Adler-Bardeen theorem in path integral formulation

The higher order contributions to Jacobian in Fujikawa’s path integral framework is considered and the form of anomaly equation in higher orders is established. An argument for the Adler-Bardeen theorem in this formulation is given.     

矢量基尔霍夫公式经典证明的漏洞与新的严格证明

矢量基尔霍夫积分公式是电磁理论的一个重要公式,更是光学衍射理论的基础.然而,我们发现经典著作中这个公式的证明普遍存在漏洞.本文将逐一指出这些漏洞,在此基础上给出一个新的严格证明.最后用数值实验验证我们的结论.     

A new proof of Wigner's theorem

In our paper we present a new, elementary proof for Wigner's famous unitary-antiunitary theorem.     

Direct proof of the mirror theorem for projective hypersurfaces up to degree 3 rational curves

In this paper, we directly derive a generalized mirror transformation of projective hypersurfaces of up to degree 3, genus 0 Gromov-Witten invariants by comparing the Kontsevich’s localization formula with residue integral representation of the virtual structure constants. We can easily generalize our method for the rational curves of arbitrary degree, except under combinatorial complexities.     

Theory of atom optics: Feynman''s path integral approach (Ⅱ)

In the preceding paper, we discussed eight configurations of particle Fraunhofer diffraction. In this paper, we consider nine configurations of particle Fresnel diffraction, and have derived the wave functions of particles in these configurations. Furthermore, the author presents a new inertial navigator principle. Its devised accuracy is not lower than that of the inertial navigator based on cold atom interferometer. The new inertial navigator is named as ABSE inertial navigator. ABSE is abbreviation for Aharonov-Bohm-Sagnac effect.      

Incompatibility of gauge invariance and nonrelativistic locality in path integral formulation

It is shown that a modification of the usual gauge transformations is essential to the path integral formulation of nonrelativistic quantum mechanics as a consequence of defining the locality condition as follows: The contribution from each path comes entirely from the points on the path. Arguments are based on the similarity between Wiener and Feynman functional (path) integrals.     

Topological anomalies from the path integral measure in superspace

A fully quantum version of the Witten-Olive analysis of the central charge in the N=1 Wess-Zumino model in d=2 with a kink solution is presented by using path integrals in superspace. We regulate the Jacobians with heat kernels in superspace, and obtain all superconformal anomalies as one Jacobian factor. The conserved quantum currents differ from the Noether currents by terms proportional to field equations, and these terms contribute to the anomalies. We identify the particular variation of the superfield which produces the central charge current and its anomaly; it is the variation of the auxiliary field. The quantum supersymmetry algebra which includes the contributions of superconformal anomalies is derived by using the Bjorken-Johnson-Low method instead of semi-classical Dirac brackets. We confirm earlier results that the BPS bound remains saturated at the quantum level due to equal anomalies in the energy and central charge.     

A path integral approach to electronic friction of a nanometer-sized tip scanning a metal surface

In this work, we study the dissipation mechanism and frictional force of a nanometer-sized tip scanning a metal surface via a path integral approach. The metal, with internal degrees of freedom (c, c^†) and a tip with an internal degree of freedom (d, d^†) couple with one another by means of an exchanged potential, V. Having integrated out all internal degrees of freedom, we obtain the in-out amplitude. Moreover, we calculate the imaginary part of the in-out amplitude and the frictional force. We find the imaginary part of the in-out amplitude to be positive, and correlated to the sliding velocity in most cases. The frictional force is proportional to the sliding velocity for the case where v < 0.01. However, for cases where v > 0.01, the frictional force demonstrates nonlinear dependence on sliding velocity.     

Establishing path integral in the entangled state representation for Hamiltonians in quantum optics

Based on two mutually conjugate entangled state representations, we establish the path integral formalism for some Hamiltonians of quantum optics in entangled state representations. The Wigner operator in the entangled state representation is presented. Its advantages are explained.     

在工科物理中引入费曼路径积分理论

用费曼的路径积分理论来表述量子力学,将经典力学作为Ｓ〉〉ｈ的极限情况纳入量子力学的理论框架。     

Effect of a force-free end on the mechanical property of a biopolymer——A path integral approach

We study the effect of a force-free end on the mechanical property of a stretched biopolymer.The system can be divided into two parts.The first part consists of the segment counted from the fixed point(i.e.,the origin) to the forced point in the biopolymer,with arclength L_f.The second part consists of the segment counted from the forced point to the force-free end with arclength △L.We apply the path integral technique to find the relationship between these two parts.At finite temperature and without any constraint at the end,we show exactly that if we focus on the quantities related to the first part,then we can ignore the second part completely.Monte Carlo simulation confirms this conclusion.In contrast,the effect for the quantities related to the second part is dependent on what we want to observe.A force-free end has little effect on the relative extension,but it affects seriously the value of the end-to-end distance if △L is comparable to L_f.     

A taxonomy of integral reaction path analysis

Achieving understanding through combustion modelling is limited by the ability to recognize the implications of what has been computed and to draw conclusions about the elementary steps underlying the reaction mechanism. This difficulty can be overcome in part by making better use of reaction path analysis in the context of multidimensional flame simulations. Following a survey of current practice, an integral reaction flux is formulated in terms of conserved scalars that can be calculated in a fully automated way. Conditional analyses are then introduced, and a taxonomy for bidirectional path analysis is explored. Many examples illustrate the resulting path analyses and uncover some new results about laminar non-premixed methane-air jets.     

A riemann integral approach to Feynman's path integral

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrödinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) practical calculations — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrödinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in practical calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.     

Thermoalgebras and path integral

Using a representation for Lie groups closely associated with thermal problems, we derive the algebraic rules of the real-time formalism for thermal quantum field theories, the so-called thermo-field dynamics (TFD), including the tilde conjugation rules for interacting fields. These thermo-group representations provide a unified view of different approaches for finite-temperature quantum fields in terms of a symmetry group. On these grounds, a path integral formalism is constructed, using Bogoliubov transformations, for bosons, fermions and non-abelian gauge fields. The generalization of the results for quantum fields in topology is addressed.     

A path integral way to option pricing

An efficient computational algorithm to price financial derivatives is presented. It is based on a path integral formulation of the pricing problem. It is shown how the path integral approach can be worked out in order to obtain fast and accurate predictions for the value of a large class of options, including those with path-dependent and early exercise features. As examples, the application of the method to European and American options in the Black–Scholes model is illustrated. A particularly simple and fast semi-analytical approximation for the price of American options is derived. The results of the algorithm are compared with those obtained with the standard procedures known in the literature and found to be in good agreement.     

一维无限深势阱中粒子运动的路径积分解法

用路径积分的分析方法求得了一维无限深势阱中粒子的传播函数,并由传播函数导出了粒子的波函数和能量,展示了路径积分与传统方法的等价性,同时还介绍了一种有用的数学函数——雅可比θ_3函数．     

关于卡诺定理证明的教学探讨

介绍并讨论了卡诺定理的一种新的证明方法，在这种方法中没有直接使用扳证法，证明过程中的各主要中间结果物理涵义清楚，直观，容易理解。     

Permutation blocking path integral Monte Carlo simulations of degenerate electrons at finite temperature

We analyse the simulation of strongly degenerate electrons at finite temperature using the recently introduced permutation blocking path integral Monte Carlo (PB‐PIMC) method [T. Dornheim et al., New J. Phys. 17 , 073017 (2015)]. As a representative example, we consider electrons in a harmonic confinement and carry out simulations for up to P = 2000 so‐called imaginary‐time propagators – an important convergence parameter within the PIMC formalism. This allows us to study the P‐dependence of different observables of the configuration space in the Monte Carlo simulations and of the fermion sign problem. We find a surprisingly persisting effect of the permutation blocking for large P, which is explained by comparing different length scales. Finally, we touch upon the uniform electron gas in the warm dense matter regime.