Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 301 / 2 / d

Use the Clifford algebra to write ${\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }{I}_4-{\gamma }^{\mu }{\gamma }^{\nu }$. The Lorentz generator from gamma-matrix commutators is thereforeThis also checks the diagonal case $\mu =\nu$, where $S^{\mu \mu }=0$ because $({\gamma }^{\mu }{)}^2={\eta }^{\mu \mu }{I}_4$. The metric term in the printed expression is understood to multiply the identity matrix. The generators are antisymmetric in their two spacetime indices.

The commutator with the scalar metric term vanishes. Reorder the third gamma matrix using the Clifford algebra twice:Subtract the last term and multiply by $1/2$ to obtainThis is the infinitesimal vector transformation law implemented by the Spinor representation of the Lorentz group. The first contraction is ${\eta }^{\nu \rho }$ as printed in the PDF; the TeX conversion's ${\eta }^{\rho \rho }$ is erroneous and would fail, for example, when $\mu =0,\nu =1,\rho =2$: the actual commutator is zero.

Apply the commutator derivation identity $[S,AB]=[S,A]B+A[S,B]$ and the preceding gamma matrix transformation identity. Keeping the matrix order unchanged givesThusThe order of the gamma matrices matters, since they generally do not commute. The PDF has the four ordinary metric contractions shown here; the conversion's stray powers such as ${\eta }^{\nu }{\rho }^{\sigma }$ are not mathematical expressions to retain.

Since $S^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$, apply the commutator derivation identity to this commutator and insert part (ii):This is exactly the printed Lorentz algebra relation, including its final two signs; their alternative appearance in other formulas comes from $S^{\rho \mu }=-S^{\mu \rho }$. Therefore the matrices S form a representation of the Lorentz algebra. Their definition contains no extra factor of $i$, so the spatial rotation generators below are represented by skew-Hermitian matrices. Adding a factor of $i$ would change the convention for the Lie algebra structure constants and cannot be done silently.