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advanced_tools:group_theory:group_contraction [2017/12/04 09:01]
127.0.0.1 external edit
advanced_tools:group_theory:group_contraction [2018/10/11 16:23] (current)
jakobadmin [Student]
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-====== Group Contraction ======+====== Group Contraction ​and Deformation======
  
 <tabbox Why is it interesting?> ​ <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> ​ <tabbox Layman> ​
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 <tabbox Student> ​ <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> ​ <tabbox Researcher> ​
Line 31: Line 92:
 <tabbox Examples> ​ <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).//"
 +
 <-- <--
   ​   ​
advanced_tools/group_theory/group_contraction.1512374469.txt.gz · Last modified: 2017/12/04 08:01 (external edit)