Dirac Matrices

A review of the basic properties of Dirac Matrices. $\text{They are defined by }\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }\text{1}\text{ with 1 being the unit matrix in the Dirac space }={\delta }_{\alpha \beta }$ We also define a 5th matrix ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3;({\gamma }^5{)}^2=\text{1};\{{\gamma }^{\mu },{\gamma }^5\}=0$ It is helpful to go through the traces of Dirac Matrices \text{i) }Tr[\gamma^\mu]=0$$$$\text{ii) }Tr[\gamma^\mu\gamma^\nu]=4g^{\mu\nu}$$$$\text{iii) }Tr[odd \# \gamma^\mu]=0$$$$\text{iv) }Tr[\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma]=4(g^{\mu\nu}g^{\rho\sigma}-g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})$$$$\text{v) }Tr[\gamma^5]=0$$$$\text{vi) }Tr[\gamma^\mu\gamma^\nu\gamma^5]=0$$$$\text{vii) }Tr[\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma^5]=4i\epsilon^{\mu\nu\rho\sigma}\;;\;\text{where }\epsilon^{\mu\nu\rho\sigma}\text{ is the completely anti-symmetric Levi-Civita tensor in 4D} We will also need contractions of Dirac matrices in D dimensions, $\text{i) }{\gamma }^{\mu }{\gamma }_{\mu }=D\text{ 1}$ $\text{ii) }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }_{\mu }=(2-D){\gamma }^{\nu }$ $\text{iii) }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }_{\mu }=(D-4){\gamma }^{\nu }{\gamma }^{\rho }+4{\gamma }^{\rho }{\gamma }^{\nu }$ $\text{iv) }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }{\gamma }_{\mu }=(4-D){\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }-2{\gamma }^{\sigma }{\gamma }^{\rho }{\gamma }^{\nu }$