Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field

Ichinose, Takashi; Tamura, Hiroshi

doi:10.1007/bf01211101

Cited by 79 publications

(78 citation statements)

References 18 publications

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“…If the magnetic field A ≡ 0, it seems that the first work which considered the existence of solutions for problem (1.1) in the subcritical case with ε = 1, formally α = 1 and K = 0 was [16]. For more details on fractional magnetic operators we refer to [19][20][21] for related physical background. If the magnetic field A ≡ 0, the above operator is consistent with the usual notion of fractional Laplacian, which may be viewed as the infinitesimal generators of a Lévy stable diffusion processes (see [1]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Fractional NLS equations with magnetic field, critical frequency and critical growth

2017

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Abstract. The paper is devoted to the study of a singularly perturbed fractional Schrödinger equations involving critical frequency and critical growth in the presence of a magnetic field. By using variational methods, we obtain the existence of mountain pass solutions uε which tend to the trivial solution as ε → 0. Moreover, we get infinitely many solutions and sign-changing solutions for the problem in absence of magnetic effects under some extra assumptions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Fractional NLS equations with magnetic field, critical frequency and critical growth

2017

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“…where (X t ) t 0 is a Lévy process with 1-symbol H(p) = p 2 + m 2 − m. The last equation is due to Ichinose and Tamura [25].…”

Section: Definition 32 the Hamiltonian Feynman Path Integralmentioning

confidence: 99%

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

Böttcher

Butko

Schilling

et al. 2011

Russ. J. Math. Phys.

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This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.

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“…The results obtained open a possibility to study dynamics of the corresponding systems by passing to imaginary values of time and employing properties of the processes. It was realized in [7], [9], where a path integral representation of the semi-group exp(−tH), t > 0, H being a relativistic Schrödinger operator, was constructed. This representation allows one to describe 'imaginary-time' evolution of wave functions ψ t = exp(−tH)ψ 0 by mean of path integrals.…”

Section: Introductionmentioning

confidence: 99%

Irreducibility of dynamics and representation of KMS states in terms of Lévy processes

Kozitsky

2005

Arch. Math.

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Quantum systems described by the Schrödinger operators H = N j =1(p j ) + W (x 1 , . . . , x N ), p j = −ı∇ j , x j ∈ R ν with being continuous functions such that the pseudodifferential operators (p j ) generate Lévy processes, are considered. It is proven that the linear span of the operators α t 1 (F 1 ) · · · α t n (F n ) is dense in the algebra of all observables in the σ -strong and hence in the σ -weak and strong topologies. Here α t (F ) = exp(ıtH)F exp(−ıtH) are time automorphisms and the F 's are taken from families of multiplication operators obeying conditions described in the paper. This result implies that a linear functional continuous in either of these topologies is fully determined by its values on such products. In the case of KMS states this yields a representation of such states in terms of path integrals.1. Introduction. Gibbs (equilibrium) states of infinite-particle quantum systems described by unbounded Schrödinger operators cannot be constructed directly as KuboMartin-Schwinger (KMS) states. The only way here is to use path integrals to represent local Gibbs states (describing finite subsystems) in terms of probability measures. Then the Gibbs states of the whole system are constructed as probability measures with the help of the DLR technique known in classical statistical mechanics, see [5]. For models like quantum crystals, the corresponding approach is well elaborated, see e.g., [1]. In the present article we make first steps in developing a similar technique for a wider class of quantum systems by proving the basic statement for representing local Gibbs states in terms of probability measures.We consider a system of interacting quantum particles with the kinetic energy of a particle being a pseudo-differential operator (p), p = −ı∇. Such operators appear if one 'quantizes' the relativistic expression [p 2 c 2 + m 2 c 4 ] 1/2 − mc 2 and were used in particular in studying problems of stability of matter, see [12]-[14] and the references therein. At the same time, such Schrödinger operators generate stochastic processes, see [4], [8], which

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Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field

Cited by 79 publications

References 18 publications

Fractional NLS equations with magnetic field, critical frequency and critical growth

Fractional NLS equations with magnetic field, critical frequency and critical growth

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

Irreducibility of dynamics and representation of KMS states in terms of Lévy processes

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