Group Contraction and Deformation

Prerequisites: Lie Algebras. Full path (4 pages)

Why is it interesting?
Intuitive
Concrete
Abstract
Examples

Why is it interesting?

"Lie-type deformations provide a systematic way of generalising the symmetries of modern physics."

https://arxiv.org/pdf/1512.04339.pdf

"Contractions are important in physics because they explain in terms of Lie algebras why some theories arise as a limit regime of more ‘exact’ theories."

On Deformations and Contractions of Lie Algebras by A. Fialowski and M. de Montigny

"From a physical point of view, ‘contractions’ can be thought of as ‘limits’ of Lie groups as some parameter approaches a specified value. The easiest example is what might be called the ‘Columbus contraction’, in which the parameter of interest is the radius of a spherical Earth. For any value of the radius, the group of symmetries is the rotation group SO(3), but if radius becomes infinite, the group suddenly becomes the Euclidean group of the plane, ISO(2)."

http://math.ucr.edu/home/baez/thesis_wise.pdf

"deformations play a role whenever one tries to find generalisations, extensions, or “perturbations” of a given physical theory or setup. […] the passage from Newtonian mechanics to special relativity or from classical to quantum mechanics can be understood as a deformation of the underlying algebraic structures."

http://www.aei.mpg.de/~gielen/report.pdf

"The mechanism which is at work, according to well established results of QFT, goes under the general name of spontaneous breakdown of symmetry and involves the physical phenomena of the Bose condensation and the mathematical structure of the (Ïnonü–Wigner) group contraction" from Group Contraction in Quantum Field Theory by Giuseppe Vitiello

Intuitive

Many physical theories contain a constant that, when sent to a limit, turns them into an older theory. Letting the speed of light become infinitely large turns special relativity into Newtonian mechanics. Letting Planck's constant go to zero turns quantum mechanics into classical mechanics. The symmetry groups follow along: in the limit of infinite speed of light the Lorentz boosts become Galilean boosts, and the Poincaré group becomes the Galilean group. A group contraction is the precise version of such a limit for symmetry groups.

The standard picture is a sphere. Its symmetries are the rotations of three-dimensional space, and any two rotations about different axes fail to commute. Now let the radius grow while we only look at a patch of fixed size. Rotations that move the patch along the surface start to look like translations of a flat plane, and their failure to commute shrinks with the inverse square of the radius. In the limit of infinite radius they commute exactly, and the rotation group of the sphere has become the symmetry group of the plane. The number of independent symmetries stays three in the limit, and only their commutation relations change.

A deformation goes the other way. We start from a group or algebra and ask in which ways its rules for combining transformations can be bent slightly while still giving a consistent algebra. The Galilean algebra can be deformed into the Poincaré algebra, with one over the speed of light squared as the small parameter, and the Poincaré algebra can be deformed into the de Sitter or anti-de Sitter algebra, with the cosmological constant as the small parameter. Read in this direction, deformations suggest how a known theory might be generalized.

Concrete

Deformation: Continuously modify the structure constants!
Contraction: Generators are multiplied with contraction parameters that are then sent to zero or infinity.

Both concepts are mutually the opposite. However while one can always deform to a group where we contracted from, the opposite procedure is not always possible.

To deform a Lie algebra, we redefine the Lie brackets as a power series in some parameter $t$ $f_t(a,b)=[a,b]+tF_1(a,b)+t^2 F_2(a,b)+\ldots,\quad a,b\in\frak{g}\,,$ and either treat it as a formal power series or demand that the series converges in some neighbourhood of the origin.

"There exists a plethora of definitions for both contractions and deformations. […] [W]e discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras."

On Deformations and Contractions of Lie Algebras by A. Fialowski and M. de Montigny

A nice discussion can be found here: 20100209000459_wigner-inonu_contraction.pdf
See also Deformation and Contraction of Lie Algebras by Levy-Nahas

Abstract

Fix a real or complex vector space $V$ of dimension $n$ . A Lie algebra structure on $V$ is an antisymmetric bilinear map $\mu: V\times V\to V$ satisfying the Jacobi identity. The set $\mathcal{L}_n$ of all such $\mu$ is an algebraic variety in $\Lambda^2 V^*\otimes V$ , and $GL(V)$ acts on it by change of basis, $(g\cdot\mu)(x,y) = g\,\mu(g^{-1}x, g^{-1}y).$ Two structures are isomorphic exactly when they lie in the same orbit.

Contraction. Let $U_\epsilon \in GL(V)$ for $\epsilon\in(0,1]$ and define $\mu_\epsilon(x,y) = U_\epsilon^{-1}\,\mu(U_\epsilon x, U_\epsilon y).$ Every $\mu_\epsilon$ is isomorphic to $\mu$ . If $\mu_0(x,y) = \lim_{\epsilon\to 0}\mu_\epsilon(x,y)$ exists for all $x,y$ , then $\mu_0$ satisfies the Jacobi identity by continuity, and $(V,\mu_0)$ is called a contraction of $(V,\mu)$ . Geometrically, $\mu_0$ lies in the closure of the orbit $GL(V)\cdot\mu$ . If $\mu_0$ is not isomorphic to $\mu$ , its orbit has strictly smaller dimension, so $\mu_0$ has a strictly larger algebra of derivations than $\mu$ . The dimension of the derived algebra $[V,V]$ can only drop in the limit, since the rank of $\mu$ as a linear map $\Lambda^2V\to V$ is lower semicontinuous.

Inönü-Wigner contractions. Let $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{p}$ as a vector space, with $\mathfrak{h}$ a subalgebra, and take $U_\epsilon = \mathrm{id}$ on $\mathfrak{h}$ and $U_\epsilon = \epsilon\,\mathrm{id}$ on $\mathfrak{p}$ . Writing $[\,\cdot,\cdot\,]_{\mathfrak{h}}$ and $[\,\cdot,\cdot\,]_{\mathfrak{p}}$ for the components, one finds for $x\in\mathfrak{h}$ and $y,z\in\mathfrak{p}$ $\begin{aligned} \mu_\epsilon(x,y) &= \epsilon\,[x,y]_{\mathfrak{h}} + [x,y]_{\mathfrak{p}}, \\ \mu_\epsilon(y,z) &= \epsilon^2\,[y,z]_{\mathfrak{h}} + \epsilon\,[y,z]_{\mathfrak{p}} . \end{aligned}$ In the limit, $\mathfrak{p}$ becomes an abelian ideal and the contracted algebra is the semidirect sum $\mathfrak{h}\ltimes\mathfrak{p}$ , with $\mathfrak{h}$ acting on $\mathfrak{p}$ through $x\cdot y = [x,y]_{\mathfrak{p}}$ . For a symmetric pair, $[\mathfrak{h},\mathfrak{p}]\subset\mathfrak{p}$ and $[\mathfrak{p},\mathfrak{p}]\subset\mathfrak{h}$ , this is the isotropy representation, and the contraction replaces the symmetric space $G/H$ by its tangent space at the base point. Examples are $\mathfrak{so}(3)\to\mathfrak{iso}(2)$ , $\mathfrak{so}(4,1)\to\mathfrak{iso}(3,1)$ with $\mathfrak{h}=\mathfrak{so}(3,1)$ , and the Poincaré algebra to the Galilean algebra with $\mathfrak{h}$ spanned by rotations and time translations.

Deformation and cohomology. Conversely, a formal deformation $\mu_t = \mu + tF_1 + t^2F_2 + \dots$ as in the Concrete section satisfies the Jacobi identity to first order in $t$ if and only if $F_1$ is a 2-cocycle of the Chevalley-Eilenberg complex of $\mathfrak{g}$ with values in the adjoint representation. Deformations that come from a $t$ -dependent change of basis give coboundaries at first order. Infinitesimal deformations up to this equivalence are therefore classified by $H^2(\mathfrak{g},\mathfrak{g})$ , and the obstructions to extending them to higher order lie in $H^3(\mathfrak{g},\mathfrak{g})$ . If $H^2(\mathfrak{g},\mathfrak{g}) = 0$ the algebra is rigid. By Whitehead's second lemma, every semisimple Lie algebra over a field of characteristic zero has $H^2(\mathfrak{g},\mathfrak{g}) = 0$ and is rigid. This is the algebraic content of the statement that $\mathfrak{so}(4,1)$ and $\mathfrak{so}(3,2)$ are stable while the Poincaré and Galilean algebras are not.

The family $\mu_\epsilon$ of an Inönü-Wigner contraction is polynomial in $\epsilon$ , so read backwards it is a deformation of the contracted algebra $\mu_0$ whose members for $\epsilon\neq 0$ are all isomorphic to $\mu$ .

Examples

Classical Mechanics -> Quantum Mechanics

In Deformations, stable theories and fundamental constants by R Vilela Mendes the author discusses how the algebra of quantum mechanics can be computed from the algebra of classical mechanics by deforming it.

To achieve this a different kind of deformation than the usual one is needed, because one must consider non-linear transformations of the generators. This a generalization of the classical theory of deformations, which is only concerned with the deformation of the structure constants of finite-dimensional Lie algebras.

There are two possibilities.

1.) We deform the Poisson algebra of functions in phase-space $\begin{equation} \{f,g\} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]. \end{equation}$

In the deformed algebra the Poisson bracket gets replaced by a so called Moyal algebra, which reads $\begin{equation} \{f,g\}_M=\{f,g\}-\frac{\hbar^2}{4\cdot 3!}\sum_{{{i_1,i_2,i_3}\atop{j_1,j_2,j_3}}}\omega^{i_1 j_1}\omega^{i_2 j_2}\omega^{i_3 j_3}\partial_{i_1 i_2 i_3}(f)\partial_{j_1 j_2 j_3}(g)+\ldots\,. \end{equation}$ The Poisson algebra is infinite-dimensional (because the space of functions is infinite-dimensional).

2.) Alternatively, we can consider the phase space coordinates as elements of an Abelian Lie algebra and deform this algebra. This yields the Heisenberg algebra:

$\begin{aligned} &\left[ \hat{x}_i, \hat{x}_j \right] = \left[ \hat{p}_i , \hat{p}_j \right] = 0 \\ &\left[ \hat{x}_i, \hat{p}_j \right] = i\hbar \, \delta_{ij} \end{aligned}$

To achieve this, a deformation is not enough. Instead, we must additionally perform a central extension together with the deformation.

Deformations of the Poincare Group

In Deformation and Contraction of Lie Algebras by Monique Levy‐Nahas it is "shown that the only groups which can be contracted in the Poincaré group are $SO(4, 1)$ and $SO(3, 2)$ "

Deformations of the static Lie algebra

In "Possible Kinematics" and Classification of ten‐dimensional kinematical groups with space isotropy the authors derived all possible deformations of the static Lie algebra.

Deformations of the Galilean algebra

All possible deformations of the Galilean algebra were derived in DEFORMATIONS OF THE GALILEAN ALGEBRA by Jose M. Figueroa-O’Farrill

Deformation of general relativity

Deformation of general relativity, as described in chapter 3 of https://arxiv.org/pdf/1103.0731v1.pdf: "Instead of viewing Minkowski space as R d−1,1 , we will view it as a homogeneous space E(d − 1, 1)/SO(d − 1, 1). A description in terms of Cartan geometry will allow us to deform general relativity by replacing E(d − 1, 1) by its deformation, the de Sitter group SO(d, 1)."