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The most general proper Lorentz transformation Λ(v, θ) includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, Λ(0, θ) = R(θ) and Λ(v, 0) = B(v). An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here.
(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]
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.
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.
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,
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]
The transformation properties of the electromagnetic field tensor, including invariance of electric charge, are then used to transform to the lab frame, and the resulting expression (again Lorentz force law) is interpreted in the spirit of Newton's second law, leading to the correct expression for the relativistic three- momentum. The ...
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.