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sk1105

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## Homework Statement

The free Dirac equation is given by ##(i\gamma ^\mu \partial _\mu -m)\psi = 0## where ##m## is the particle's mass and ##\gamma ^\mu## are the Dirac gamma matrices. Show that for the equation to be consistent with Relativity, the gamma matrices must satisfy ##[\gamma ^\mu ,\gamma ^\nu]=2g^{\mu \nu}##.

## Homework Equations

Dirac equation

Gamma matrices

##E^2=|\vec p|^2 + m^2## in natural units

We use the ##+---## metric.

## The Attempt at a Solution

I know that the Dirac Hamiltonian is ##\vec {\alpha} \cdot \vec p +\beta m##, so I have equated it to the energy-momentum relation as follows:

##E^2 = (\vec {\alpha} \cdot \vec p + \beta m)^2 = (\vec {\alpha} \cdot \vec p)^2 + 2\beta m\vec {\alpha} \cdot \vec p + \beta^2m^2 = |\vec p|^2 + m^2##

It's clear that this leads to ##\beta^2=1##. I also know that we get ##\alpha^i\beta + \beta\alpha^i = 0##, although I'm less sure why. The bit that really puzzles me though is that we're supposed to get ##\alpha^i\alpha^j + \alpha^j\alpha^i = 2\delta^{ij}##, and I can't see how it follows from the equation above.

Then even with those three pieces I'm not sure how to arrive at the required commutator. Any help is much appreciated.