- #1

Jason Bennett

- 49

- 3

- Homework Statement
- below

- Relevant Equations
- below

I am trying to understand the following:

$$

\epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp}

$$

Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices representation.

The pattern I recognize is that, the only term to remain is the one where the two indices of both the delta and the Lorentz algebra matrix are in the same "slot" of the Levi-Civita symbol. Notably, the m and the p of the delta are both in the first "slot" of the L-C symbol, and the n and q are both in the second "slot" of the L-C symbol.

Can someone help me by pointing out which property of the L-C symbols I ought to be using?

Some additional points that may be on the right track

- anti-symmetric times symmetric = 0

- L-C = anti-sym,

- delta = sym (?), and

- the Lorentz matrices = anti-sym

$$

\epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp}

$$

Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices representation.

The pattern I recognize is that, the only term to remain is the one where the two indices of both the delta and the Lorentz algebra matrix are in the same "slot" of the Levi-Civita symbol. Notably, the m and the p of the delta are both in the first "slot" of the L-C symbol, and the n and q are both in the second "slot" of the L-C symbol.

Can someone help me by pointing out which property of the L-C symbols I ought to be using?

Some additional points that may be on the right track

- anti-symmetric times symmetric = 0

- L-C = anti-sym,

- delta = sym (?), and

- the Lorentz matrices = anti-sym