A Levi Civita - SO(4) Group Theory: Proving Relation in Landau and Lifshitz

[Nusc]

Landau and Lifshitz, second volume - Classical Theory of Fields, page 7

$$e_mu,nu,alpha,beta e^alpha, beta, gamma, sigma = -2 ( delta^gamma_mu * delta^sigma_nu delta delta^sigma_mu * delta^gamma_nu )
$$
If for example I calculate the following:
$$
e^0,1_alpha,beta e^alpha,beta_0,1 = e_0123 e^2301 + e_0132 e^3201
= 1(+1) +(-1)(-1) = +2$$
If we use LL:

$$-2(delta^0_0 delta^1_1 - delta^0_1 delta^1_0) = -2$$
and one can do that for

$e^1,0_alpha,beta e^alpha,beta_0,1$ and you get the opposite result
Same with

$e^0,1_alpha,beta e^alpha,beta_1,0$ and you get the opposite result

I don't think LL is correct.

I have been told that this relation can be proved using group theory, in particular, methods for SO(4)

I don't think it's true but I wanted to know if anyone here could do it since I don't know group theory.

 

[haushofer]

I strongly suggest you use Latex for your formulas and give a little context and conventions.

Did you try the wikipedia page on epsilon symbols,

https://en.wikipedia.org/wiki/Levi-Civita_symbol

at "Four dimensions".? Also, be aware of the normalisation constants in antisymmetrization. E.g., a lot of authors define $T_{[ab]} \equiv \frac{1}{2!} \Bigl(T_{ab} - T_{ba} \Bigr)$.

 

[Nusc]

I calculated that relation by brute force and I am off by a negative sign. That's why I want the proof.$$e_{\mu,\nu,\alpha,\beta} e^{\alpha, \beta, \gamma, \sigma} = -2 ( \delta^{\gamma}_{\mu} * \delta^{\sigma}_{\nu} -\delta^\sigma_\mu * \delta^\gamma_\nu )
$$

$$
e_{0,1,\alpha,\beta} e^{\alpha,\beta,0,1} = e_{0123} e^{2301} + e_{0132} e^{3201}

= 1(+1) +(-1)(-1) = +2
$$If we use LL:
$$-2(\delta^0_0 \delta^1_1 - \delta^0_1 \delta^1_0) = -2$$

<mentor edit>

 

[Orodruin]

Your index placement is impossible to understand without proper LaTeX. Anyway, you seem to be missing a sign from lowering/raising some indices in your brute force computation (depending on what exactly your definitions are).

 

[jim mcnamara]

I tried a latex edit, poor results, I'm asking for some more help from others.

 

[haushofer]

How did you raise the indices on the epsilon symbol? You shouldn't use the Minkowski metric, as we're talking SO(4) here.

 

[Orodruin]

How did you raise the indices on the epsilon symbol? You shouldn't use the Minkowski metric, as we're talking SO(4) here.

I think the exact problem is that he did not use the Minkowski metric, while Landau-LIfshitz probably do.

 

[haushofer]

Ah, yes, I read that LL used SO(4), but it was TS his own comment. That could be a reason, indeed.