Lorentz Algebra in Boosts for the spin-1/2 Dirac Field

1. Jul 14, 2009

maverick280857

Hi,

What is the origin of the following commutation relation in Lorentz Algebra:

$$[J^{\mu\nu}, J^{\alpha\beta}] = i(g^{\nu\alpha}J^{\mu\beta}-g^{\mu\alpha}J^{\nu\beta}-g^{\nu\beta}J^{\mu\alpha}+g^{\mu\beta}J^{\nu\alpha})$$

This looks a whole lot similar to the commutation algebra of angular momentum in O(4). But how does the Metric Tensor enter here? I encountered this while studying about boosts in the context of the Dirac Equation.

Thanks.

2. Jul 14, 2009

Hans de Vries

This is the "Master commutation" rule of the Lorentz section of the PoincarĂ© group

$J^{\mu\nu}$ can be both a rotation generator as well as a boost generator, so this is the most
general way of defining the commutation rules between these generators.

It's far simpler to look at the individual rules like $[J^i,J^i]$ and $[K^i,K^i]$ or $[J^i,K^i]$ where
J and K are the rotation and boost generators respectively.

You can find these in many books like Ryder, Weinberg (vol 1) or P&S

Regards, Hans

3. Jul 14, 2009

maverick280857

Thanks.

4. Jul 18, 2009

RedX

Lorentz algebra is SO(4) algebra, synonymous almost. To be exact, Lorentz algebra is SO(3,1) algebra. So you can derive the commutation relations for SO(4), and try to turn SO(4) into SO(3,1) by converting Krockner deltas into metric tensors by raising or lowering.