     Next: Projection Operators for Energy Up: Plane-Wave Solutions Previous: Plane-Wave Solutions

## Spin

We have just seen how one can create a state of arbitrary momentum by Lorentz transforming the solution for a particle at rest. In a similar manner, we can create a state of arbitrary spin polarized along the direction by applying the rotation operator, (5.198)

to the solution for a particle at rest and polarized in the direction. The defining relationship for such a state is (5.199)

where the spinor corresponds to a particle polarized along the direction of the unit vector . Remember that the spin operator is and thus the eigenvalue of the spin operator in this case is really , as we might expect.

Let denote a spinor of positive energy, momentum , and spin . The spin vector is defined as (5.200)

where is the polarization unit vector in the rest frame and is a Lorentz transformation from the rest frame. Thus we also have , where . This tells us that (5.201)

and (5.202)

which is true in any frame since they are Lorentz scalars. Thus in the rest frame (5.203)

Let denote a spinor of negative energy, with polarization in the reset frame. In this case, (5.204)

Therefore in an arbitrary Lorentz frame            (5.205)

where is a 4-vector, which in the rest frame is . An arbitrary spinor is thus specified by the sign of the energy, its momentum , and polarization in the rest frame .     Next: Projection Operators for Energy Up: Plane-Wave Solutions Previous: Plane-Wave Solutions
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18