Abstract
We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard “configuration space” formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.
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Bastos C., Bertolami O., Dias N.C., Prata J.N.: Weyl–Wigner formulation of noncommutative quantum mechanics. J. Math. Phys. 49, 072101 (2008)
Bastos C., Dias N.C., Prata J.N.: Wigner measures in noncommutative quantum mechanics. Commun. Math. Phys. 299(3), 709–740 (2010)
Bastos C., Bertolami O., Dias N.C., Prata J.N.: Phase-space noncommutative quantum cosmology. Phys. Rev. D 78, 023516 (2008)
Bastos C., Bertolami O., Dias N.C., Prata J.N.: Black holes and phase-space noncommutativity. Phys. Rev. D 80, 124038 (2006)
Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. I. Deformation of symplectic structures. Ann. Phys. 111, 6–110 (1978)
Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. II Physical applications. Ann. Phys. 110, 111–151 (1978)
Bertolami O., Rosa J.G., de Aragão C.M.L., Castorina P.P., Zappalà D.: Noncommutative gravitational quantum well. Phys. Rev. D (3) 72(2), 025010–025018 (2005)
Bopp F.: La mécanique quantique est-elle une mé canique statistique particulière?. Ann. Inst. H. Poincaré 15, 81–112 (1956)
Bracken A., Watson P.: The quantum state vector in phase space and Gabor’s windowed Fourier transform. J. Phys. A: Math. Theor. 43, 395304 (2010)
Carroll S.M., Harvey J.A., Kostelecký V.A., Lane C.D., Okamoto T.: Noncommutative field theory and Lorentz violation. Phys. Rev. Lett. 87(14), 141601–141605 (2001)
Dias N.C., de Gosson M., Luef F., Prata J.N.: A deformation quantization theory for non-commutative quantum mechanics. J. Math. Phys. 51, 072101 (2010)
Dias N.C., de Gosson M., Luef F., Prata J.N.: A pseudo-differential calculus on non-standard symplectic space; spectral and regularity results in modulation spaces. J. Math. Pure Appl. 96, 423–445 (2011)
Douglas M.R., Nekrasov N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977–1029 (2001)
de Gosson M.: Extended Weyl calculus and application to the phase space Schrödinger equation. J. Phys. A Math. Gen. 38, L325–L329 (2005)
de Gosson M.: Symplectic Geometry and Quantum Mechanics. Birkh äuser, Basel (2006)
de Gosson M.: Spectral properties of a class of generalized landau operators. Commun. Partial Differ. Oper. 33(11), 2096–2104 (2008)
de Gosson, M.: Symplectic Methods in Harmonic Analysis. Applications to Mathematical Physics. Pseudo-Differential Operators, Theory and Applications. Birkhäuser, Basel (2011)
de Gosson M., Luef F.: A new approach to the \({\star}\) -genvalue equation. Lett. Math. Phys. 85, 173–183 (2008)
de Gosson M., Luef F.: Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization. J. Pseudo-Differ. Oper. Appl. 1(1), 3–34 (2010)
de Oliveira C.R.: Intermediate Spectral Theory and Quantum Dynamics. Birkhäuser, Basel (2009)
Gröchenig K.: Foundations of Time–Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001)
Henneaux C.: Teitelboim, Quantization of Gauge Systems. Princeton University Press, NJ (1992)
Hörmander L.: The Weyl calculus of pseudo-differential operators. Commun. Pure Appl. Math. 32, 359–443 (1979)
Maillard J.M.: On the twisted convolution product and the Weyl transformation of tempered distributions. J. Geom. Phys. 3(2), 232–261 (1986)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin, 1st edn (1987); 2nd edn (2001) [original Russian edition in Nauka, Moskva, 1978]
Stein E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, NJ (1993)
Torres-Vega G., Frederick J.: A quantum mechanical representation in phase space. J. Chem. Phys. 98(4), 3103–3120 (1993)
Szabo R.J.: Quantum field theory on noncommutative spaces. Phys. Rep. 378, 207 (2003)
Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Self-adjoint differential operator associated with self-adjoint differential expressions (2006). quant-ph/0603187
Weyl, H.: Gruppentheorie und Quantenmechanik. Transl. by H.P. Robertson, The Theory of Groups and Quantum Mechanics. Dover, NY (1931) (reprinted 1950)
Wong M.W.: Weyl Transforms. Springer, Berlin (1998)
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Dias, N.C., de Gosson, M., Luef, F. et al. Quantum mechanics in phase space: the Schrödinger and the Moyal representations. J. Pseudo-Differ. Oper. Appl. 3, 367–398 (2012). https://doi.org/10.1007/s11868-012-0054-9
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DOI: https://doi.org/10.1007/s11868-012-0054-9