Noncommutative configuration space. Classical and quantum mechanical aspects
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| Format: | Artículo científico |
| Sprache: | en |
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Sociedade Brasileira de Física
2006
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| _version_ | 1876449137353818112 |
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| author | F. J. Vanhecke |
| author_facet | F. J. Vanhecke |
| contents | Noncommutative configuration space. Classical and quantum mechanical aspects F. J. Vanhecke C. Sigaud A. R. da Silva Física, Astronomía y Matemáticas Quantization Noncommutativity Symplectic mechanics In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and itsquantisation. In coordinates fqi; pkg the canonical symplectic two-form is w0 = dqi ^d pi. It is well known insymplectic mechanics [5, 6, 9] that the interaction of a charged particle with a magnetic field can be described ina Hamiltonian formalism without a choice of a potential. This is done by means of a modified symplectic twoformw=w0¡eF, where e is the charge and the (time-independent) magnetic field F is closed: dF=0. With thissymplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: fpk; plg =eFkl(q). Similarly a closed two-form in p-space G may be introduced. Such a dual magnetic field G interactswith the particles dual charge r. A new modified symplectic two-form w = w0¡eF+rG is then defined. Now,both p- and q-variables will cease to Poisson commute and upon quantisation they become noncommutingoperators. In the particular case of a linear phase space R2N, it makes sense to consider constant F and G fields.It is then possible to define, by a linear transformation, global Darboux coordinates: fxi;pkg = dik. These canthen be quantised in the usual way [bxi;bpk] = i~dik. The case of a quadratic potential is examined with somedetail when N equals 2 and 3. 2006 artículo científico 0103-9733 https://www.redalyc.org/articulo.oa?id=46436212 en http://www.redalyc.org/revista.oa?id=464 Brazilian Journal of Physics application/pdf Sociedade Brasileira de Física Brazilian Journal of Physics (Brasil) Num.1B Vol.36 |
| format | Artículo científico |
| id | redalyc_46436212 |
| institution | Redalyc |
| language | en |
| publishDate | 2006 |
| publisher | Sociedade Brasileira de Física |
| spellingShingle | Noncommutative configuration space. Classical and quantum mechanical aspects F. J. Vanhecke Física, Astronomía y Matemáticas Quantization Noncommutativity Symplectic mechanics Noncommutative configuration space. Classical and quantum mechanical aspects F. J. Vanhecke C. Sigaud A. R. da Silva Física, Astronomía y Matemáticas Quantization Noncommutativity Symplectic mechanics In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and itsquantisation. In coordinates fqi; pkg the canonical symplectic two-form is w0 = dqi ^d pi. It is well known insymplectic mechanics [5, 6, 9] that the interaction of a charged particle with a magnetic field can be described ina Hamiltonian formalism without a choice of a potential. This is done by means of a modified symplectic twoformw=w0¡eF, where e is the charge and the (time-independent) magnetic field F is closed: dF=0. With thissymplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: fpk; plg =eFkl(q). Similarly a closed two-form in p-space G may be introduced. Such a dual magnetic field G interactswith the particles dual charge r. A new modified symplectic two-form w = w0¡eF+rG is then defined. Now,both p- and q-variables will cease to Poisson commute and upon quantisation they become noncommutingoperators. In the particular case of a linear phase space R2N, it makes sense to consider constant F and G fields.It is then possible to define, by a linear transformation, global Darboux coordinates: fxi;pkg = dik. These canthen be quantised in the usual way [bxi;bpk] = i~dik. The case of a quadratic potential is examined with somedetail when N equals 2 and 3. 2006 artículo científico 0103-9733 https://www.redalyc.org/articulo.oa?id=46436212 en http://www.redalyc.org/revista.oa?id=464 Brazilian Journal of Physics application/pdf Sociedade Brasileira de Física Brazilian Journal of Physics (Brasil) Num.1B Vol.36 |
| title | Noncommutative configuration space. Classical and quantum mechanical aspects |
| topic | Física, Astronomía y Matemáticas Quantization Noncommutativity Symplectic mechanics |
| url | https://www.redalyc.org/articulo.oa?id=46436212 |