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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 ...
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 ...
Specifically, its Lie algebra (which captures the group’s algebraic structure) is itself a Cartan subalgebra. When we consider the identity component of a subgroup, it shares the same Lie algebra. However, there isn’t a universally agreed-upon definition for which subgroup with this property should be called the ‘Cartan subgroup ...
Let be a representation i.e. a homomorphism: of a group where is a vector space over a field.If we pick a basis for , can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called a matrix representation.
According to Bargmann's theorem, every projective unitary representation of the Poincaré group comes from an ordinary unitary representation of its universal cover, which is a double cover. (Bargmann's theorem applies because the double cover of the Poincaré group admits no non-trivial one-dimensional central extensions .)
In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz invariant (i.e., they transform under the trivial representation). An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use ...
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group. PGL(V) = GL(V) / Z(V)