Quantum group symmetric Bargmann–Fock space: Integral kernels, Green functions, driving forces
TLDR
In this article, the deformed commutation relations of raising and lowering operators that transform under the SUq(n)−quantum group get deformed decomposition relations and are represented as adjoint operators on a Hilbert space of noncommutative holomorphic functions.Abstract:
Raising and lowering operators that transform under the SUq(n)‐quantum group get deformed commutation relations. They are represented as adjoint operators on a Hilbert space of noncommutative holomorphic functions. Through the algebraically defined integral on this function space, every operator on the Fock space can also be represented as an integral kernel. The Green function for free harmonic oscillators and spin‐1/2’s in a constant magnetic field is given. Further on it is studied how such systems react on the switching on of a driving force. Calculating the vacuum–vacuum transition amplitude, it is found that the deviations from the undeformed case grow with the strength of the driving force. Throughout all calculations the bosonic and the fermionic cases are considered simultaneously.read more
Citations
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Uncertainty relation in quantum mechanics with quantum group symmetry
TL;DR: In this article, the commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations.
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Uncertainty Relation in Quantum Mechanics with Quantum Group Symmetry
TL;DR: In this article, the authors studied the commutation relations, uncertainty relations and spectra of position and momentum operators within the framework of quantum group % symmetric Heisenberg algebras and their (Bargmann-) Fock representations.
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Maximal localization in the presence of minimal uncertainties in positions and in momenta
Haye Hinrichsen,Achim Kempf +1 more
TL;DR: In this paper, the authors extended this treatment to the case with minimal uncertainties both in positions and in momenta, and calculated the physical states and their properties for the case of minimal uncertainties in positions only.
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On quantum field theory with nonzero minimal uncertainties in positions and momenta
TL;DR: In this article, the ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in positions and momenta were studied for non-commutative geometric spaces, and the case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered.
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Algebraic q‐integration and Fourier theory on quantum and braided spaces
Achim Kempf,Shahn Majid +1 more
TL;DR: An algebraic theory of integration on quantum planes and other braided spaces is introduced in this article, where it is shown that the definite integral ∫x∞−x ∞ can also be evaluated algebraically as multiples of the integral of a q•Gaussian, with x remaining as a bosonic scaling variable associated with the q•deformation.
References
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Book
Quantum Field Theory
TL;DR: In this article, a modern pedagogic introduction to the ideas and techniques of quantum field theory is presented, with a brief overview of particle physics and a survey of relativistic wave equations and Lagrangian methods.
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On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q
TL;DR: The quantum group SU(2)q is discussed in this paper by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose.
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The quantum group SUq(2) and a q-analogue of the boson operators
TL;DR: In this article, a new realisation of the quantum group SUq(2) is constructed by means of a q-analogue to the Jordan-Schwinger mapping, determining thereby both the complete representation structure and qanalogues to the Wigner and Racah operators.
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Covariant differential calculus on the quantum hyperplane
TL;DR: In this paper, a differntial calculus on the quantum hyperplane covariant with respect to the action of the quantum group GLq(n) is developed, which is a concrete example of noncommutative differential geometry.
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Quasitriangular Hopf Algebras and {Yang-Baxter} Equations
TL;DR: The theory of quasitriangular Hopf algebras and its connections with physics are reviewed in this paper. But the main focus of this paper is on the Yang-Baxter equations.