David W. Dreisigmeyer, Peter M. Young
We show that, for a class of systems described by a Lagrangian L(x,\dot{x},t) = 1/2\dot{x}^{2} - V(x,t) the propagator can be reduced via Noether's Theorem to a standard path integral multiplied by a phase factor. Using Henstock's integration technique, this path integral is given a firm mathematical basis. Finally, we recast the propagator as an evolutionary semigroup.
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We prove a Feynman-Kac formula for Schrodinger operators with potentials V(x) that obey (for all ε> 0): V(x) \geq - ε|x|^2 - C_ε. Even though e^{-tH} is an unbounded operator, any ϕ, ψ\in L^2 with compact support lie in D(e^{-tH}) and <ϕ, e^{-tH}ψ> is given by a Feynman-Kac formula.
A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of $n$-fold iterated integrals of some elementary functions as $n\to\infty$. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman--Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae gi…
This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number $τ$) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains functions which are second order polynomials with respect to the momentum variable and also some other functions. The considered semigroups are represented as limits of $n$-fold iterated integrals when $n$ tends to infinity (such representations are called Feynman formulae). Some of…
Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains
本文为包含Fröhlich哈密顿量的自伴算子半群证明Feynman-Kac公式，处理多极化子模型，涵盖奇异磁场、静电势及受限系统，通过新方法克服声子色散常数带来的困难。
Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
本文研究了由离开给定领域时被杀死的Feller过程生成的半群，这些半群对应于一类演化方程的Cauchy-Dirichlet型初始-外部值问题。通过Chernoff定理对这些半群进行近似，并将构造的Chernoff近似转化为Feynman公式，即n重积分的极限。此外，提出了一种近似时间分数阶（包括分布阶时间分数阶）演化方程解的方法，该方法基于时间分数阶与时间非分数阶演化方程之间的联系以及Chernoff近似。进一步，该方法导致时间分数阶演化方程解的Feynman公式。通过考虑一类分布阶时间分数阶扩散方程，获得了对应的Cauchy和Cauchy-Dirichlet问题的Feynman公式。
A parabolic partial differential equation $u'_t(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent coefficients, which may depend on $x$. We assume that the spatial coordinate $x$ belongs to a finite- or infinite-dimensional real separable Hilbert space $H$. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup by a Feynman formula, i.e. we write it in the form of the limit of a multiple integral over $H$ as the multiplicity of the integral tends to infinity…
We prove two assumptions made in an article by Ya.A. Butko, M. Grothaus, O.G. Smolyanov concerning the existence of a strongly continuous operator semigroup solving a Cauchy-Dirichlet problem for an elliptic differential operator in a bounded domain and the existence of a smooth contractive embedding of a core of the generator of the semigroup into the space $C_c^{2,α}(\R^n)$. Based on these assumptions a Feynman formula for the solution of the Cauchy-Dirichlet problem is constructed in the article mentioned above. In this article we show that the assumptions are fulfilled for domains with $C^…
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy opera…
We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.
In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. (The theory is constructive in that, operators acting at different times, actually commute.) We first develop an operator version of the Henstock-Kurzweil integral, and a new Hilbert space that allows us to construct the elementary path integral in the manner originally envisioned by Feynman. After developing our time-ordered operator theory we extend a few of the important theorems of semigroup theory, including the Hille-Yosida theorem. As an application, we unify and extend the theory of ti…
By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of…
The main aim of this article is to show some intimate relations among the following three notions: (1) the metaplectic representation of $Sp(2n,\mathbb{R})$ and its extension to some semigroups, called the Olshanski semigroup for $Sp(2n,\mathbb{R})$ or Howe's oscillator semigroup, (2) antinormally-ordered quantizations on the phase space $\mathbb{R}^{2m}\cong\mathbb{C}^{m}$, (3) path integral quantizations where the paths are on the phase space $\mathbb{R}^{2m}\cong\mathbb{C}^{m}$. In the Main Theorem, the metaplectic representation $ρ(e^{X})$ ($X\in\mathfrak{sp}(2n,\mathbb{R})$) is expressed …
The earlier proposed Dynamics-Generating Approach (DGA) is reviewed and extended. Starting from an arbitrarily chosen group or semigroup which have structure similar to the structure of Galilei group, DGA allows one to construct phenomenological description of dynamics of the corresponding "elementary quantum object" (a particle or non-local object of special type). A class of Galilei-type semigroups, with semigroup of trajectories (parametrized paths) instead of translations, allows one to derive Feynman path integrals in the framework of DGA. The measure of path integrating (exponential of t…
Characterizing Cohen-Macaulay One-Loop Feynman Integrals
本文研究了单环费曼积分的Cohen-Macaulay性质，通过整数线性规划和图形描述确定其特征，并推广了相关环的正常性结果。
Exotic B-series representation of the Feller semigroup for Itô diffusions and the MSR path integral
本文研究了一维伊藤扩散费勒半群的时间级数展开，通过异类树的扩展，推导出组合因子的显式表达，建立异类Butcher级数表示，并展示其与MSR路径积分形式的等价性。
Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985) 2239] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space $L^2(R^2)$ of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultra-diffusive limit of the unde…
Probabilistic Representations of Ordered Exponentials: Vector-Valued Schrödinger Semigroups and the Combinatorics of Anderson Localization
本文利用矩阵有序指数的概率表示推广 Feynman–Kac 公式，研究向量值随机 Schrödinger 半群的联合矩，并通过随机游走自交与互交解释含时及静态 Gaussian 噪声下的 Anderson 局域化机制。
Tamed Feynman-Kac diffusion processes: Killing-branching intertwine
研究漂移布朗运动平衡松弛的Feynman-Kac核，通过引入杀死和分支速率解释负势区域的驯化效应，并用数值模拟验证一维非线性模型。
Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman-Kac formula is taken here by subordinated Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular…
Markovian Repeated Interaction Quantum Systems
本文研究了由马尔可夫链驱动的随机量子动力学系统涌现的动力学半群，通过分析其生成元的谱性质提取系统大时间行为，并应用于系统与热探针的重复相互作用，推导出热交换的涨落定理与线性响应公式。
On the physical and mathematical foundations of quantum physics via functional integrals
本文基于 Banach 空间和 Henstock-Kurzweil 积分构建严格测度，构造新 Hilbert 空间 KS2 以保持 Feynman 路径积分的计算优势，并证明历史时间以扩展含核的时间相关群与半群。
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