Real pseudo-orthogonal groups and the canonical commutation relations

Patrick Moylan ()
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Patrick Moylan: The Pennsylvania State University, Abington College, Department of Physics

A chapter in Physical and Mathematical Aspects of Symmetries, 2017, pp 271-277 from Springer

Abstract: Abstract Let $$ {W}_{n} (\mathbb{R})$$ be the Weyl algebra of index n over $$ \mathbb{R} $$ and let $$ \tilde{\mathfrak{D}} $$ (so(2,1)) be a certain extension of the skew field of fractions of U(so(2;1)), the universal enveloping algebra of $$ \mathfrak{U} $$ so(2,1). In a previous work we have established a skew field isomorphism between $$ \tilde{\mathfrak{D}} $$ (so(2,1)) and $$ \mathfrak{D}_{(1,1)} (\mathbb{R}) $$ where $$ \mathfrak{D}_{(1,1)} (\mathbb{R}) $$ is the fraction field of $$ {W}_{1,1} (\mathbb{R})\simeq {W}_{1} (\mathbb{R})\bigotimes(\mathbb{R})_{(y)}$$ with $$ (\mathbb{R})_{(y)}$$ being the ring of polynomials over $$ \mathbb{R} $$ in the indeterminate . Using this isomorphism, we were able to construct, out of unitary and irreducible representations of the universal covering group of SO0(2,1), representations of $$ {W}_{1} (\mathbb{R})$$ with all of the desired properties required by physics, including hermicity of the momentum and position operators. Thus, we have obtained the canonical commutation relations and acceptable representations of them out of the so(2,1) symmetry. In this work we investigate generalizations of the above results to higher dimensions. In particular, we describe generalizations with $$ \mathfrak{D} $$ (so(2,1)) replaced by $$ \mathfrak{D} $$ (so(p,q)) and $$ \mathfrak{D}_{(1,1)} (\mathbb{R}) $$ replaced by $$ \mathfrak{D}_{p+q}-{2,1} (\mathbb{R}) $$ . As in the so(2,1) case, we make use of our results to obtain applications to representations.

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-69164-0_40

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DOI: 10.1007/978-3-319-69164-0_40

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