Lorentz group

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The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. It is the subgroup of the Poincaré group consisting of all isometries that leave the origin fixed. Sometimes this group is referred to as the homogeneous Lorentz group while the larger Poincaré group is called the inhomogeneous Lorentz group.

Mathematically, the Lorentz group is the generalized orthogonal group O(1, 3), (or O(3, 1) depending on the sign convention). It is a 6-dimensional noncompact Lie group which is not connected, and whose connected components are not simply connected. Its subgroups include the rotation group SO(3).

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Components

The Lorentz group O(1, 3) has four connected components. The elements in each component are characterized by whether or not they reverse the orientation of space and/or time. A Lorentz transformation which reverses either the orientation of time or space (but not both) has determinant −1, while the rest have determinant +1. A proper Lorentz transformation is one with determinant +1. The subgroup of proper Lorentz transformations is denoted SO(1, 3). A transformation which preserves the orientation of time (relative to the orientation of space) is called orthochronous. The subgroup of orthochronous transformations is often denoted O+(1, 3) or something similar. The identity component of the Lorentz group is the set of all Lorentz transformations preserving both the orientation of space and time. It is called the proper, orthochronous Lorentz group, and is denoted by SO+(1, 3).

Warning: Often times people will refer to SO(1, 3) or even O(1, 3) when they actually mean SO+(1, 3).

Every element in O(1,3) can be written as the product of a proper, orthochronous transformation and an element of the discrete group {1, P, T, PT} where P and T are the space inversion and time reversal operators:

P = diag(1, −1, −1, −1)
T = diag(−1, 1, 1, 1)

The proper, orthochronous Lorentz group

As stated above the proper, orthochronous Lorentz group SO+(1, 3) — also called the restricted Lorentz group — is the identity component of the Lorentz group. That is, it consists of all Lorentz transformations which can be continuously connected to the identity.

The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction). The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO(3). The set of all boosts, however, does not form a subgroup (composing two boosts does not, in general, result in another boost). However, a boost in any one spatial direction does generate a one-parameter subgroup. An arbitrary rotation is specified by 3 real parameters, as is an arbitrary boost. Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost it takes 6 parameters to describe an arbitrary transformation.

The covering spin group

Like the rotation group SO(3), the restricted Lorentz group is not simply connected; rather, it is doubly connected. That is, the fundamental group of SO+(1, 3) is isomorphic to Z2.

The universal cover of the restricted Lorentz group can be identified with the special linear group SL(2, C). The restricted Lorentz group is isomorphic to the quotient group SL(2, C)/{±1} also known as the projective linear group, PSL(2, C). This group shows up in another guise as the group of all Möbius transformations of the Riemann sphere.

In applications to quantum mechanics the group SL(2, C) is sometimes called the Lorentz group.

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