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For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{\bar{γ}_μ,\bar{γ}_ν\} = 2 g_{μν} I?$$

For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*g_{μμ}}γ_μ$$ satisfied these conditions (the minkowski simply adds negatives to cancel the spacelike gammas). However for metrics which are not diagonalizable at every point in spacetime I've been having trouble.

$$+---$$ is being used here.

Playing around with Tetrads seems like the way to go but I havent had as much luck this far. Thanks in advance to any help!

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    $\begingroup$ The spacetime manifold should be a spin manifold. $\endgroup$ – Qmechanic♦ Oct 30 '18 at 3:49
  • $\begingroup$ @Qmechanic Are you sure about that? Where there exist obstructions to the construction of a spin structure, the construction of the Clifford algebra bundle is functorial, so that all pseudo-Riemannian manifolds have a Clifford bundle. In any case, I think the OP is thinking of Minkowski space but representations by possibly non-orthonormal matrices $\gamma_\mu$. $\endgroup$ – doetoe Oct 31 '18 at 11:26
  • $\begingroup$ @doetoe I'm thinking more spacetimes like the Kerr metric. Metrics for which there is no coordinate system which can diagonalize the metric everywhere. $\endgroup$ – Craig Oct 31 '18 at 14:37
  • $\begingroup$ @Craig OK. Could we read your question as "Let a manifold be given with a pseudo-Riemannian metric $g_{\mu\nu}$. Is it possible to find a smoothly varying set of linearly independent matrices $\gamma_\mu$ for which $$\{\gamma_\mu,\gamma_\nu\} = 2 g_{\mu\nu} I$$ holds at every point?" $\endgroup$ – doetoe Oct 31 '18 at 14:51
  • $\begingroup$ @doetoe yes this is precisely what I meant :) $\endgroup$ – Craig Oct 31 '18 at 22:14
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Yes, tetrads are the way to go. Suppose we have Dirac matrices for Minkowski space, $$ \{ \gamma^a, \gamma^b \} = 2 \eta^{ab} I, $$ and we also have a tetrad/vierbein, $e_a^\mu$, [1] $$ g_{\mu\nu}\ e_a^\mu\ e_b^\nu = \eta_{ab}, \qquad \eta^{ab}\ e_a^\mu\ e_b^\nu = g^{\mu\nu}. $$ Then we can define $\bar{\gamma}^\mu$: $$ \bar{\gamma}^\mu = e_a^\mu\ \gamma^a, \qquad \{ \bar{\gamma}^\mu, \bar{\gamma}^\nu \} = e_a^\mu\ e_b^\nu\ \{ \gamma^a, \gamma^b \} = e_a^\mu\ e_b^\nu\ 2 \eta^{ab} I = 2 g^{\mu\nu} I. $$ When dealing with spinors in curved space, people usually stick to the tetrad formalism.

References:
Carroll, Spacetime and geometry (the last appendix).
Weinberg, The Quantum Theory of Fields: Volume 3 (Section 31.1)

[1] As alluded to by Qmechanic and doetoe in the comments above, there can be topological problems with finding a tetrad or defining spinors. I'd ask them for references if you want to know more about this.

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  • $\begingroup$ Wow thank you for answering this months after it was posted! I really do appreciate it! :) $\endgroup$ – Craig Mar 28 at 14:26

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