gamma^{a} + 2 \eta^{a \mu} - \eta^{a \mu}) \gamma^{\nu}] \\ &= 4 c (\eta^{a \mu} \gamma^{\nu} - \eta^{a \nu} \gamma^{\mu}). \end{align} This gives the result $c = i/4$. The generator of Lorentz transformations of gamma matrices is \begin{align} \Sigma^{\mu \nu} &= \dfrac{i}{4} [\gamma^{\mu},\gamma^{\nu}] \\ &= \dfrac{i}{2}(\gamma^{\mu} \gamma^{\nu} - \eta^{\mu \nu}) \ \ \text{i.e.} \\ S &= I - \frac{i}{2} \omega_{\mu \nu} (\frac{i}{4} [\gamma^{\mu},\gamma^{\nu}]) \\ &= I + \dfrac{1}{8} \omega_{\mu \nu} [\gamma^{\mu},\gamma^{\nu}]. \end{align} Using the fact that the gamma matrices transform as a vector under the spinor representation of an infinitesimal Lorentz transformation, \begin{align} [\Sigma^{\mu \nu}, \gamma^{\rho}] = i (\gamma^{\mu} \eta^{\nu \rho} - \gamma^{\nu} \eta^{\mu \rho}) \end{align} we can show the spinor representation of a Lorentz transformation satisfies the Lorentz algebra commutation relations, since for $\rho \neq \sigma$ \begin{align} [\Sigma^{\mu \nu},\Sigma^{\rho \sigma}] &= \frac{i}{2}[\Sigma^{\mu \nu},\gamma^{\rho} \gamma^{\sigma}] \\ &= \frac{i}{2}([\Sigma^{\mu \nu},\gamma^{\rho} ] \gamma^{\sigma} + \gamma^{\rho} [\Sigma^{\mu \nu}, \gamma^{\sigma}]) \\ &= \frac{i}{2}\{ i (\gamma^{\mu} \eta^{\nu \rho} - \gamma^{\nu} \eta^{\mu \rho}) \gamma^{\sigma} + \gamma^{\rho} i (\gamma^{\mu} \eta^{\nu \sigma} - \gamma^{\nu} \eta^{\mu \sigma}) \} \\ &= - \frac{1}{2}\{ \gamma^{\mu} \eta^{\nu \rho} \gamma^{\sigma} - \gamma^{\nu} \eta^{\mu \rho} \gamma^{\sigma} + \gamma^{\rho} \gamma^{\mu} \eta^{\nu \sigma} - \gamma^{\rho} \gamma^{\nu} \eta^{\mu \sigma} \} \\ &= \frac{i}{2}\{ \eta^{\nu \rho} (2 \Sigma^{\mu \sigma} + \eta^{\mu \sigma}) - \eta^{\mu \rho} (2 \Sigma^{\nu \sigma} - \eta^{\nu \sigma}) + (2 \Sigma^{\rho \mu} - \eta^{\rho \mu}) \eta^{\nu \sigma} - (2 \Sigma^{\rho \nu}) - \eta^{\rho \nu}) \eta^{\mu \sigma} \} \\ &= i ( \eta^{\nu \rho} \Sigma^{\mu \sigma} - \eta^{\mu \rho} \Sigma^{\nu \sigma} + \Sigma^{\rho \mu} \eta^{\nu \sigma} - \Sigma^{\rho \nu} \eta^{\mu \sigma} ). \end{align} This method generalizes from $SO(3,1)$ to $SO(N)$, see e.g. Kaku QFT Sec. 2.6, and the underlying reason for doing any of this in the first place is that one seeks to find projective representations which arise due to the non-simple-connectedness of these orthogonal groups. Regarding your question about arbitrary metrics $g_{\mu \nu}$, this method applies to, and arises due to the non-simple-connectedness of, special orthogonal groups, you can't generalize to arbitrary metrics, this is a problem which can be circumvented in supergravity and superstring theory using veilbein's.
References: