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  • Print publication year: 2020
  • Online publication date: May 2020

Appendix A - Lorentz Transformation and Covariance of the Dirac Equation

Summary

Lorentz Transformations

Lorentz transformation (L.T.) is the rotation in Minkowski space. Minkowski space is the mathematical representation of the space time in Einstein's theory of relativity. Unlike the Euclidean space, Minkowski space treats space and time on a different footing and the space time interval between the two events would be the same in all frames of reference.

If an event is observed in an inertial frame S at the coordinates (x, y, z, t) (Fig.A.1) and in another inertial frame at the coordinates and the frame is moving with a constant velocity v with respect to S in the direction of x, then the measurements in S and S are related by

where

The L.H.S. of these equations describe the transformation of the coordinates from S to S and the R.H.S. of the equations describe the transfer of coordinates from S to S. Equation (A.1) may also be written as

The invariant length element squared ds2 is defined as

where is a contravariant vector and is a covariant vector. The metric tensor is defined as

One may rewrite Eq. (A.2) as

Where is the Lorentz transformation matrix. Any set of quantities which have four components and transform like Eq. (A.5) under L.T. form a four vector, that is,

with the properties

For an infinitesimal may also be written as The tensor generates the infinitesimal L.T. and it is considered to be a quantity that is smaller than unity. The major contributor is (unit matrix) and generates infinitesimal boost or infinitesimal rotation or both. This can be understood as demonstrated in the next section.

Rotation of coordinates around the zaxis in 3D

If we consider the rotation around the z-axis by an angle θ, then

In the matrix form, Eq. (A.7) can be written as

For an infinitesimal rotation, that is,

where represents an infinitesimal rotation if. Equation (A.7) can be rewritten as

The matrix representing the infinitesimal change is antisymmetric. Reconsidering Eq. (A.2),

When and Now and is given by

For an infinitesimal boost or rotation, that is,

Therefore, Eq. (A.11) implies that

Now,

The second term on the R.H.S. of Eq. (A.13) can be neglected as it is the product of two infinitesimal numbers, that is,

Since should be an invariant quantity,

that is, Eq. (A.14) shows antisymmetry. Therefore, the generator of an L.T. must be an antisymmetric matrix.