Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
For example, the following laws, equations, and theories respect Lorentz symmetry:

The kinematical laws of special relativity
The local structure of general relativity
Maxwell's field equations in the theory of electromagnetism
The Dirac equation in the theory of the electron
The Standard Model of particle physics

The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variation is negligible, physical laws are Lorentz-invariant.

Basic properties

The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave a single point fixed. Thus, the Lorentz group is the isotropy subgroup with respect to a point of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations.

Physics definition

Assume two inertial reference frames and, and two points,, the Lorentz group is the set of all the transformations between the two reference frames that preserve the speed of light propagating between the two points:
In matrix form these are all the linear transformations such that:
These are then called Lorentz transformations.

Mathematical definition

Mathematically, the Lorentz group may be described as the indefinite orthogonal group, the matrix Lie group that preserves the quadratic form
on . This quadratic form is, when put on matrix form, interpreted in physics as the metric tensor of Minkowski spacetime.

Note on notation

Both and are in common use for the Lorentz group. The first refers to matrices which preserve a metric of signature with, and the second refers to a metric of signature. Because the overall sign of the metric is irrelevant in the defining equation, the resulting groups of matrices are identical. There appears to be a modern push from some sectors to adopt notation versus, but the latter still finds plenty of use in current practice, and a great deal of the historical literature employed it. Everything described in this article applies to notation as well, mutatis mutandis. These considerations extend to related definitions as well (ex. vs.

Mathematical properties

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. The identity component of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted. The restricted Lorentz group consists of those Lorentz transformations that preserve both the orientation of space and the direction of time. Its fundamental group has order 2, and its universal cover, the indefinite spin group, is isomorphic to both the special linear group and to the symplectic group. These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably spinors. Thus, in relativistic quantum mechanics and in quantum field theory, it is very common to call the Lorentz group, with the understanding that is a specific representation of it.
A recurrent representation of the action of the Lorentz group on Minkowski space uses biquaternions, which form a composition algebra. The isometry property of Lorentz transformations holds according to the composition property.
Another property of the Lorentz group is conformality or preservation of angles. Lorentz boosts act by hyperbolic rotation of a spacetime plane, and such "rotations" preserve hyperbolic angle, the measure of rapidity used in relativity. Therefore, the Lorentz group is a subgroup of the conformal group of spacetime.
Note that this article refers to as the "Lorentz group", as the "proper Lorentz group", and as the "restricted Lorentz group". Many authors use the name "Lorentz group" for rather than. When reading such authors it is important to keep clear exactly which they are referring to.

Connected components

Because it is a Lie group, the Lorentz group is a group and also has a topological description as a smooth manifold. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.
The four connected components can be categorized by two transformation properties its elements have:

Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector
Some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein

Lorentz transformations that preserve the direction of time are called . The subgroup of orthochronous transformations is often denoted. Those that preserve orientation are called proper, and as linear transformations they have determinant. The subgroup of proper Lorentz transformations is denoted.
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.
The set of the four connected components can be given a group structure as the quotient group, which is isomorphic to the Klein four-group. Every element in can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group
where P and T are the parity and time reversal operators:
Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite-dimensional Lie groups.

Restricted Lorentz group

The restricted Lorentz group is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a continuous curve lying in the group. The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension, in this case with dimension six.
The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts. Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost, it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six-dimensional.
The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group. The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. A boost in some direction, or a rotation about some axis, generates a one-parameter subgroup.

Surfaces of transitivity

If a group acts on a space, then a surface is a surface of transitivity if is invariant under and for any two points there is a such that. By definition of the Lorentz group, it preserves the quadratic form
The surfaces of transitivity of the orthochronous Lorentz group, acting on flat spacetime are the following:

is the upper branch of a hyperboloid of two sheets. Points on this sheet are separated from the origin by a future time-like vector.
is the lower branch of this hyperboloid. Points on this sheet are the past time-like vectors.
is the upper branch of the light cone, the future light cone.
is the lower branch of the light cone, the past light cone.
is a hyperboloid of one sheet. Points on this sheet are space-like separated from the origin.
The origin.

These surfaces are, so the images are not faithful, but they are faithful for the corresponding facts about. For the full Lorentz group, the surfaces of transitivity are only four since the transformation takes an upper branch of a hyperboloid to a lower one and vice versa.

As symmetric spaces

An equivalent way to formulate the above surfaces of transitivity is as a symmetric space in the sense of Lie theory. For example, the upper sheet of the hyperboloid can be written as the quotient space, due to the orbit-stabilizer theorem. Furthermore, this upper sheet also provides a model for three-dimensional hyperbolic space.

Representations of the Lorentz group

These observations constitute a good starting point for finding all infinite-dimensional unitary representations of the Lorentz group, in fact, of the Poincaré group, using the method of induced representations. One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are called little groups by physicists. The problem is then essentially reduced to the easier problem of finding representations of the little groups. For example, a standard vector in one of the hyperbolas of two sheets could be suitably chosen as. For each, the vector pierces exactly one sheet. In this case the little group is, the rotation group, all of whose representations are known. The precise infinite-dimensional unitary representation under which a particle transforms is part of its classification. Not all representations can correspond to physical particles. Standard vectors on the one-sheeted hyperbolas would correspond to tachyons. Particles on the light cone are photons, and more hypothetically, gravitons. The "particle" corresponding to the origin is the vacuum.