Group Contraction and Deformation [The Physics Travel Guide]

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--- advanced_tools:group_theory:group_contraction [2017/12/04 09:01]
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-====== Group Contraction ======
+====== Group Contraction and Deformation======
 <tabbox Why is it interesting?>
+<blockquote>
+"Lie-type deformations provide a systematic way of generalising the symmetries of modern physics."
+<cite>https://arxiv.org/pdf/1512.04339.pdf</cite>
+</blockquote>
+<blockquote>
+"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."
+<cite>[[http://www.emis.de/journals/SIGMA/2006/Paper048/|On Deformations and Contractions of Lie Algebras]] by A. Fialowski and M. de Montigny</cite>
+</blockquote>
+<blockquote>
+"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)."
+<cite>http://math.ucr.edu/home/baez/thesis_wise.pdf</cite>
+</blockquote>
+<blockquote>
+"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."
+<cite>http://www.aei.mpg.de/~gielen/report.pdf</cite>
+</blockquote>
+<blockquote>"//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//" <cite>from Group Contraction in Quantum Field Theory by Giuseppe Vitiello</cite></blockquote>
 <tabbox Layman>
@@ Line 11: / Line 46: @@
 <tabbox Student>
-A nice discussion can be found here: {{ :advanced_tools:group_theory:20100209000459_wigner-inonu_contraction.pdf |}}
+  * **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 demand that the series converges in some neighbourhood of the origin.
+----
+<blockquote>
+"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."
+<cite>[[http://www.emis.de/journals/SIGMA/2006/Paper048/|On Deformations and Contractions of Lie Algebras]] by A. Fialowski and M. de Montigny</cite>
+</blockquote>
+  * A nice discussion can be found here: {{ :advanced_tools:group_theory:20100209000459_wigner-inonu_contraction.pdf |}}
+  * See also [[advanced_tools:group_theory:https://aip.scitation.org/doi/10.1063/1.1705338|Deformation and Contraction of Lie Algebras]] by Levy-Nahas
 <tabbox Researcher>
@@ Line 31: / Line 92: @@
 <tabbox Examples>
---> Example1#
+-->  Classical Mechanics -> Quantum Mechanics#
+In [[http://label2.ist.utl.pt/vilela/Papers/DeforJPA94.pdf|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.
 <--
---> Example2:#
+--> Deformations of the Poincare Group#
+In [[http://aip.scitation.org/doi/abs/10.1063/1.1705338|Deformation and Contraction of Lie Algebras by Monique Levy‐Nahas]] it is "s//hown 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 "[[http://aip.scitation.org/doi/abs/10.1063/1.1664490|Possible Kinematics]]" and [[http://aip.scitation.org/doi/abs/10.1063/1.527306|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 [[http://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/deform.pdf|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).//"
 <--

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