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In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz. For example, the following laws, equations, and theories respect Lorentz symmetry:
This notation is related to the notation O + (1, 3) for the orthochronous Lorentz group, where the + refers to preserving the orientation on the first (temporal) dimension. The group O( p , q ) is also not compact , but contains the compact subgroups O( p ) and O( q ) acting on the subspaces on which the form is definite.
The action of the Lorentz group on the space of field configurations (a field configuration is the spacetime history of a particular solution, e.g. the electromagnetic field in all of space over all time is one field configuration) resembles the action on the Hilbert spaces of quantum mechanics, except that the commutator brackets are replaced ...
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.
[R 43] [R 44] However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the Lorentz group. [R 45] In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group Con(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.
The commutation relations of the Hamiltonian, and the Lorentz generators, guarantee that Lorentz invariance implies rotational invariance, so that any state can be rotated by 180 degrees. Since a sequence of two CPT reflections is equivalent to a 360-degree rotation, fermions change by a sign under two CPT reflections, while bosons do not.
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.