Lorentz group

In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity.

There appears to be a modern push from some sectors to adopt (1,3) notation versus (3,1), but the latter still finds plenty of use in current practice, and a great deal of the historical literature employed it.

These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably spinors.

A recurrent representation of the action of the Lorentz group on Minkowski space uses biquaternions, which form a composition algebra.

For the full Lorentz group, the surfaces of transitivity are only four since the transformation T takes an upper branch of a hyperboloid (cone) to a lower one and vice versa.

For example, the upper sheet of the hyperboloid can be written as the quotient space SO+(1, 3) / SO(3), due to the orbit-stabilizer theorem.

This presentation, the Weyl presentation, satisfies Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as a real vector space) with Minkowski spacetime, in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime.

is the Hermitian transpose of S. This action preserves the determinant and so SL(2, C) acts on Minkowski spacetime by (linear) isometries.

[b] It is important to observe that this pair of coverings does not survive quantization; when quantized, this leads to the peculiar phenomenon of the chiral anomaly.

The classical (i.e., non-quantized) symmetries of the Lorentz group are broken by quantization; this is the content of the Atiyah–Singer index theorem.

is Written as the four-vector, the relationship is This transforms as Taking one more transpose, one gets The symplectic group Sp(2, C) is isomorphic to SL(2, C).

An example of each type is given in the subsections below, along with the effect of the one-parameter subgroup it generates (e.g., on the appearance of the night sky).

The one-parameter subgroup it generates is obtained by taking θ to be a real variable, the rotation angle, instead of a constant.

The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles.

The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about the z axis as θ increases.

Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin.

The spinor map converts this to the Lorentz transformation The one-parameter subgroup this generates is obtained by replacing η + iθ with any real multiple of this complex constant.

The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles).

The spinor map converts this to the matrix (representing a Lorentz transformation) This generates a two-parameter abelian subgroup, which is obtained by considering α a complex variable rather than a constant.

Differentiating this transformation with respect to the now real group parameter α and evaluating at α = 0 produces the corresponding vector field (first order linear partial differential operator), Apply this to a function f(t, x, y, z), and demand that it stays invariant; i.e., it is annihilated by this transformation.

The solution of the resulting first order linear partial differential equation can be expressed in the form where F is an arbitrary smooth function.

The case c3 = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes.

A particular null line lying on the light cone is left invariant; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above.

In the above explicit example, a massless particle moving in the z direction, so with 4-momentum P = (p, 0, 0, p), is not affected at all by the x-boost and y-rotation combination Kx − Jy defined below, in the "little group" of its motion.

This is evident from the explicit transformation law discussed: like any light-like vector, P itself is now invariant; i.e., all traces or effects of α have disappeared.

(The other similar generator, Ky + Jx as well as it and Jz comprise altogether the little group of the light-like vector, isomorphic to E(2).

Then a given point on the celestial sphere can be associated with ξ = u + iv, a complex number that corresponds to the point on the Riemann sphere, and can be identified with a null vector (a light-like vector) in Minkowski space or, in the Weyl representation (the spinor map), the Hermitian matrix The set of real scalar multiples of this null vector, called a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars.

From the Möbius side, SL(2, C) acts on complex projective space CP1, which can be shown to be diffeomorphic to the 2-sphere – this is sometimes referred to as the Riemann sphere.

It may be helpful to briefly recall here how to obtain a one-parameter group from a vector field, written in the form of a first order linear partial differential operator such as The corresponding initial value problem (consider

with some initial conditions) is The solution can be written or where we easily recognize the one-parameter matrix group of rotations exp(iλJz) about the z-axis.