Search results
Results from the WOW.Com Content Network
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 ...
For certain types of Lie groups, namely compact [2] and semisimple [3] groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such groups, a typical goal of representation theory is to classify all finite-dimensional irreducible representations of ...
Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
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.
The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) representation of the group SO(3). (This fact is the origin of statements to the effect that "if you rotate the wave function of an electron by 360 degrees, you get the negative of the original wave function.")
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 ...
Representation theory of SL2(R) Representation theory of SU(2) Representation theory of the Galilean group; Representation theory of the Lorentz group; Representation theory of the Poincaré group; Representations of classical Lie groups; Representative function