﻿ Special Relativity - Groups Contents

## Notes on Groups

In the context of special relativity, groups always refer to a defining set of transformations to an inertial grid. There are 4 basic kinds of transformations from one inertial grid to another:

• displacement in space or time: change the place selected for the origin (x y z)=(0 0 0), or the moment in time for which t=0.
• reflection in space or time: swap the plus and minus directions for either 1 or 3 of the xyz axes. The case of swapping 2 of the xyz axes is actually the same as a reorientation, so it's not counted as a reflection. You can reflect the time as well, simply by having the clocks go backwards into the future. These operations are often referred to as P for parity, and T for time-reversal.
• reorientation in space: change the directions of the spatial xyz axes, to point to some other fixed directions.
• boost: let the grid move along one of the xyz axes (a fixed direction), at a constant speed with respect to a given grid (this is the only change that involves one grid moving with respect to another).

Somewhat confusingly, the jargon related to the groups used in special relativity varies among authors.

As a good starting point, here are the definitions used by Steven Weinberg in his book Quantum Theory of Fields. Be warned that other variations exist.

Weinberg
Transformation123
Displacement in space-time yn-
Reflection in space-time (P, T) yyn
Reorientation in space yyy
Boost yyy

S. Weinberg, The Quantum Theory of Fields, vol I:

• Weinberg 1: Poincaré Group, or Inhomogenous Lorentz Group.
• Weinberg 2: Homogeneous Lorentz Group. Leaves out displacements.
• Weinberg 3: Proper Orthochronous Lorentz Group. Leaves out parity and time-reversal. Displacement may or may not be included as an additional criterion.

According to Weinberg, the full Poincaré Group corresponds to the full set of transformations that leave dxidxi invariant. It has 10 continuous parameters: 4 for translation, 6 for space-time reorientations. The previous statement leaves out the discrete symmetries of P and T. The space-time reorientations can be broken down into 3 space reorientations, and 3 boosts.

According to Weinberg, the topology of Minkowski space-time is R(4) x R(3) x S(3)/Z2, and its doubly-connected. Here, S(3)/Z2 is the 3-sphere, but with opposite points identified.

Transformation Min. Lan. Hawk.
Displacement in space-time y y y
Reflection in space-time (P, T) n n n
Reorientation in space y y y
Boost y y y

• H. Minkowski, Space and Time. He names the group Gc. He uses geometrical constructions instead of the algebraic equations for a boost.
• Landau and Lifshitz, The Classical Theory of Fields. No group is named. They state the desire to derive the coordinate transformation between any two inertial grids, using the invariance of the squared-interval. The transformations are enumerated, but leave out the reflections.
• Hawking and Ellis, The Large-Scale Structure of Space-Time. They name the group the Inhomogeneous Lorentz Group. They use Killing vectors as a starting point. Note that this differs from Weinberg.
• Robert Wald, General Relativity. He refers to the Poincaré transformations as consisting in all linear transformations which leave the square-interval unchanged (as does Weinberg). He doesn't enumerate the transformations.

Two groups related to the Poincaré Group:

• Conformal Group
• ISL(2,C)

The Conformal Space-Time Group contains the Poincaré Group as a subgroup. It's defined by insisting that only light-like intervals are invariant. It transforms a light cone into itself, and leaves the speed limit invariant. More generally, it preserves light-like, time-like, and space-like separations (causal structure). It has 15 parameters. In this case, dxidxi is in general proportional, but not equal, between grids. (I think the "constant" of proportionality is allowed to vary across space-time.) Geometrically, conformal transformations always preserve angles. "The maximal spacetime symmetry group of massless particles is the conformal group. (link)"

The ISL(2,C) group is a covering group of the Poincaré Group. It leads to spinors, which are, I believe, a generalization of tensors.

In geometrodynamics, knowledge of the light-cone goes most of the way towards giving you the metric - what's missing is a conformal factor that varies from event to event.

Weinberg defines the Lorentz Transformation to be the full affine form, with no restriction on reflections. Other authors exclude displacements and/or reflections from being included in what they call Lorentz Transformations. So the definition of what exactly constitutes a Lorentz Transformation varies between authors.

Many physicists refer to affine transformations (which include displacements) as linear transformations. Mathematicians would never do that.

In his 1972 book Gravitation and Cosmology, Weinberg does not use the term orthochronous. Instead, he uses the term proper to refer to both P and T.

The non-continuous transformations P and T seem to have a special status. Some fundamental interactions in physics are P-invariant and T-invariant, but others are not.

Some other group names that I've seen (again, this may vary between authors):
Transformation O(1,3)SO(1,3)SO+(1,3)R(4) SO(3)
Displacement in space-time n n n y n
Reflection in space P y n n n n
Reflection in time T y y n n n
Rotation in space y y y n y
Boost y y y n n

SL(2,C) are 2x2 complex matrices with det=+1, 3 complex params and 6 real params. SL(2,C) is a covering group of the Homogeneous Lorentz Group O(1,3).

U(n) are nxn Hermitian (inverse = conjugate) matrices.

SU(n) are nxn Hermitian matrices with det=+1.

SU(2) is a double-cover of SO(3).

Wikipedia: "A Lie group G is said to be simply connected if every loop in G can be shrunk continuously to a point in G.... An example of a simply connected group is the special unitary group SU(2), which as a manifold is the 3-sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics."

In a chat with Eric Weinstein, Roger Penrose talks about a connection between relativistic aberration, 2-spinors, and conformal transformations of the Riemann sphere onto itself (stereographic projection). This relates to the Penrose paper about the invariance of circular outlines of objects, as the intersection of a past light cone of a detector with an object's history. That circularity apparently has as its source the same math of the stereographic projection. It might be useful to note that a circular outline is an invariant property, since there don't seem to be many invariant properties in optics.

Link: "The double cover of the Poincaré group is the fundamental spacetime symmetry of modern physics and is a crucial component of the standard model of particle physics."

"The Hilbert space of one-particle states is always an irreducible representation space of the Poincare group... T he construction of the unitary irreducible representations of the Poincare group is probably the most successful part of special relativity (in particle physics, not in gravitation theory, for which it is a disaster). It permits us to classify all kinds of particles and implies the main conservation laws (energy-momentum and angular momentum)."