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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 ...
Lorentz covariance has two distinct, but closely related meanings: A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors.
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.
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
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.
A bispinor field () transforms according to the rule ′ (′) = [] (′) = [] ()where is a Lorentz transformation.Here the coordinates of physical points are transformed according to ′ =, while , a matrix, is an element of the spinor representation (for spin 1/2) of the Lorentz group.
It was a key step towards the modern classification scheme of particle types, according to which particle types are partly characterized by which representation of the Lorentz group under which it transforms. The Lorentz group is a symmetry group of every relativistic quantum field theory. Wigner's early work laid the ground for what many ...