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group_theory:notions:group_deformations_and_contractions

Group Deformations and Contractions

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.

“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

Deformations

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 demand that the series converges in some neighbourhood of the origin.

Why are they important?

“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

Important Examples

Classical Mechanics $\leftrightarrow$ 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{array} &\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{array}

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

Further Examples

c.f. https://en.wikipedia.org/wiki/Group_contraction

group_theory/notions/group_deformations_and_contractions.txt · Last modified: 2017/12/06 09:33 (external edit)