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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 ...
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 ...
It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space —the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events.
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 ...
It forms an indefinite orthogonal group called the Lorentz group O(1,n), while the case det g=+1 forms the restricted Lorentz group SO(1,n). The quadratic form becomes the Lorentz interval in terms of an indefinite quadratic form of Minkowski space (being a special case of pseudo-Euclidean space ), and the associated bilinear form becomes the ...
The group of translational symmetries of the associated phase space, , is the commutative group . In the usual quantum mechanical picture, the symmetry is not implemented by a unitary representation of . After all, in the quantum setting, translations in position space and translations in momentum space do not commute.
Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector ^ = (,,), and a rotation angle θ about a three-dimensional unit vector ^ = (,,) defining an axis, so ^ = (,,) and ^ = (,,) are together six parameters of the Lorentz group (three for rotations and three for boosts). The ...
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.