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Tensor products of representations appear at the Lie algebra level as either of [nb 18]. Here, the latter interpretation, which follows from G6 , is intended. Explicit realizations and group representations are given later. Here the tensor product is interpreted in the former sense of A0.

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These representations are concretely realized below. These are, up to a similarity transformation , uniquely given by [nb 20]. These are explicitly given as [68]. The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. From a practical point of view, it is important whether the first formula in G2 can be used for all elements of the group. The Lorentz group is doubly connected , i.

The trace and determinant conditions imply: [77]. See the section spinors.

From the relations. Define the set. This means that p A belongs to the full Lorentz group SO 3; 1.

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They can be and usually are written down from scratch. The holomorphic group representations meaning the corresponding Lie algebra representation is complex linear are related to the complex linear Lie algebra representations by exponentiation. They can be exponentiated too. These are usually indexed with only one integer but half-integers are used here.

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of i and there is no factor of i in the exponential mapping compared to the physics convention used elsewhere. Accordingly, the corresponding projective representation of the group is never unitary. In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations.

The kernel of u is a normal subgroup of G. Since G is simple, ker u is either all of G , in which case u is trivial, or ker u is trivial, in which case u is faithful. This would mean that u G is an embedded non-compact Lie subgroup of the compact group U V.

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian. It is the SO 3 -invariant subspaces of the irreducible representations that determine whether a representation has spin. It cannot be ruled out in general, however, that representations with multiple SO 3 subrepresentations with different spin can represent physical particles with well-defined spin.

It may be that there is a suitable relativistic wave equation that projects out unphysical components , leaving only a single spin. The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:.

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Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. This follows from that complex conjugation commutes with addition and multiplication. For simplicity, consider only the "discrete part" of End H , that is, given a basis for H , the set of constant matrices of various dimension, including possibly infinite dimensions.

The induced 4-vector representation of above on this simplified End H has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. In other words,. The []. Moreover, they have the commutation relations of the Lorentz Lie algebra, []. For details, see bispinor and Dirac algebra. There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations.

Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e. These representations are in general not irreducible. The Lorentz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations.

The reducible representations will therefore not be discussed. It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. This can hold only if A i and B i have the same dimensions, i.

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It must be specified separately. By explicitly including a representative for T , as well as one for P , a representation of the full Lorentz group O 3; 1 is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. These are interpreted as generators of translations. The time-component P 0 is the Hamiltonian H. The operator T satisfies the relation []. Such states do not exist. When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force , must be formulated in terms of Weyl spinors.

Without space parity inversion, it is not an irreducible representation. The third discrete symmetry entering in the CPT theorem along with P and T , charge conjugation symmetry C , has nothing directly to do with Lorentz invariance. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces. The subgroup SO 3 of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space.

An arbitrary square integrable function f one the unit sphere can be expressed as []. The Lorentz group action restricts to that of SO 3 and is expressed as.

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This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection , the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.

The Riemann P-functions , solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as []. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is []. 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. The resulting function is again a Riemann P-function.

The new function is expressed as. The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. Walter Rudin. Marshall Hall. William S. Gordon James. Serge Lang.