The rotation and Lorentz groups and their representations for physicists

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Representation theory of the Lorentz group

The Lorentz group starts with a group of four-by-four matrices performing Lorentz transformations on the four-dimensional Minkowski space of t, z, x, y. There are three generators of rotations and three boost generators. Thus, the Lorentz group is a six-parameter group. The Lorentz group starts with a group of four-by-four matrices performing Lorentz transformations on the four-dimensional Minkowski space of.

The transformation leaves invariant the quantity. It was Einstein who observed that this Lorentz group is also applicable to the four-dimensional energy and momentum space of. In this way, he was able to derive his Lorentz-covariant energy—momentum relation commonly known as.

This transformation leaves invariant.

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In other words, the particle mass is a Lorentz-invariant quantity. Let us start with rotations applicable to the coordinates.

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The four-by-four matrix for this operation is. The matrix J 3 is known as the generator of the rotation around the z -axis. It is not difficult to write the generators of rotations around the x - and y -axes, and they can be written as J 1 and J 2 , respectively, with. It is then possible to write the matrices for the generators K 1 and K 2 , as.

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There are six generators of the Lorentz group and they satisfy the three sets of commutation relations given in 1. It is said that the Lie algebra of the Lorentz group consists of these sets of commutation relations. These commutation relations are invariant under Hermitian conjugation. While the rotation generators are Hermitian, the boost generators are anti-Hermitian.

Thus, it is possible to construct two four-by-four representations of the Lorentz group, one with K i and the other with For this purpose we shall use the notation Berestetskii , Kim and Noz Since there are two representations, transformations with K i are called the covariant transformations, while those with are called contravariant transformations.

Let us define. These two-by-two matrices satisfy the Lie algebra of the Lorentz group given in 1.

1.2. Two-by-two representation of the Lorentz group

Since its determinant is fixed and is equal to one, there are six independent parameters. This six-parameter group is commonly called. Since the Lorentz group has six generators, this two-by-two matrix can serve as a representation of the Lorentz group. It is said in the literature that serves as the covering group for the Lorentz group. For each G -matrix of , there exists one four-by-four Lorentz transformation matrix.

We can start with the Minkowskian four-vector written as. The correspondence between the two-by-two and four-by-four representations of the Lorentz group along with the generators are given in table 1. These representations can be used for coordinate or momentum transformations, as well as other four-vector quantities such as electromagnetic four-potentials. We can now consider the transformation. The transformation of 1. Lowering the index of spinors of higher rank. An observation on tensors. A bispinor of the first rank. Irreducible representations of the general group. The ClebschGordan coefficients.

Section 7. The additional formula in the matrix elements. Expansion of arbitrary quantities.

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Definition of invariant equations. Solution of invariant equations. Solution of Dirac equations. Invariant equation with x 0. The ClebschGordan coefficients for the case when. Surfaces in fourdimensional space which are transi. The relation between the proper Lorentz group and. The relation between representations of the proper. The unique and twovalued representations of the proper. Unitary irreducible representations of the proper.

The determination of the matrices Lo 1 1 La L. Equations invariant with respect to the complete. The invariant Lagrangian function. Quantities formed from the wave function and. The Daffine equation for vector particles. The PauliFierz equation.

Mod-02 Lec-21 Finite dimensional representations of the Lorentz group, Euclidean and Galilean groups

The rest system The rest mass. The polarization of particles with zero rest mass.