Feynman path integral

We use the Feynman path integral approach to nonrelativistic quantum mechanics twofold. First, we derive the lagrangian for a spinless particle moving in a uniformly but not necessarily constantly accelerated reference frame; then, applying the strong equivalence principle (SEP) we obtain the Schroedinger equation for a particle in an inertial frame and in the presence of a uniform and constant gravity field. Second, using the associated Feynman propagator, we propagate an initial gaussian wave packet, with the final wave function and probability density depending on the ratio m , where m is the inertial mass of the particle, thus exhibiting the fact that the weak equivalence principle (WEP) is violated by quantum mechanics. Although due to rapid oscillations the wave function does not exist in the classical limit, the probability density is well defined and mass independent when → 0, showing the recovery of the WEP. Finally, at the quantum level, a heavier particle does not necessarily falls faster than a lighter one; this depends on the relations between the initial and final common positions and times of the particles.

Google: chip cuántico indica existencia de otros universos https://www.dw.com/es/google-dice-que-su-nuevo-chip-cu%C3%A1ntico-apunta-aexistencia-de-universos-paralelos/a-71038849?maca=es-Whatsapp-sharing El fundador de Google Quantum AI sugirió que el chip toma potencia de cálculo de universos paralelos por su rapidez. Los escépticos, sin embargo, sostienen que la afirmación es especulativa.

We use the influence functional path-integral method to derive an exact master equation for the quan- tum Brownian motion of a particle linearly coupled to a general environment (ohmic, subohmic, or supraohmic) at arbitrary temperature and apply it to study certain aspects of the loss of quantum coher- ence.

We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E, = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P,(y) is identical to 1 YO(~)l' in the quantum theory.

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R 2n . Our main results include a connection of the oscillation numbers of the given Lagrangian path with the Lidskii angles of a special symplectic orthogonal matrix. We also present Sturmian type comparison and separation theorems for the difference of the oscillation numbers of two continuous Lagrangian paths. These results, as well as the definition of the oscillation number itself, are based on the comparative index theory . The applications of these results are directed to the theory of Maslov index of two continuous Lagrangian paths. We derive a formula for the Maslov index via the Lidskii angles of a special symplectic orthogonal matrix, and hence we express the Maslov index as the oscillation number of a certain transformed Lagrangian path. The results and methods are based on a generalization of the recently introduced oscillation numbers and dual oscillation numbers for conjoined bases of linear Hamiltonian systems 2020) and on the connection between the comparative index and Lidskii angles of symplectic matrices ( Šepitka and Šimon Hilscher, 2020).

A perturbation theorem for unitary groups generated by form sums of operators is established. This result is used to derive a general dominated convergence theorem for Feynman path integrals. It is then applied to the modified Feynman integral which is shown to enjoy optimal stability properties. The fact that the modified Feynman integral -previously introduced by the author -converges in a rather general setting, is used in an essential manner. Other results concerning less general definitions of the Feynman integral are also given.

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly.

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly.

Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to the meta-monogenic (massive) Dirac equation is constructed which form a Hilbert space. A new integral solution is proposed which involves application of a kernel to the right side of the function, instead of to the left as usual. This allows for the introduction of a multivector generalization of the Feynman Path Integral formulation, which shows that particular "geometric groupings" of solutions evolve in the manner to which we ascribe the term "quantum particle". Further, it is shown that the role of usual i is subplanted by the unit time basis vector, applied on the right side of the functions.

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory.

This paper describes our progress on a neutron interferometry search for the Aharonov-Casher (A-C) effect. Unpolarized neutrons are passed through a 40 kV/mm vacuum electrode system. The spin-dependent phase is set to maximum sensitivity with a magnetic field, and the spin-independent phase is adjusted to zero (modulo 2~r) using the Earth's gravitational field. Upon reversal of the electric field, the predicted A-C phase shift is 0.2 °. Currently, our results are statistically limited.

The geometric construction of the functional integral over coset spaces M/G is reviewed. The inner product on the cotangent space of infinitesimal deformations of M defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber G, the functional measure on the coset space M/G is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev-Popov determinant of the more traditional gauge fixed approach in non-abelian gauge theory. If the general construction is applied to the case where G is the group of coordinate reparametrizations of spacetime, the continuum functional inte

The Feynman path integral approach to quantum mechanics is examined in the case where the configuration space is curved. It is shown how the ambiguity that is present in the choice of path integral measure may be resolved if, in addition to general covariance, the path integral is also required to be consistent with the Schwinger action principle. On this basis it is argued that in addition to the natural volume element associated with the curved space, there should be a factor of the Van Vleck-Morette determinant present. This agrees with the conclusion of an approach based on the link between the path integral and stochastic differential equations.

A phenomenological model is introduced for the dissipative quantum dynamics of the phase p across a current-biased Josephson junction. The model is invariant under p &+2m. This enables us to restrict p to the interval 0 to 2x, equating p+2x with p, and study the role played by the resulting nontrivial topology. Using Feynman's influence functional theory it is shown that the dissipation suppresses interference between paths with different winding numbers. For Ohmic dissipation this interference is completely destroyed, and p can effectively be treated as an extended coordinate. This justifies the use of the usual washboard potential description of a currentbiased junction even in the quantum case, provided an Ohmic dissipation mechanism is present. In the past several years there has been much theoretical'2 and experimental3 5 interest in the possibility of observing "secondary" macroscopic quantum effects in small-capacitance Josephson junctions. In these junctions the charging energy is comparable to the Josephson coupling energy, and quantum-mechanical fluctuations of the phase difference p across the junction become important. This necessitates treating the phase as a quantum operator P, which is canonically conjugate to the operator n which transfers n Cooper pairs across the junction; [), ti1 i Since p is a phase variable, p and p+2tr are physically identical states. 7 This fact is of little consequence for macroscopic quantum tunneling since the important changes in p are small compared to 2tr. Indeed p is usually treated as an extended coordinate, 's p e [-,~]. Recently a new type of macroscopic quantum phenomena has been discussed in the literature, " which is the possible existence of "Bloch" oscillations in current-biased Josephson junctions. Under appropriate conditions, an applied dc current is predicted to cause voltage oscillations

Super Yang-Mills on the Noncommutative Torus 22 2.1 Matrix Compactification 23 2.2 Twisted Quantum Bundles on T 2 30 2.3 Twisted Quantum Bundles on Tori 39 2.4 Adjoint Sections on Twisted• Bundles 42 2.5 Two and Three Dimensional Solutions 48

A dominating feature of knot theory is the problem of knot classification. In this paper, we hope to simplify the task of classification through strong connections between Arnold's work with knot invariants and that of Xiao-Song Lin and Zhenghan Wang. We show that the defect of a plane curve associated with a flattened alternating knot may be considered an invariant

A dominating feature of knot theory is the problem of knot classification. In this paper, we hope to simplify the task of classification through strong connections between Arnold's work with knot invariants and that of Xiao-Song Lin and Zhenghan Wang. We show that the defect of a plane curve associated with a flattened alternating knot may be considered an invariant for alternating knots. We also give simple method of computing a certain Vassiliev knot invariant from Arnold's plane curve invariants and signed crossings.

In this paper, we provide the decay of correlations for random dynamical systems. Precisely, we consider the uniformly C 2 piecewise expanding maps defined on the unit interval satisfying As a principal tool of these studies, we use a coupling method for analyzing the coupling time of observables with bounded variation.

We apply techniques developed for strings to the case of the spinless point particle. The Polyakov path integral with ghosts is used to obtain the propagator and one-loop vacuum amplitude. The propagator is shown to correspond to the Green's function for the BRST field theory in Siegel gauge. The reparametrization invariance of the Polyakov path integral is shown to lead automatically to the correct "trace log" result for the one-loop diagram, despite the fact that "naive sewing" of the ends of a propagator would give an incorrect answer. This type of failure of "naive sewing" is identical to that found in the string case. The present treatment provides, in the simplified context of the point particle, a pedagogical introduction to Polyakov path integral methods with and without ghosts.

The notion of approximate inertial manifold (AIM) has shown it's usefulness in the construction of approximate solutions for a class of parabolic PDEs generating dissipative dynamical systems. Thus, by the use of AIMs, the so-called non-linear Galerkin methods and post processed non-linear Galerkin methods were conceived, improving the well-known Galerkin method. These new methods proved to yield much lower errors at the same dimension of the projection space than the classical Galerkin method. In a previous paper we presented a new modified Galerkin method, related to the non-linear and post-processed Galerkin methods, but different of these. Our method leads to accurate approximate solutions using very low dimensional projection spaces. In the present paper we use this method for an advection-diffusion problem.

A formula for the complex phase shift, pertaining to the one-turning-point scattering problem, where the turning point may be very close to the pole at the origin, is obtained in a general form based on a kind of arbitrary-order phase-integral approximation. For sufficiently small I and high energies this formula gives a significant correction to the one obtained previously by N. Froman and the present author. The accuracy of the respective formula is illustrated by application to a particular screened Coulomb potential.

A formula for the complex phase shift, pertaining to the one-turning-point scattering problem with a complex potential, is given in. a general form based on a kind of arbitrary-order phase-integral approximation. The accuracy of the formula is illustrated by application to a real Coulomb potential supplemented by an imaginary potential term proportional to 1/r'. For that case the arbitrary-order phase-integral expression for the complex phase shift is evaluated in a closed, analytical form, which displays the simplicity and accuracy of the higher-order corrections.

Electron attachment of water clusters was explored by the quantum path-integral molecular-dynamics method, demonstrating that the energetically favored localization mode involves a surface state of the excess electron. The cluster size dependence, the energetics, and the charge distribution of these novel electron-cluster surface states are explored.

Molecular dynamics simulation is currently the theoretical technique eligible to simulate a wide range of systems from soft condensed matter to biological systems. However, of the excellent results that the technique has arrogated, this approach remains computationally expensive, but with the emergence of the new supercomputing technologies bases on graphics processing units graphical processing units-based systems GPUs, the perspective has changed. The GPUs allow performing large and complex simulations at a significantly reduced time. In this work, we present recent innovations in the acceleration of molecular dynamics in GPUs to simulate non-Hamiltonian systems. In particular, we show the performance of measure-preserving geometric integrator in the canonical ensemble, that is, at constant temperature. We provide a validation and performance evaluation of the code by calculating the thermodynamic properties of a Lennard-Jones fluid. Our results are in excellent agreement with rep...

Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $\frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in many different ways, including other subgraphs, minimum degree conditions, etc. This article deals with a generalization to edge-colored multigraphs, which can be viewed as a union of simple graphs, each corresponding to an edge-color class. The case of two colors is the same as the original setting: Diwan and Mubayi proved that any two graphs $G_1$ and $G_2$ on the same set of $n$ vertices, each containing more than $\frac{1}{4}n^2$ edges, give rise to a triangle with one edge from $G_1$ and two edges from $G_2$. The situation is however different for three colors. Fix $\tau=\frac{4-\sqrt{7}}{9}$ and split the $n$ vertices into three sets $A$, $B$ and $C$, such that $|B|=|C|=\lfloor\tau n\rfloor$ and $|A|=n-|B|-|C|$. The graph $G_1$ contains all edges inside $A$ and inside $B$, the g...

The geometrical model of an electrical charge is proposed. This model has the "nake" charge shunted with "fur-coat" consisting of virtual wormholes. The 5D wormhole solution in the Kaluza-Klein's theory is the "nake" charge. The splitting off the supplementary coordinates happens on the two spheres (null surfaces) bounding this 5D wormhole. This allows to sew two Reissner-Nordström's black holes to it on both sides. Virtual wormholes entrap a part of the electrical force lines outcoming from "nake" charge. This effect can essentially decrease the charge visible at infinity up to real relation m 2 < e 2. The analogical construction for colour SU (2) gauge charge is made.

Copyright © 2014 Abdul Rauf Nizami et al. This is an open access article distributed under the Creative Commons Attribution Li-cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intel-lectual property Abdul Rauf Nizami et al. All Copyright © 2014 are guarded by law and by SCIRP as a guardian. We give a general formula of the quantum 2sl-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra 2g sl = in its standard two-dimensional representation. We also recover the Jones polynomial of these knots as a special case of this quantum invariant.

In this letter we provide evidence that quantum mechanics can be interpreted as a rational algorithm for finding the least complex description for the correlations in the outputs of sensors in a large array. In particular, by comparing the selforganization approach to solving the Traveling Salesman Problem with a solution based on taking the classical limit of a Feynman path integral, we are led to a connection between the quantum mechanics of motion in a magnetic field and selforganized information fusion.

We provide evidence that quantum mechanics can be interpreted as a rational algorithm for finding the least complex description for the correlations in the outputs of sensors in a large array. In particular, by comparing the self-organization approach to solving the Traveling Salesman Problem with a solution based on taking the classical limit of a Feynman path integral, we are

Natural modalities are often analysed from an abstract point of view where they are associated with putative laws of nature. However, the way possibilities are represented in physics is more complex. Lagrangian mechanics, for instance, involves two different layers of modalities: kinematical and dynamical possibilities. This paper examines the status of these two layers, both in the classical and quantum case. The quantum case is particularly problematic: we identify four possible interpretive options. The upshot is that a close inspection of the way possibilities are represented in physics could lead to new ways of thinking about natural modalities. Keywords Lagrangian mechanics • Modalities • Laws of nature • Feynman path integral • Kinematic possibilities

We consider the conductance of an Andreev interferometer, i.e., a hybrid structure where a dissipative current flows through a mesoscopic normal (N) sample in contact with two superconducting (S) ''mirrors.'' Giant conductance oscillations are predicted if the superconducting phase difference is varied. Conductance maxima appear when is on odd multiple of due to a bunching at the Fermi energy of quasiparticle energy levels formed by Andreev reflections at the N-S boundaries. For a ballistic normal sample the oscillation amplitude is giant, and proportional to the number of open transverse modes. We estimate, using both analytical and numerical methods, how scattering and mode mixing-which tend to lift the level degeneracy at the Fermi energy-effect the giant oscillations. These are shown to survive in a diffusive sample at temperatures much smaller than the Thouless temperature, provided there are potential barriers between the sample and the normal electron reservoirs. Our results are in good agreement with previous work on conductance oscillations of diffusive samples, which we propose can be understood in terms of a Feynman path-integral description of quasiparticle trajectories. ͓S0163-1829͑98͒06815-5͔

We have used the concept of De Broglie's matter wave associated with particles to derive a inverse square law of gravitation like the newton's law of gravitation at the plank's length with a slight modification. Obtaining the Newtonian form we have further extrapolated it to formulate Einstein's field equation, which generally is embedded with a slight modification.The Schwarzschild solution is calculated from the modified field equation, which gives us the Schwarzschild radius of a miniature black hole. Using the solution, the entropy and the hawking temperature is also modeled theoretically for the miniature black hole at the plank's order.

This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted LK for a link K , and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. L0= 1. 3. ¿"y~ =aL' L-(5"~ = a~lL-4-¿a< +¿X. =2(¿s^ +L5C)• Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from L by writhe-normalization.

An apparent difference between formulating mean field perturbation theory for @* field theory via path integrals or via functional differential equations when there are external sources present is shown not to exist when mean lield theory is considered as the N= 1 limit of the O(N) @" field theory. A simple method is given for determining the l/N expansion for the Green's functions in the presence of external sources by directly solving the functional differential equations order by order in l/N. The l/N expansion for the effective action r(& 1) is obtained by directly integrating the functional differential equations for the fields 4 and x (= jL/N 4,@ ~ p2) in the presence of two external sources j =-&f/&4. S =-bT/b~.

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251].

We present novel path modeling techniques suitable for use in the Path-Integral formulation of Quantum Mechanics. Our proposed platform aims to address existing challenges encountered in Monte Carlo and other similar path modeling methods. By introducing 'smooth' path modeling techniques, we demonstrate how they can be seamlessly integrated with current approaches, facilitating more accessible amplitude estimations in this invaluable formulation of Quantum Mechanics.

The vortex system in a high-T c superconductor has been studied numerically using the mapping to 2D bosons and the path-integral Monte Carlo method. We find a single first-order transition from an Abrikosov lattice to an entangled vortex liquid. The transition is characterized by an entropy jump ∆S ≈ 0.4 k B per vortex and layer (parameters for YBCO) and a Lindemann number c L ≈ 0.25. The increase in density at melting is given by ∆ρ = 6.0×10 −4 /λ(T) 2. The vortex liquid corresponds to a bosonic superfluid, with ρ s = ρ even in the limit λ → ∞.

We have found that the Regge gravity [1, 2], can be represented as a superposition of less complicated theory of random surfaces with Euler character as an action. This extends to Regge gravity our previous result [6], which allows to represent the gonihedric string [7] as a superposition of less complicated theory of random paths with curvature action. We propose also an alternative linear action A(M 4) for the four and high dimensional quantum gravity. From these representations it follows that the corresponding partition functions are equal to the product of Feynman path integrals evaluated on time slices with curvature and length action for the gonihedric string and with Euler character and gonihedric action for the Regge gravity. In both cases the interaction is proportional to the overlapping sizes of the paths or surfaces on the neighboring time slices. On the lattice we constructed spin system with local interaction, which have the same partition function as the quantum gravity. The scaling limit is discussed.

We discuss the canonical derivation of the Feynman rules for relativistic, real-time covariant calculations in field theory at finite temperature. The resulting rules are equivalent to the ones derived using the path-integral method, with the particular value zero for the time-path gauge parameter. The canonical approach complements the path-integral formulation in a way analogous to the relation between the operator and the path-integral formalisms in ordinary field theory. This is illustrated by a simple rederivation of the relations satisfied by the exact propagators of the theory. Special emphasis is given to formulate the results in a way readily useful for practical calculations, with particular attention to the connection between the effective action calculated with the Feynman rules and the classical field equations that are of interest for macroscopic physics. Several examples are provided to illustrate various aspects of the issues involved.

There were 25 talks in the workshop in September 17-21, and there were 27 talks in the seminars in the other weeks of September. Each speaker was requested to give his/her open problems in a short problem session after his/her talk, and many interesting open problems were given and discussed by the speakers and participants in the workshop and the seminars. Contributors of the open problems were also requested to give kind expositions of history, background, significance, and/or importance of the problems. This problem list was made by editing these open problems and such expositions. 1 The logo for the workshop and the seminars was designed by N. Okuda. 1 Open problems on the Rozansky-Witten invariant were written in a separate manuscript [349]. Some fundamental problems are quoted from other problem lists such as [188], [220], [262], [285], [286], [388, Pages 571-572]. 2 For example, directions related to other areas such as hyperbolic geometry via the volume conjecture and the theory of operator algebras via invariants arising from 6j-symbols.