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Qmechanic ♦
Qmechanic ♦
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We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ig_{\mu \lambda} M_{\nu\sigma} + ig_{\mu \nu} M_{\mu\sigma} - ig_{\nu \sigma} M_{\mu \lambda} $$ and from this we can obtain the conmutationcommutation relations of $J_i$ and $K_i$ using $$ M_{ij}=\varepsilon_{ijk}J_k, \qquad M_{0i}=K_i $$

I want to make the inverse way: how can we obtain $ \left[ M_{\mu \nu},M_{\sigma \lambda} \right]$ knowing the conmutationcommutation relations of $J_i$ and $K_i$?

We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ig_{\mu \lambda} M_{\nu\sigma} + ig_{\mu \nu} M_{\mu\sigma} - ig_{\nu \sigma} M_{\mu \lambda} $$ and from this we can obtain the conmutation relations of $J_i$ and $K_i$ using $$ M_{ij}=\varepsilon_{ijk}J_k, \qquad M_{0i}=K_i $$

I want to make the inverse way: how can we obtain $ \left[ M_{\mu \nu},M_{\sigma \lambda} \right]$ knowing the conmutation relations of $J_i$ and $K_i$?

We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ig_{\mu \lambda} M_{\nu\sigma} + ig_{\mu \nu} M_{\mu\sigma} - ig_{\nu \sigma} M_{\mu \lambda} $$ and from this we can obtain the commutation relations of $J_i$ and $K_i$ using $$ M_{ij}=\varepsilon_{ijk}J_k, \qquad M_{0i}=K_i $$

I want to make the inverse way: how can we obtain $ \left[ M_{\mu \nu},M_{\sigma \lambda} \right]$ knowing the commutation relations of $J_i$ and $K_i$?

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Carlos
Carlos
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On the generators of the Lorentz group

We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ig_{\mu \lambda} M_{\nu\sigma} + ig_{\mu \nu} M_{\mu\sigma} - ig_{\nu \sigma} M_{\mu \lambda} $$ and from this we can obtain the conmutation relations of $J_i$ and $K_i$ using $$ M_{ij}=\varepsilon_{ijk}J_k, \qquad M_{0i}=K_i $$

I want to make the inverse way: how can we obtain $ \left[ M_{\mu \nu},M_{\sigma \lambda} \right]$ knowing the conmutation relations of $J_i$ and $K_i$?