Poincare Group

Prerequisites: Lorentz Group. Full path (8 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

$\color{red}{P(1, 3)} = \color{blue}{T(4)} \color{magenta}{\rtimes} \color{green}{SO(1, 3)}$

Intuitive

The Poincare group is the mathematical tool that we use to describe the symmetry of special relativity.

The starting point for Einstein on his road towards what is now called special relativity was the experimental observation that the speed of light has the same value in all inertial frames of reference. This curious fact of nature was discovered by the famous Michelson-Morley experiment.

A symmetry is a transformation that we can perform without changing something. Therefore, the invariance of the speed of light under arbitrary changes of the frame of reference is a symmetry and we call this symmetry the Poincare group. The Poincare group contains all transformations that we can perform without changing the speed of light.

Concrete

The $\color{red}{\text{Poincare group}}$ consists of $\color{blue}{\text{translations}}$ $\color{magenta}{\text{plus}}$ $\color{green}{\text{rotations and boosts}}$.

• A nice discussion can be found here Representations of Lorentz and Poincar´e groups by Joseph Maciejko and
• here: Representations of the Poincar´e group Prahar Mitra and
• here: http://pages.cs.wisc.edu/~guild/symmetrycompsproject.pdf
• see also page 211ff in Quantum Field Theory by Mark Srednicki

The Poincare Algebra

$$\begin{array}{rcl} {}[J_i,J_j]&=&i \epsilon_{ijk}J_k, \\ {} [J_i,K_j]&=& i\epsilon_{ijk} K_k, \\ {} [K_i,K_j]&=& -i \epsilon_{ijk}J_k , \\ {} [J_i,P_0]&=&0, \\ {} [J_i,P_j]&=& i\epsilon_{ijk} P_k, \\ {} [P_0,P_i]&=&0, \\ {} [K_i,P_0]&=&i P_i, \\ {}[K_i,P_j]&=& i\ P_0 \delta_{ij} , \end{array}$$

• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Lorentz and Poincaré Transformations"

Abstract

The Poincaré group is the isometry group of Minkowski space $\mathbb{R}^{1,3}$, the semidirect product $$ISO(1,3) = \mathbb{R}^{1,3}\rtimes O(1,3), \qquad (a,\Lambda)\,(a',\Lambda') = (a + \Lambda a',\ \Lambda\Lambda'),$$ acting on points as $x\mapsto\Lambda x + a$. The translations form an abelian normal subgroup, and Minkowski space is the homogeneous space $ISO(1,3)/O(1,3)$. The component connected to the identity is $\mathbb{R}^{1,3}\rtimes SO^+(1,3)$. Its universal cover is $\mathbb{R}^{1,3}\rtimes SL(2,\mathbb{C})$, where $SL(2,\mathbb{C})$ acts on four-vectors through Hermitian $2\times 2$ matrices, $x^\mu\sigma_\mu\mapsto A\,x^\mu\sigma_\mu A^\dagger$.

Why the cover. Quantum states are rays, so a symmetry acts by a projective unitary representation. By Bargmann's theorem, projective representations of a connected Lie group come from true unitary representations of a central extension of its universal cover, and the central extensions that matter are classified by $H^2(\mathfrak{g},\mathbb{R})$. For the Poincaré algebra this group vanishes, so the projective representations of the connected Poincaré group are exactly the unitary representations of $\mathbb{R}^{1,3}\rtimes SL(2,\mathbb{C})$. (For the Galilean algebra it does not vanish, and the extra central element is the mass.)

Wigner-Mackey classification. The unitary irreducible representations are obtained by induction. We use the signature $(+,-,-,-)$. Since translations are abelian, a representation decomposes into joint eigenstates of the momenta $P_\mu$, and $SL(2,\mathbb{C})$ permutes the eigenvalues $p$ along orbits in momentum space. An irreducible representation is fixed by one orbit together with an irreducible unitary representation of the stabiliser of a point $p$ on it, the little group:

• $p^2 = m^2 > 0$, $p^0>0$: little group $SU(2)$, labelled by the spin $s\in\{0,\frac12,1,\dots\}$.
• $p^2 = 0$, $p^0>0$: little group $\widetilde{ISO}(2)$, the double cover of the Euclidean group of the plane. Its finite-dimensional unitary irreducible representations are one-dimensional and labelled by the helicity $\lambda\in\frac12\mathbb{Z}$. Its infinite-dimensional ones give the continuous spin representations.
• $p = 0$: little group $SL(2,\mathbb{C})$ itself.
• $p^2<0$: little group a double cover of $SO(1,2)$ (tachyonic representations).

The physical particle representations are those with $p^0\geq 0$ on the spectrum (the spectrum condition).

Casimirs. The centre of the enveloping algebra is generated by $P^2 = P_\mu P^\mu$ and $W^2 = W_\mu W^\mu$, with the Pauli-Lubanski vector $$W_\mu = \frac12\,\epsilon_{\mu\nu\rho\sigma}J^{\nu\rho}P^\sigma .$$ On a massive representation $P^2 = m^2$ and $W^2 = -m^2 s(s+1)$. On a massless representation with finite-dimensional little group representation, $W_\mu = \lambda P_\mu$, where the overall sign depends on the convention for $\epsilon_{0123}$.

• For a modern discussion of the Poincare group, see D. Giulini, The Poincare group: Algebraic, representation-theoretic, and geometric

aspects.

Why is it interesting?

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

The Poincare group is the set of all transformations that leave the speed of light invariant. Thus, the Poincare group yields all possible transformations between allowed frames of reference. This is incredibly useful, when we want to write down fundamental laws of nature. The fundamental laws should be valid in all allowed frames of reference, otherwise they would be quite useless.

In practice, we can use our knowledge of all transformations inside the Poincare group to write down equations that are invariant under all these transformations. These equations then hold in all allowed frames of reference. This is such a strong restriction on the possible equations that is almost enough to derive the most important equations of fundamental physics: the Dirac equation, the Klein-Gordon equation and the Maxwell-Equations.

"The Hilbert space of one-particle states is always an irreducible representation space of the Poincare group. […] The 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). […] The translation generators are responsible for the energy-momentum conservation laws, the rotation generators of the conservation of angular momentum, and the boost generators of the conservation of initial position. " from Reflections on the Evolution of Physical Theories by Henri Bacry

"The enlargement of the Lorentz group to the Poincare group was proposed [ 13] as a way of describing the quantum states of relativistic particles without using the wave equations. The states of a free particle are then given by the unitary irreducible representations of the Poincare group." from Deformed Poincare containing the exact Lorentz algebra by Alexandros A. Kehagias et. al.

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Newton's Second Law
5. Vector Calculus
6. Newtonian Mechanics
7. Special Relativity
8. Lorentz Group

Leads to: Conformal Group, De Sitter Group, DeSitter Group, Supersymmetry, The Coleman-Mandula Theorem, Weinberg-Witten Theorem, Wigner's Little Groups.