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The full Lorentz group O(1, 3), the proper Lorentz group SO(1, 3) and the proper orthochronous Lorentz group SO + (1, 3) (the component connected to the identity) all have the same Lie algebra, which is typically denoted (,) .
Under a proper orthochronous Lorentz transformation x → Λx in Minkowski space, all one-particle quantum states ψ j σ of spin j with spin z-component σ locally transform under some representation D of the Lorentz group: [12] [13] () where D(Λ) is some finite-dimensional representation, i.e. a matrix.
In the context of the Dirac equation and Weyl equation, the Weyl spinors satisfying the Weyl equation transform under the simplest irreducible spin representations of the Lorentz group, since the spin quantum number in this case is the smallest non-zero number allowed: 1/2.
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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 simple and thus semisimple, but is not connected, and none of its components are simply connected. Furthermore, the Lorentz group is not compact. [31]
In physics, the Lorentz group O(1,3) is of central importance, being the setting for electromagnetism and special relativity. (Some texts use O(3,1) for the Lorentz group; however, O(1,3) is prevalent in quantum field theory because the geometric properties of the Dirac equation are more natural in O(1,3).)
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
Lorentz covariance has two distinct, but closely related meanings: A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors.