Make the Dirac Equation Consistent with Relativity

[sk1105]

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.

 

[blue_leaf77]

$[\gamma ^\mu ,\gamma ^\nu]=2g^{\mu \nu}$.

That should be an anticommutator.

The bit that really puzzles me though is that we're supposed to get αiαj+αjαi=2δij\alpha^i\alpha^j + \alpha^j\alpha^i = 2\delta^{ij}, and I can't see how it follows from the equation above.

Expand $(\alpha \cdot \mathbf{p})^2$ and make it equal to $|\mathbf{p}|^2$

$2\beta m\vec {\alpha} \cdot \vec p$

By the way you shouldn't write it like that since you know that $\beta$ does not commute with $\alpha_i$.

 

[sk1105]

That should be an antimcommutator.

Yes you're right, my bad.

Expand $(α⋅p)2(\alpha \cdot \mathbf{p})^2$ and make it equal to |p|2

That makes sense, I was just getting muddled with all the vector components. Thanks for your help.