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Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string ...
The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected. [1] The identity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted SO ...
A complex representation of a group is an action by a group on a finite-dimensional vector space over the field .A representation of the Lie group G, acting on an n-dimensional vector space V over is then a smooth group homomorphism
In the case of the Lorentz group, the exponential map is just the matrix exponential. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.
The boost and rotation generators have representations denoted D(K) and D(J) respectively, the capital D in this context indicates a group representation. For the Lorentz group, the representations D(K) and D(J) of the generators K and J fulfill the following commutation rules.
This is a spin representation. When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the S(Λ) of above are of this form. The 6 dimensional space the σ μν span is the representation space
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos.It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
Let be a representation i.e. a homomorphism: of a group where is a vector space over a field.If we pick a basis for , can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called a matrix representation.