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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 ...
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 ...
A representation of the Lie algebra so(3,1) of the Lorentz group O(3,1) will emerge among matrices that will be chosen as a basis (as a vector space) of the complex Clifford algebra over spacetime. These 4×4 matrices are then exponentiated yielding a representation of SO(3,1) + .
The Lorentz group is a symmetry group of every relativistic quantum field theory. Wigner's early work laid the ground for what many physicists came to call the group theory disease [1] in quantum mechanics – or as Hermann Weyl (co-responsible) puts it in his The Theory of Groups and Quantum Mechanics (preface to 2nd ed.), "It has been rumored ...
The existence of this invariant symbol follows from a result in the representation theory of the Lorentz group or more properly its Lie algebra. Labeling irreducible representations by ( j , ȷ ¯ ) {\displaystyle (j,{\bar {\jmath }})} , the spinor and its complex conjugate representations are the left and right fundamental representations
The satisfy the Lorentz algebra, and turn out to exponentiate to a representation of the spin group (,) of the Lorentz group (,) (strictly, the future-directed part (,) + connected to the identity). The S μ ν {\displaystyle S^{\mu \nu }} are then the spin generators of this representation.
showing that the quantity of γ μ can be viewed as a basis of a representation space of the 4 vector representation of the Lorentz group sitting inside the Clifford algebra. The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group , which is η Λ T η = Λ − 1 , {\displaystyle ...
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.