# Conserved quantity due to invariance under temporal rotations

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## Main Question or Discussion Point

Hi,

I was looking at the so(3,3) Lie algebra which has 3 temporal rotation generators as well as the normal 3 spatial rotation generators. When I try to use Noether's Theorem to determine what the conserved quantity is, due to invariance under temporal rotations, I seem to get an integral where the integrand is of the form

[(energy)*(time) - (energy)*(time)]

The equivalent integrand corresponding to spatial rotations is

[(momentum)*(distance) - (momentum)*(distance)]

which is normal because the conserved quantity turns out to be angular momentum. But (energy)*(time) has the same units as angular momentum which seems to imply that invariance under temporal rotations and invariance under spatial rotations both lead to conserved quantities with the units of Planck's constant. I'm not sure if this can be right.

My calculations are on the back of an envelope so I was hoping that someone might know if this is correct or if there is a paper that deals more rigorously with this question.

Thanks.

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Orodruin
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But (energy)*(time) has the same units as angular momentum which seems to imply that invariance under temporal rotations and invariance under spatial rotations both lead to conserved quantities with the units of Planck's constant. I'm not sure if this can be right.
Why would this be an issue?

In the context of so(3,3), if the above is correct then there might be a second type of "spinor" representation. The normal spinor representation has the form

{(J+iK), (J+iK), (J+iK)}

where the Js are spatial rotation generators and the Ks are boost generators. The second type of representation would have the form

{(T+iK), (T+iK), (T+iK)}

where the Ts are temporal rotation generators.

It might open the door to an additional mathematical description for a spin 1/2 particle.