The Poincaré group $ISO(d,1)$ is the isometry group of Minkowski spacetime of dimension $d+1$. The classical case is $ISO(3, 1)$, the group of affine transformations on $\mathbb{R}^4$ which preserve the Minkowski "metric", i.e., the group of maps $f: \mathbb{R}^4 \to \mathbb{R}^4$ of the form $f(\vec{x}) = M \vec{x} + \vec{b}$ such that
where the quadratic form $Q(t, x, y, z) = t^2 - x^2 - y^2 - z^2$ is often called the Minkowski “norm”. The group elements are multiplied by composing maps.
The Poincaré group $G$ may also be described as a semidirect product
where $O(1, 3)$, the Lorentz group, consists of all linear transformations $L: \mathbb{R}^4 \to \mathbb{R}^4$ that preserve the Minkowski inner product of signature $(1, 3)$.
The Lorentz group is a 6-dimensional Lie group. It has four connected components; the connected component of the identity is called the special Lorentz group and is denoted $SO^+(1, 3)$.
to itself, and the $S$ indicates of course group elements which have determinant 1 as $4 \times 4$ matrices. It easily follows that elements of $SO^+(1, 3)$ preserve orientation of the spatial component $\mathbb{R}^3$.
Passage between the four components of the full Lorentz group is effected by composing with a time-reversal operator
and with a spatial inversion (or parity-reversal) operator
The special Lorentz group (also called the proper, orthochronous Lorentz group: “orthochronous” here means the forward light cone is mapped to itself, and “proper” means orientation-preserving) may be analyzed further. The subgroup of $SO^+(1, 3)$ that fixes the unit time-like vector
may be identified with the group of rotations $SO(3)$, since the restriction of the Minkowski norm to the spatial component $\mathbb{R}^3$ is minus the usual Euclidean norm, $-x^2 - y^2 - z^2$. This subgroup is of course 3-dimensional.
A general element $g \in SO^+(1, 3)$ may be decomposed uniquely in the form
where $\rho$ is a rotation and $\beta_v$ is a boost in the direction $v$ ($v \in S^2$ a unit spatial vector), mapping
for some parameter $\beta$ (called the rapidity), and acting as the identity on the spatial plane orthogonal to $v$. Thus a boost is described by a pair $(v, \beta)$, involving 3 parameters. (Warning: boosts do not compose to form a subgroup.) A boost can be thought of as a relativistic coordinate change from a “laboratory” frame of reference to the frame of reference of an observer moving inertially in the direction $v$ with speed $\tanh(\beta)$ (relative to the speed of light $c = 1$), as measured in the laboratory frame.
The universal cover of $SO^+(1, 3)$ is a double cover (the spin double cover)
constructed as follows: to each $x = (x_0, x_1, x_2, x_3)$ one associates a Hermitian matrix
whose determinant is the Minkowski norm of $x$. We thus identify $\mathbb{R}^4$ with the space $H$ of Hermitian matrices, and define an action of $SL_2(\mathbb{C})$ on $H$:
Observe that $A X A^*$ belongs to $H$. Also, $det(A \cdot X) = det(X)$ since $A$ has determinant 1, so the action preserves the Minkowski norm. Therefore the action
factors through the inclusion $O(1, 3) \hookrightarrow GL(\mathbb{R}^4)$. Furthermore, since $SL_2(\mathbb{C})$ is connected, the action $SL_2(\mathbb{C}) \to O(1, 3)$ factors through the connected component $SO^+(1, 3)$ of $O(1, 3)$. It is not hard to check that the kernel of the action is $\{I, -I\}$; therefore the map
is an open homomorphism between connected Lie groups of the same dimension, and is therefore surjective. In this way, we have produced an explicit identification
which exhibits $SL_2(\mathbb{C})$ as a double cover of $SO^+(1, 3)$; another way to say it is that $SO^+(1, 3)$ is identified with the group $PSL_2(\mathbb{C})$ of complex Moebius transformations. Finally, there is morphism of covering spaces
Here the inclusions are homotopy equivalences and the left projection is a universal covering map (as $SU(2) \cong S^3$ is simply connected), therefore $SL_2(\mathbb{C})$ is also simply connected and the projection on the right is a universal covering map. This is the spin double cover; it is crucial for getting a correct mathematical description of fermions in particle theory.
With regard to the inclusion maps above being homotopy equivalences, we remark in passing that the homogeneous space
is identified with the space of boost maps; concretely, each coset of $SO^+(1, 3)/SO(3)$ is of the form $\beta_v SO(3)$ for a unique boost map $\beta_v$. Topologically, the space of boost maps is $\mathbb{R}^3$ or a $3$-ball, hence contractible, and from the long exact homotopy sequence applied to the fibration
we deduce that the inclusion $i: SO(3) \to SO^+(1, 3)$ is a homotopy equivalence.
Similarly the lift to the double covers $SU(2) \to SL_2(\mathbb{C})$ is a homotopy equivalence. Since $SU(2) \cong S^3$ is identified with the space of unit quaternions, $SU(2)$ and $SL_2(\mathbb{C})$ are simply connected and hence the respective universal covering spaces of $SO(3)$ and $SO^+(1, 3)$.
As for any Lie group, there are various mechanisms for describing the Lie algebras of the Lorentz group and of the Poincaré group: by left-invariant vector fields, or by studying “infinitesimal generators” of 1-parameter subgroups, etc. We begin with the Lorentz group. See also Poincaré Lie algebra.
In the vector field picture, one often chooses a basis of the Lorentz algebra? consisting of six elements: the first three
describe rotational flows (around the $z$-, $x$-, and $y$-axes respectively), and the last three
describe hyperbolic or boost flows, (where the boosts are in the directions of the $x$-, $y$-, and $z$-axes, respectively). One may easily compute the commutators by hand and reproduce the standard gobbledygook formula given in physics texts:
where lower-case Greek letters range over $0, 1, 2, 3$ and $\eta$ is the $4 \times 4$ matrix representing the Minkowski quadratic form.
(I could be off by a sign here. It depends on whether $\eta$ has one plus and three minuses, or one minus and three pluses.)
By integrating the vector fields, we obtain in each of these cases flows or 1-parameter subgroups, e.g.,
$\,$
Differentiating these maps at $s = 0$ gives matrix representations for the Lie algebra elements $M_{\mu\nu}$.
For the 10-dimensional Poincaré algebra, we need to give, in addition to infinitesimal generators for rotations and boosts, four more elements which generate translations. In the vector field picture, these Lie algebra elements are represented by
and these of course commute; the brackets with the $M_{\mu\nu}$ are
Integration of the vector fields $P_\rho$ leads to the expected translations, e.g.,
The Poincaré group is basic to relativistic physics, since the fundamental principle of relativity is that physical laws are required to be invariant with respect to the action of the Poincaré group on spacetime. (There is some fine print here: in some cases, e.g., where physical laws are not invariant under space or time inversions, one must restrict to the action of the group $\mathbb{R}^4 \rtimes SO^+(1, 3)$. If one is dealing with fermions, one considers invariance with respect to an action of the universal cover $\mathbb{R}^4 \rtimes SU_2(\mathbb{C})$.)
Such physical laws may be classical or quantum, according to the description of physical states and observables in the theory. For example, in classical mechanics, pure states correspond to points in a symplectic phase space such as the cotangent bundle of Minkowski 4-space; in quantum mechanics, pure states are described by unit vectors in a suitable Hilbert space such as $L^2(\mathbb{R}^4)$. In either case, a relativistic theory will involve an action or representation of the Poincaré group, together with a structure which governs the dynamics of the theory, e.g., a Hamiltonian.
In the quantum case, a fundamental relativistic condition is that probability amplitudes $\langle \psi|\phi \rangle$, obtained by pairing an initial state $\phi$ with a final state $\psi$, are invariant under the action of the Poincaré group. This condition says
for every $g$ in the Poincaré group. Thus the representation of the Poincaré group on Hilbert space is required to be unitary. Due to the noncompactness of the Poincaré group, unitary representations on finite-dimensional Hilbert spaces are scarce; one must really pass to unitary representations on infinite-dimensional Hilbert spaces to get anything interesting.
In particular, an elementary particle in quantum physics is sometimes defined to be an irreducible unitary representation of the Poincaré group on $L^2(\mathbb{R}^4)$.
Euclidean group?
group | symbol | universal cover | symbol | higher cover | symbol |
---|---|---|---|---|---|
orthogonal group | $\mathrm{O}(n)$ | Pin group | $Pin(n)$ | Tring group | $Tring(n)$ |
special orthogonal group | $SO(n)$ | Spin group | $Spin(n)$ | String group | $String(n)$ |
Lorentz group | $\mathrm{O}(n,1)$ | $\,$ | $Spin(n,1)$ | $\,$ | $\,$ |
anti de Sitter group | $\mathrm{O}(n,2)$ | $\,$ | $Spin(n,2)$ | $\,$ | $\,$ |
Narain group | $O(n,n)$ | ||||
Poincaré group | $ISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |
super Poincaré group | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |