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| The '''Schrödinger group''' is the [[symmetry group]] of the free particle [[Schrödinger equation]].
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| ==Schrödinger algebra==
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| The Schrödinger algebra is the [[Lie algebra]] of the Schrödinger group. | |
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| It contains [[Galilean transformation|Galilei algebra]] with central extension.
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| :<math>[J_i,J_j]=i \epsilon_{ijk} J_k,\,\!</math> | |
| :<math>[J_i,P_j]=i \epsilon_{ijk} P_k,\,\!</math>
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| :<math>[J_i,K_j]=i \epsilon_{ijk} K_k,\,\!</math>
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| :<math>[P_i,P_j]=0, [K_i,K_j]=0, [K_i,P_j]=i \delta_{ij} M,\,\!</math>
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| :<math>[H,J_i]=0, [H,P_i]=0, [H,K_i]=i P_i.\,\!</math>
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| Where ''J_i'', ''P_i'', ''K_i'', ''H'' are generators of rotations ([[angular momentum operator]]), spatial translations ([[momentum operator]]), Galilean boosts and time translation ([[Hamiltonian mechanics#Mathematical formalism|Hamiltonian]]) correspondingly. [[Central extension (mathematics)|Central extension]] ''M'' has interpretation as non-relativistic [[mass]] and corresponds to the symmetry of [[Schrödinger equation]] under phase transformation (and to the conservation of probability).
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| There are two more generators which we will denote by ''D'' and ''C''. They have the following commutation relations:
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| :<math>[H,C]=i D, [C,D]=-2i C, [H,D]=2i H,\,\!</math>
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| :<math>[P_i,D]=i P_i, [K_i,D]=-iK_i,\,\!</math>
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| :<math>[P_i,C]=-iK_i,[K_i,C]=0,\,\!</math>
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| :<math>[J_i,C]=[J_i,D]=0.\,\!</math>
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| The generators ''H'', ''C'' and ''D'' form the sl(2,R) algebra.
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| ==The role of the Schrödinger group in mathematical physics==
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| Though the Schrödinger group is defined as symmetry group of the free particle [[Schrödinger equation]], it is realised in some interacting non-relativistic systems (for example cold atoms at criticality).
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| The Schrödinger group in d spatial dimensions can be embedded into relativistic [[conformal group]] in d+1 dimensions SO(2,d+2). This embedding is connected with the fact that one can get the [[Schrödinger equation]] from the massless [[Klein-Gordon equation]] through [[Kaluza-Klein theory|Kaluza-Klein compactification]] along null-like dimensions and Bargmann lift of [[Newton-Cartan theory]].
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| == References ==
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| * C. R. Hagen, Scale and Conformal Transformations in Galilean-Covariant Field Theory, Phys. Rev. D 5, 377–388 (1972)
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| * Arjun Bagchi, Rajesh Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 0907:037,2009
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| * D.T.Son, Toward an AdS/cold atoms correspondence: A geometric realization of the Schrödinger symmetry, Phys. Rev. D 78, 046003 (2008)
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| ==See also==
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| *[[Schrödinger equation]]
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| *[[Galilean transformation]]
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| *[[Poincaré group]]
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| {{DEFAULTSORT:Schrodinger group}}
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| [[Category:Theoretical physics]]
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| [[Category:Quantum mechanics]]
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| [[Category:Lie groups]]
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