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# Proof of Covariance

To prove Lorentz covariance two conditions must be satisfied:

1. If then .
2. Given of observer , there must be a prescription for observer to compute , which describes to the same physical state.

It can be shown that all matrices (with hermitian and anti-hermitian) are equivalent up to a unitary transformation:

 (5.113)

where . We drop the distinction between and and write

 (5.114)

where .

We require that the transformation between and be linear since the Dirac equation and Lorentz transformation are linear.

 (5.115)

is a matrix which depends only on the relative velocities of and . has an inverse if and also . The inverse is

 (5.116)

or we could write

 (5.117) (5.118)

We can now write

 (5.119)

Using we have

 (5.120)

Therefore we require

 (5.121)

or

 (5.122)

This relationship defines only up to an arbitrary factor. this factor is further restricted to a sign if we require that the form a representation of the Lorentz group. We obtain thus the two-valued spinor representation, in agreement with our previous assumptions. A wave function transforming according to equation 5.115 and equation 5.116 by means of equation 5.122 is a four-component Lorentz spinor. Such a spinor is also frequently called a bi-spinor, since it consists of two 2-component spinors, known to us from the Pauli equation.

Consider an infinitesimal proper Lorentz transformation

 (5.123)

where is anti-symmetric for an invariant proper time interval. Each of the six independent non-vanishing generates an infinitesimal Lorentz transformation.

 (5.124)

for a transformation to a coordinate system moving with velocity along the -direction.

 (5.125)

for a rotation through an angle about the -direction.

We expand in powers of to first order,

 (5.126) (5.127)

with .

We now solve for . Equation 5.122 becomes

 (5.128)

Also

 (5.129)

Combining equation 5.128 and equation 5.129 gives,

 (5.130)

We must find six matrices which satisfy the above equation. We try the anti-symmetric product of two matrices

 (5.131)

Substituting (5.131) into the right-hand side of (5.130) gives

 (5.132)

which is the left-hand side of (5.130). Therefore

 (5.133)

is a solutions to equation 5.130.

Thus

 (5.134)

We now construct finite proper transformations. We define

 (5.135)

where is an infinitesimal parameter of the Lorentz group. is a matrix for a general unit space-time rotation around an axis in the direction labelled by . For proper Lorentz transformations has the property . labels the row and labels the column.

We can write the finite transformation using as

 (5.136)

For Lorentz translations, and we can write

 (5.137)

Similarly for space-rotations, and we can write

 (5.138)

Turning now to the construction of a finite spinor transformation , we have

 (5.139)

The following sections consider finite transformations for a rotation in 3-space, a general Lorentz boost, and spatial inversion.

Subsections

Next: Rotations Up: Dirac Equation Previous: Covariant Form of the
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18