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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. 相似文献
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B. Dragovich A. Yu. Khrennikov S. V. Kozyrev I. V. Volovich 《P-Adic Numbers, Ultrametric Analysis, and Applications》2013,5(1):83-86
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. 相似文献
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Zeta-nonlocal scalar fields 总被引:1,自引:0,他引:1
B. Dragovich 《Theoretical and Mathematical Physics》2008,157(3):1671-1677
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
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 364–372, December, 2008. 相似文献
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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. 相似文献
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Branko Dragovich Zoran Rakić 《P-Adic Numbers, Ultrametric Analysis, and Applications》2010,2(4):322-340
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. 相似文献
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V. S. Anashin B. Dragovich A. N. Kochubei S. V. Kozyrev I. V. Volovich 《P-Adic Numbers, Ultrametric Analysis, and Applications》2018,10(4):344-347
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. 相似文献
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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. 相似文献