Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

Symbology of Feynman integrals from twistor geometries

We study the symbology of planar Feynman integrals in dimensional regularization by considering geometric configurations in momentum twistor space corresponding to their leading singularities(LS). Cutting propagators in momentum twistor space amounts to intersecting lines associated with loop and external dual momenta, including the special line associated with the point at infinity, which breaks dual conformal symmetry. We show that cross-ratios of intersection points on these lines, especially those on the infinity line, naturally produce symbol letters for Feynman integrals in D = 4-2∈, which include and generalize their LS. At one loop, we obtain all symbol letters using intersection points from quadruple cuts for integrals up to pentagon kinematics with two massive corners, which agree perfectly with canonical differential equation(CDE) results. We then obtain all two-loop letters, for up to four-mass box and one-mass pentagon kinematics, by considering more intersections arising from two-loop cuts. Finally we comment on how cluster algebras appear from this construction, and importantly how we may extend the method to non-planar integrals.

Second-order interference of two independent and tunable single-mode continuous-wave lasers

The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman＇s path integral theory. It is concluded that whether the second-order temporal interference pattern can or cannot be retrieved via two-photon coincidence counting rate is dependent on the resolution time of the detection system and the frequency difference between these two lasers. Two identical and tunable single-mode continuous-wave diode lasers are employed to verify the predictions. These studies are helpful to understand the physics of two-photon interference with photons of different spectra.

Second-order temporal interference of two independent light beams at an asymmetrical beam splitter

The second-order temporal interference of classical and nonclassical light at an asymmetrical beam splitter is discussed based on two-photon interference in Feynman＇s path integral theory. The visibility of the second-order interference pattern is determined by the properties of the superposed light beams, the ratio between the intensities of these two light beams, and the reflectivity of the asymmetrical beam splitter. Some requirements about the asymmetrical beam splitter have to be satisfied in order to ensure that the visibility of the second-order interference pattern of nonclassical light beams exceeds the classical limit. The visibility of the second-order interference pattern of photons emitted by two independent single-photon sources is independent of the ratio between the intensities. These conclusions are important for the researches and applications in quantum optics and quantum information when an asymmetrical beam splitter is employed.

One-loop contributions for h→ℓℓγ and e−e+→hγ in the U(1)_(B−L) extension of the standard model

We present one-loop contributions for h→ℓℓγ with ℓ=νe,μ,τ,e,μ and e−e+→hγ in the U(1)_(B−L) extension of the standard model. In the phenomenological results, the signal strengths for h→ℓℓγ at the Large Hadron Collider and for e−e+→hγ at future lepton colliders are analyzed in the physical parameter space for both the vector and chiral B−L models. We found that the contributions from the neutral gauge boson Z′ to the signal strengths are rather small. Consequently, the effects will be difficult to probe at future colliders. However, the impacts of charged Higgs and CP-odd Higgs in the chiral B−L model on the signal strengths are significant and can be measured with the help of the initial polarization beams at future lepton colliders.

Intersection theory rules symbology

We propose a novel method to determine the structure of symbols for any family of polylogarithmic Feynman integrals.Using the d log-bases and simple formulas for the leading order and next-to-leading contributions to the intersection numbers,we give a streamlined procedure to compute the entries in the coefficient matrices of canonical differential equations,including the symbol letters and the rational coefficients.We also provide a selection rule to decide whether a given matrix element must be zero.The symbol letters are deeply related to the poles of the integrands and also have interesting connections to the geometry of Newton polytopes.Our method can be applied to many cutting-edge multi-loop calculations.The simplicity of our results also hints at the possible underlying structure in perturbative quantum field theories.

General one-loop reduction in generalized Feynman parametrization form

The search for an effective reduction method is one of the main topics in higher loop computation.Recently,an alternative reduction method was proposed by Chen in[1,2].In this paper,we test the power of Chen’s new method using one-loop scalar integrals with propagators of higher power.More explicitly,with the improved version of the method,we can cancel the dimension shift and terms with unwanted power shifting.Thus,the obtained integrating-by-parts relations are significantly simpler and can be solved easily.Using this method,we present explicit examples of a bubble,triangle,box,and pentagon with one doubled propagator.With these results,we complete our previous computations in[3]with the missing tadpole coefficients and show the potential of Chen’s method for efficient reduction in higher loop integrals.

Quantum Measurement and Extended Feynman Path Integral

Quantum measurement problem has existed many years and inspired a large of literature in both physics and philosophy,but there is still no conclusion and consensus on it.We show it can be subsumed into the quantum theory if we extend the Feynman path integral by considering the relativistic effect of Feynman paths.According to this extended theory,we deduce not only the Klein-Gordon equation,but also the wave-function-collapse equation.It is shown that the stochastic and instantaneous collapse of the quantum measurement is due to the "potential noise" of the apparatus or environment and "inner correlation" of wave function respectively.Therefore,the definite-status of the macroscopic matter is due to itself and this does not disobey the quantum mechanics.This work will give a new recognition for the measurement problem.

Analytic Feynman Integrals of Functionals in a Banach Algebra Involving the First Variation

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.

Review of Feynman’s Path Integral in Quantum Statistics:from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation.It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems.In this Review,the molecular Schrödinger equation will first be introduced,together with the BornOppenheimer approximation that decouples electronic and internuclear motions.Some effective semiclassical potentials,e.g.,centroid potential,which are all formulated in terms of Feynman’s path integral,will be discussed and compared.These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues.As a result,path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques.To complement these techniques,we will examine how Kleinert’s variational perturbation(KP)theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential.To enable the powerful KP theory to be practical for many-body molecular systems,we have proposed a new path-integral method:automated integrationfree path-integral(AIF-PI)method.Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method,we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions.The computational procedure of using our AIF-PI method,along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy(AMAZE),are also highlighted in this review.

One-loopformulas for H→Zν_(ι)ν_(ι),forιe,μ,τin't Hooft-Veltman gauge

In this paper,we present analytical results for one-loop contributions to the decay processes H-Zvivi(for I=e,μ,T).The calculations are performed within the Standard Model framework in the't Hooft-Veltman gauge.One-loop form factors are then written in terms of scalar one-loop functions in the standard notations of LoopTools.As a result,one-loop decay rates for the decay channels can be evaluated numerically by using the package.Furthermore,we analyze the signals of H→Zvivi via the production processes e-e+→ZH*-Z(H*→Zν_(ι)ν_(ι)),including the initial beam polarizations at future lepton colliders.The Standard Model backgrounds,such as the processes e-e+→ν_(ι)ν_(ι)ZZ,are also examined in this study.Numerical results indicate that one-loop corrections make contributions of approximately 10%to the decay rates.These are sizeable contributions and should be taken into account at future colliders.We show that the signals H-Zν_(ι)ν_(ι)are clearly visible at the center-of-mass energy√s=250 GeV and are difficult to probe in higher-energy regions owing to the dominant backgrounds.

One loop integrals reduction

By further examining the symmetry of external momenta and masses in Feynman integrals, we fulfilled the method proposed by Battistel and Dallabona, and showed that recursion relations in this method can be applied to simplify Feynman integrals directly.

Hopf algebras and Dyson-Schwinger equations

In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.

Complete reduction of integrals in two-loop five-light-parton scattering amplitudes

We reduce all the most complicated Feynman integrals in two-loop five-light-parton scattering amplitudes to basic master integrals,while other integrals can be reduced even easier.Our results are expressed as systems of linear relations in the block-triangular form,very efficient for numerical calculations.Our results are crucial for complete next-to-next-to-leading order quantum chromodynamics calculations for three-jet,photon,and/or hadron production at hadron colliders.To determine the block-triangular relations,we develop an efficient and general method,which may provide a practical solution to the bottleneck problem of reducing multiloop multiscale integrals.