Papers by Maciej Przanowski
Journal of physics, Oct 27, 2000
Deformation quantization of bosonic strings is considered. We show that the light-cone gauge is t... more Deformation quantization of bosonic strings is considered. We show that the light-cone gauge is the most convenient classical description to perform the quantization of bosonic strings in the deformation quantization formalism. Similar to the field theory case, the oscillator variables greatly facilitates the analysis. The mass spectrum, propagators and the Virasoro algebra are finally described within this deformation quantization scheme.
Deformation quantization (the Moyal deformation) of SDYM equation for the algebra of the area pre... more Deformation quantization (the Moyal deformation) of SDYM equation for the algebra of the area preserving diffeomorphisms of a 2-surface $\Sigma_{2}$, sdiff($\Sigma_{2}$), is studied. Deformed equation we call the master equation (ME) as it can be reduced to many integrable nonlinear equations in mathematical physics. Two sets of concerved charges for ME are found. Then the linear systems for ME (the Lax pairs) associated with the conserved charges are given. We obtain the dressing operators and the infinite algebra of hidden symmetries of ME. Twistor construction is also done.
Annals of Physics, 2008
Deformation quantization for any Grassmann scalar free field is described via the Weyl-Wigner-Moy... more Deformation quantization for any Grassmann scalar free field is described via the Weyl-Wigner-Moyal formalism. The Stratonovich-Weyl quantizer, the Moyal ⋆-product and the Wigner functional are obtained by extending the formalism proposed recently in [35] to the fermionic systems of infinite number of degrees of freedom. In particular, this formalism is applied to quantize the Dirac free field. It is observed that the use of suitable oscillator variables facilitates considerably the procedure. The Stratonovich-Weyl quantizer, the Moyal ⋆-product, the Wigner functional, the normal ordering operator, and finally, the Dirac propagator have been found with the use of these variables.
Modern Physics Letters A, 1996
It is demonstrated that the action of SU(N) principal chiral model leads in the limit N →∞ to the... more It is demonstrated that the action of SU(N) principal chiral model leads in the limit N →∞ to the action for the Park-Husain (P-H) heavenly equation. The principal chiral model in the Hilbert space L 2(ℝ1) is considered and it is shown, that in this case the chiral equation is equivalent to the Moyal deformation of the P-H heavenly equation. A new method of searching for solutions to this latter equation, via Lie algebra representations in L 2(ℝ1) is given.
Journal of Mathematical Physics, 2017
The general Weyl–Wigner formalism in finite dimensional phase spaces is investigated. Then this f... more The general Weyl–Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd-dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number–phase Wigner function in quantum optics. Analogous results for the “almost” symmetric ordering in an even-dimensional Hilbert space are obtained. Relations between the discrete Wigner functions introduced in our paper and some other discrete Wigner functions appearing in literature are studied.
Fortschritte der Physik, 2019
The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is... more The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space L 2 (R 3) ⊗ H (s+1) and functions on the phase space R 3 × R 3 × {0, ..., s} × {0, ..., s} is found. The expressions for the Stratonovich-Weyl quantizer, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin 1 2 nonrelativistic particle as well as the magnetic resonance for a spin 1 2 nonrelativistic uncharged particle are analysed.
Journal of Physics A: Mathematical and General, 2002
Second quantization of a classical nonrelativistic one-particle system as a deformation quantizat... more Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrödinger spinless field is considered. Under the assumption that the phase space of the Schrödinger field is C ∞ , both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is CP ∞ endowed with the Fubini-Study Kählerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one.
The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is... more The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. This formalism is then used to find the Landau levels and the corresponding Wigner functions for a spin 1/2 nonrelativistic particle. Magnetic resonance for a spin 1/2 nonrelativistic uncharged particle is also studied in terms of that formalism.
Physical Review A, 2021
It is shown that the photon position operator ~̂ X with commuting components can be written in th... more It is shown that the photon position operator ~̂ X with commuting components can be written in the momentum representation as ~̂ X = i ~̂ D, where ~̂ D is a flat connection in the tangent bundle T (R \ {(0, 0, k3) ∈ R : k3 ≥ 0}) over R \ {(0, 0, k3) ∈ R : k3 ≥ 0} equipped with the Cartesian structure. Moreover, ~̂ D is such that the tangent 2-planes orthogonal to the momentum are parallelly propagated with respect to ~̂ D and, also, ~̂ D is an anti-Hermitian operator with respect to the scalar product 〈Ψ|Ĥ−2s|Φ〉. The eigenfunctions Ψ ~ X(~x) of the position operator ~̂ X are found.
It is shown that the photon position operator ~̂ X with commuting components introduced by Margar... more It is shown that the photon position operator ~̂ X with commuting components introduced by Margaret Hawton can be written in the momentum representation as ~̂ X = i ~̂ D, where ~̂ D is a flat connection in the tangent bundle T (R \ {(0, 0, k3) ∈ R : k3 ≥ 0}) over R \ {(0, 0, k3) ∈ R : k3 ≥ 0} equipped with the Cartesian structure. Moreover, ~̂ D is such that the tangent 2-planes orthogonal to the momentum are parallelly propagated with respect to ~̂ D and, also, ~̂ D is an anti-Hermitian operator with respect to the scalar product 〈Ψ|Ĥ−2s|Φ〉. The eigenfunctions Ψ ~ X(~x) of the position operator ~̂ X are found.
International Journal of Modern Physics A, 2001
We study the deformation quantization of scalar and Abelian gauge classical free fields. Stratono... more We study the deformation quantization of scalar and Abelian gauge classical free fields. Stratonovich–Weyl quantizer, star products and Wigner functionals are obtained in field and oscillator variables. The Abelian gauge theory is particularly intriguing since the Wigner functional is factorized into a physical part and the other one containing the constraints only. Some effects of nontrivial topology within the deformation quantization formalism are also considered.
Classical and Quantum Gravity, 1997
In 1993, a proof was published, within this journal, that there are no regular solutions to the l... more In 1993, a proof was published, within this journal, that there are no regular solutions to the linearized version of the twisting, type-N, vacuum solutions of the Einstein field equations. While this proof is certainly correct, we show that the conclusions drawn from that fact were unwarranted, namely that this irregularity caused such solutions not to be able to truly describe pure gravitational waves. In this article, we resolve the paradox-since such first-order solutions must always have singular lines in space for all sufficiently large values of r-by showing that if we perturbatively iterate the solution up to the third order in small quantities, there are acceptable regular solutions. That these solutions become flat before they become non-twisting tells us something interesting concerning the general behavior of solutions describing gravitational radiation from a bounded source.
Journal of Physics A-mathematical and General, 2000
Weyl-Underhill-Emmrich (WUE) quantization and its generalization are considered. It is shown that... more Weyl-Underhill-Emmrich (WUE) quantization and its generalization are considered. It is shown that an axiomatic definition of the Stratonovich-Weyl quantizer leads to severe difficulties. Quantization on the cylinder within the WUE formalism is discussed.
Annals of Physics, 2013
Generalized Weyl quantization formalism for the cylindrical phase space S 1 × R 1 is developed. I... more Generalized Weyl quantization formalism for the cylindrical phase space S 1 × R 1 is developed. It is shown that the quantum observables relevant to the phase of linear harmonic oscillator or electromagnetic field can be represented within this formalism by the self-adjoint operators on the Hilbert space L 2 (S 1).
Annals of Physics, 2008
The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of... more The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. This correspondence is studied by computing the relevant Stratonovich-Weyl quantizer. The Moyal ⋆-product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.
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Papers by Maciej Przanowski