Representation theory of the Lorentz group

This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations.

[2] The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, Harish-Chandra, later turned mathematician,[nb 3] in 1947.

[6] There were speculative theories,[7][8] (tensors and spinors have infinite counterparts in the expansors of Dirac and the expinors of Harish-Chandra) consistent with relativity and quantum mechanics, but they have found no proven physical application.

[9] While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT,[10] it is the case that so far all quantum field theories can be approached this way, including the standard model.

Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional unitary representations exist.

[39][40] In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Élie Cartan.

Mathematicians Hermann Weyl[41][45][37][46][47] and Harish-Chandra[48][49] and physicists Eugene Wigner[50][51] and Valentine Bargmann[52][53][54] made substantial contributions both to general representation theory and in particular to the Lorentz group.

The objects in the following list are in one-to-one correspondence: Tensor products of representations appear at the Lie algebra level as either of[nb 12] where Id is the identity operator.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers m = μ and n = ν, conventionally written as one of

For a projective representation Π of SO(3; 1)+, it holds that[72] since any loop in SO(3; 1)+ traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map.

[95] The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.

According to the above paragraph, there are subspaces with spin both ⁠3/2⁠ and ⁠1/2⁠ in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under SO(3).

are the gamma matrices, the sigmas, only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space.

If m = n it can be extended to a representation of all of O(3; 1), the full Lorentz group, including space parity inversion and time reversal.

A special case of this construction is when V is a space of functions defined on the a linear group G itself, viewed as a n-dimensional manifold embedded in

An arbitrary square integrable function f on the unit sphere can be expressed as[121] where the flm are generalized Fourier coefficients.

The Lorentz group action restricts to that of SO(3) and is expressed as where the Dl are obtained from the representatives of odd dimension of the generators of rotation.

The connection is[124] The set of constants 0, ∞, 1 in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation.

The first set of constants on the left hand side in (T1), a, b, c denotes the regular singular points of Riemann's differential equation.

also have infinite dimensional unitary representations, studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) at the instigation of Paul Dirac.

[131] In Dirac (1945) he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors.

By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions |k| + 2m + 1 with m a non-negative integer.

Since −I acts as (−1)k on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.

[61] The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties.

the triple (K1, K2, K3) ≡ K is a vector operator[148] and the Wigner–Eckart theorem[149] applies for computation of matrix elements between the states represented by the chosen basis.

where the superscript (1) signifies that the defined quantities are the components of a spherical tensor operator of rank k = 1 (which explains the factor √2 as well) and the subscripts 0, ±1 are referred to as q in formulas below, are given by[151]

The imposition of the requirement of unitarity of the corresponding representation of the group restricts the possible values for the arbitrary complex numbers j0 and ξj.

They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) VL and VR, whose elements ΨL and ΨR are called left- and right-handed Weyl spinors respectively.

[162] See Weinberg (2002, Chapter 5), Tung (1985, Section 10.5.2) and references given in these works.It should be remarked that high spin theories (s > 1) encounter difficulties.

It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in Greiner & Müller (1994, Section 15.2.