Path Integrals in Noncommutative Quantum Mechanics

We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.     

Vasiliy Sergeevich Vladimirov (09.01.1923–03.11.2012)

We present a brief biographical review of the scientific work and achievements of Vasiliy Sergeevich Vladimirov on the occasion of his death on November 3, 2012.     

Zeta-nonlocal scalar fields

We consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued Riemann zeta function through the d’Alembertian □ in its argument. The construction of the corresponding Lagrangians L starts with the exact Lagrangian for the effective field of the p-adic tachyon string, which is generalized by replacing p with an arbitrary natural number n and then summing over all n. We obtain several basic classical properties of these fields. In particular, we study some solutions of the equations of motion and their tachyon spectra. The field theory with Riemann zeta-function dynamics is also interesting in itself. Dedicated to Vasilii Sergeevich Vladimirov on his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 364–372, December, 2008.     

Noncommutative classical and quantum mechanics for quadratic Lagrangians (Hamiltonians)

Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ _eff which depends on additional noncommutativity.     

Path integrals for quadratic lagrangians on <Emphasis Type="Italic">p</Emphasis>-adic and adelic spaces

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ _p and ℚ ↔ ℚ _p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.     

Andrei Yurievich Khrennikov and his Research

This paper contains a brief review of a very diverse and vast scientific work of Andrei Yurievich Khrennikov on the occasion of his 60th birthday.     

p-Adic invariant summation of some p-adic functional series

We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer argument. In this paper, we present rather large class of such p-adic functional series with integer coefficients which contain factorials. By recurrence relations, we constructed sequence of polynomials A _k (n; x) which are a generator for a few other sequences also relevant to some problems in number theory and combinatorics.     

Efficient Large-Scale Synthesis of 2-Amino-5-methanesulfonylaminobenzenesulfonamide

A novel synthesis of 2-amino-5-methanesulfonylaminobenzenesulfonamide is described. This preparation begins with readily available 4-nitroaniline and proceeds through five steps (isolations) in 41% overall yield. The described chemistry is conducted on a 50- to 100-g scale and does not require extractive workup procedures or chromatographic purifications.     

Adelic model of harmonic oscillator

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.This work was partially supported by the Russian Foundation for Fundamental Research, Grant No. 93-011-147.Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 349–359, December, 1994.