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The Lorentz group, a Lie group on which special relativity is based, has a wide variety of representations. Many of these representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics in the description of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory. This representation theory also provides the theoretical ground for the concept of spin, which, for a particle, can be either integer or half-integer in the unit of the reduced Planck constant ℏ. Quantum mechanical wave functions representing particles with half-integer spin are called spinors. The classical electromagnetic field has spin as well. It transforms under a representation with spin one. It enters into general relativity because in small enough regions of spacetime, physics is that of special relativity. The group may also be represented in terms of a set of functions defined on the Riemann sphere. These are the Riemann P-functions, which are expressible as hypergeometric functions. The identity component of the Lorentz group is isomorphic to the Möbius group, and hence any representation of the Lorentz group is necessarily a representation of the Möbius group and vice versa.The subgroup with its representation theory form a simpler theory, but the two are related and both are prominent in theoretical physics as descriptions of spin, angular momentum, and other phenomena related to rotation. The adopted Lie algebra basis and conventions used are presented here. == Finite-dimensional representations == Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The full Lorentz group is no exception. The Lorentz group has some properties that makes it "agreeable" and others that make it "not very agreeable" within the context of representation theory. The group is semisimple, and also simple, but is not connected, and none of its components are simply connected. Perhaps most importantly, the Lorentz group is not compact.〔These facts can be found in most elementary mathematics texts, and many physics texts. See e.g. , 〕 For finite-dimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a well-developed theory. In addition, all representations are built from the irreducible ones.〔See e.g. 〕 But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to spin representations of the group.〔 Appendix D2.〕 The non-connectedness means that, for representations of the full Lorentz group, one has to deal with time reversal and space inversion separately.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Representation theory of the Lorentz group」の詳細全文を読む スポンサード リンク
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