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(The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1, 3). The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO + (1, 3). [a]
The relations between the primed and unprimed spacetime coordinates are the Lorentz transformations, each coordinate in one frame is a linear function of all the coordinates in the other frame, and the inverse functions are the inverse transformation. Depending on how the frames move relative to each other, and how they are oriented in space ...
that carry both the indices (x, α) operated on by Lorentz transformations and the indices (p, σ) operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection. [25] To exhibit the connection, subject both sides of equation to a Lorentz transformation resulting in for e.g. u,
In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.
The transformations of these functions in spacetime are given below. Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψ σ locally transform under some representation D of the Lorentz group: [8] [9]
Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ: ′ =. In index notation, the contravariant and covariant components transform according to, respectively: ′ =, ′ = in which the matrix Λ has components Λ μ ν in row μ and column ν, and the matrix (Λ −1) T has components Λ ...
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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.