Verifying Lorentz Algebra with Clifford/Dirac Algebra

In summary, the conversation discussed the difficult computation of verifying the n-dimensional representation of the Lorentz algebra, which satisfies a certain commutation relation. The person speaking had already tried various lengthy computations and was struggling to find the correct solution. They asked for help in finishing the computation and were advised to "cheat" in a way by already knowing the form of the final answer. This involved using the Clifford/Dirac algebra and the fact that the term \gamma^{\mu}\gamma^{\nu} = 2g^{\mu\nu}-\gamma^{\nu}\gamma^{\mu} would be necessary in the final answer.
  • #1
Onamor
78
0
Paraphrasing Peskin and Schroeder:

By repeated use of
[itex]\left\{ \gamma^{\mu} , \gamma^{\nu} \right\}= 2 g^{\mu\nu} \times \textbf{1}_{n \times n} [/itex] (Clifford/Dirac algebra),
verify that the n-dimensional representation of the Lorentz algebra,
[itex]S^{\mu \nu}=\frac{i}{4}\left[\gamma^{\mu},\gamma^{\nu}\right][/itex],
satisfies the commutation relation
[itex]\left[J^{\mu \nu},J^{\rho \sigma}\right]=i\left(g^{\nu \rho}J^{\mu \sigma}-g^{\mu \rho}J^{\nu \sigma}-g^{\nu \sigma}J^{\mu \rho}+g^{\mu \sigma}J^{\nu \rho}\right)[/itex].

I've tried many lengthy computations and always seem to be missing something.
Most obvious thing to try is just
[itex]\left[S^{\mu \nu},S^{\rho \sigma}\right]=S^{\mu \nu}S^{\rho \sigma}-S^{\rho \sigma}S^{\mu \nu}=\frac{-1}{16}\left(\left[\gamma^{\mu},\gamma^{\nu}\right]\left[\gamma^{\rho},\gamma^{\sigma}\right]-\left[\gamma^{\rho},\gamma^{\sigma}\right]\left[\gamma^{\mu},\gamma^{\nu}\right]\right)[/itex]
[itex]=\frac{-1}{16}\left(\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\left(\gamma^{\rho}\gamma^{\sigma}-\gamma^{\sigma}\gamma^{\rho}\right)-\left(\gamma^{\rho}\gamma^{\sigma}-\gamma^{\sigma}\gamma^{\rho}\right)\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)\right)[/itex]
[itex]=\frac{-1}{16}\left( \gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma} - \gamma^{\mu} \gamma^{\nu} \gamma^{\sigma} \gamma^{\rho} - \gamma^{\nu} \gamma^{\mu} \gamma^{\rho} \gamma^{\sigma} + \gamma^{\nu} \gamma^{\mu} \gamma^{\sigma} \gamma^{\rho} - \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu} \gamma^{\nu} + \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu} \gamma^{\nu} + \gamma^{\sigma} \gamma^{\rho} \gamma^{\mu} \gamma^{\nu} - \gamma^{\sigma} \gamma^{\rho} \gamma^{\nu} \gamma^{\mu} \right) [/itex]

and then I've tried a few different commutation relations but to no avail.
Would be very grateful for any help in finishing this off.
 
Last edited:
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  • #2


This is indeed a difficult computation. You must cheat in a way, in the sense that you already know what the final answer looks like. So the the LHS with those 8 terms must lead to the RHS which has also 8 terms (4 times J, but each J has 2 times gammas). The g's will appear when you use the clifford algebra as

[tex] \gamma^{\mu}\gamma^{\nu} = 2g^{\mu\nu}-\gamma^{\nu}\gamma^{\mu} [/tex]

So try to group the 8 terms of 4 gammas into the desired form according to how the J's indices occur in the RHS of what you're trying to prove.
 

1. What is the significance of verifying Lorentz algebra with Clifford/Dirac algebra?

The Lorentz algebra describes the symmetries of spacetime in special relativity, while the Clifford/Dirac algebra describes the properties of spinors. By verifying the equivalence of these two algebras, we can better understand the relationship between spacetime symmetries and spinors in relativistic quantum mechanics.

2. How do you verify Lorentz algebra with Clifford/Dirac algebra?

This can be done by comparing the generators of the two algebras and showing that they satisfy the same commutation relations. Additionally, one can show that the representation matrices of the generators also satisfy the same algebraic relations.

3. What are the implications of successfully verifying Lorentz algebra with Clifford/Dirac algebra?

Successfully verifying the equivalence of these two algebras would provide strong evidence for the validity of relativistic quantum mechanics and the relationship between spacetime symmetries and spinors. It could also potentially lead to further insights and developments in quantum field theory.

4. Are there any challenges or limitations to verifying Lorentz algebra with Clifford/Dirac algebra?

One challenge is the mathematical complexity of the two algebras and the calculations involved in verifying their equivalence. Additionally, the results may depend on the specific chosen representation of the algebras, which could limit the generalizability of the findings.

5. How does verifying Lorentz algebra with Clifford/Dirac algebra relate to experimental physics?

While verifying the algebraic equivalence itself may not have direct implications for experimental physics, it provides a deeper understanding of the underlying principles of spacetime symmetries and spinors in relativistic quantum mechanics, which can inform and guide experimental research in this field.

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