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

[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.

 

[Hans de Vries]

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.

This is the "Master commutation" rule of the Lorentz section of the Poincaré group

http://en.wikipedia.org/wiki/Poincaré_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

 

[maverick280857]

Thanks.

 

[RedX]

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.

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.