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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 ...
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 scalar W μ W μ is a Lorentz-invariant operator, and commutes with the four-momentum, and can thus serve as a label for irreducible unitary representations of the Poincaré group. That is, it can serve as the label for the spin , a feature of the spacetime structure of the representation, over and above the relativistically invariant label ...
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.
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
Now, the link comes as the third sentence (or something like that). What I really want to do is to write an introduction to Representation theory of the Lorentz group#Finite-dimensional representations as well as a smaller one to Representation theory of the Lorentz group#Finite-dimensional representations#The Lie algebra, if you see my point.
It is the group of orientation-preserving projective transformations of the real projective line R ∪ {∞}. It is the group of conformal automorphisms of the unit disc. It is the group of orientation-preserving isometries of the hyperbolic plane. It is the restricted Lorentz group of three-dimensional Minkowski space.