Use Of Feynman Research Articles

Educational reconstructions of quantum physics using the sum over paths approach with energy dependent propagators

Moving from the research tradition on the use of Feynman’s sum over paths approach in education, we have developed an educational reconstruction of elementary quantum physics which uses Feynman paths at fixed energy and is capable of a simple conceptual explanation of discrete energy levels in bound systems. The reconstruction has been found to offer several educational advantages, may help students think of quantum physics in a way more akin to how the subject is used by scientists, and allows to shed light on the relationship between energy and time.

Spectral Rigidity of Random Schrödinger Operators via Feynman–Kac Formulas

We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrodinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas for RSOs with multiplicative noise (Gaudreau Lamarre in Semigroups for one-dimensional Schrodinger operators with multiplicative Gaussian noise, Preprint arXiv:1902.05047v3 , 2019; Gaudreau Lamarre and Shkolnikov in Ann Inst Henri Poincare Probab Stat 55(3):1402–1438, 2019; Gorin and Shkolnikov in Ann Probab 46(4):2287–2344, 2018) by Gorin, Shkolnikov, and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $$-\frac{1}{2}\Delta +V+\xi $$ , where V is a deterministic potential and $$\xi $$ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise $$\xi $$ .

The use of Feynman diagrammatic approach for well test analysis in stochastic porous media

Transient well test analysis provides valuable information about reservoir characteristics such as permeability and the hydraulic diffusivity coefficient. It is based on the solution of diffusivity equation, which describes mass transfer in porous media. On the whole, analytical solutions are used for interpreting well test data. However, all these solutions were obtained under the condition of reservoir homogeneity. In a heterogeneous reservoir with spatially variable permeability, the exact analytical solutions are not known. The heterogeneous permeability field can be represented as the sum of two terms. The first term is a constant mean permeability value and the second one is a random function with known statistical properties. The second term can be considered a perturbation. The possibility of evaluating geostatistical parameters from well test analysis was considered by various authors and is still a challenging problem. In a randomly heterogeneous reservoir, a flow equation is formulated for the pressure, which is averaged over all the permeability realizations. It can be solved using Green’s function techniques, where the ensemble-averaged pressure is represented as an infinite perturbation series. This series can be represented graphically using Feynman diagrams and its summation can be performed following the rules that are well known in the quantum theory of solid state. For the first time, this framework was introduced to reservoir simulation in King (J. Phys. A: Math. Gen. 20, 3935–3947, 1987), where the stochastic pressure equation was solved for the steady-state case. In this study, we use diagram approach to obtain the solution of the time-dependent stochastic pressure equation which is derived for a lognormal random permeability field under the assumption of the Gaussian correlation function. The expression for transient ensemble averaged pressure is obtained with respect to high-order corrections of permeability variance. In the limit of sufficiently small variance, analytical expressions for the pressure correction are presented. The two limiting cases were considered: (i) the distance between wells is much bigger than the permeability correlation length; (ii) the opposite case where the correlation length is the smallest length parameter. The resulting solution can be used for the analysis of drawdown, build-up, and interference tests in stochastic porous media. The possibility of estimating the parameters of a random permeability field based on well test data is discussed.

Overcoming the diffraction limit by multi-photon interference: a tutorial

The nature of light, extending from the optical to the x-ray regime, is reviewed from a diffraction point of view by comparing field-based statistical optics and photon-based quantum optics approaches. The topic is introduced by comparing historical diffraction concepts based on wave interference, Dirac’s notion of photon self-interference, Feynman’s interference of space–time photon probability amplitudes, and Glauber’s formulation of coherence functions based on photon detection. The concepts are elucidated by a review of how the semiclassical combination of the disparate photon and wave concepts have been used to describe light creation, diffraction, and detection. The origin of the fundamental diffraction limit is then discussed in both wave and photon pictures. By use of Feynman’s concept of probability amplitudes associated with independent photons, we show that quantum electrodynamics, the complete theory of light, reduces in lowest order to the conventional wave formalism of diffraction. As an introduction to multi-photon effects, we then review fundamental one- and two-photon experiments and detection schemes, in particular the seminal Hanbury Brown–Twiss experiment. The formal discourse of the paper starts with a treatment of first-order coherence theory. In first order, the statistical optics and quantum optics formulations of coherence are shown to be equivalent. This is elucidated by a discussion of Zernike’s powerful theorem of partial coherence propagation, a cornerstone of statistical optics, followed by its quantum derivation based on the interference of single-photon probability amplitudes. The treatment is then extended to second-order coherence theory, where the equivalence of wave and particle descriptions is shown to break down. This is illustrated by considering two photons whose space–time probability amplitudes are correlated through nonlinear birth processes, resulting in entanglement or cloning. In both cases, the two-photon diffraction patterns are shown to exhibit resolution below the conventional diffraction limit, defined by the one-photon diffraction patterns. The origin of the reduction is shown to arise from the interference of two-photon probability amplitudes. By comparing first- and second-order diffraction, it is shown that the conventional first-order concept of partial coherence with its limits of chaoticity and first-order coherence has the second-order analogue of partial entanglement, with its limits corresponding to two entangled photons (“entangled biphotons”) and two cloned photons (“cloned biphotons”), the latter being second-order coherent. The concept of cloned biphotons is extended to the case of n cloned photons, resulting in a 1/n reduction of the diffraction limit. In the limit of nth-order coherence, all photons within the nth-order collective state are shown to propagate on particle like trajectories, reproducing the 0th-order ray-optics picture. These results are discussed in terms of the linearity of quantum mechanics and Heisenberg’s space–momentum uncertainty principle. A general concept of coherence based on photon density is developed that in first order is equivalent to the conventional wave-based picture.

Regularization of Diagrammatic Series with Zero Convergence Radius

The divergence of perturbative expansions which occurs for the vast majority of macroscopic systems and follows from Dyson's collapse argument prevents the direct use of Feynman's diagrammatic technique for controllable studies of strongly interacting systems. We show how the problem of divergence can be solved by replacing the original model with a convergent sequence of successive approximations which have a convergent perturbative series while maintaining the diagrammatic structure. As an instructive model, we consider the zero-dimensional |ψ|⁴ theory.

Coherent and incoherent tunneling in asymmetric double-well potentials

The determination of the time scale for coherent and incoherent tunneling in asymmetric double-well potentials is reconsidered according to the instanton-bounce method. In particular, by making use of Feynman's transition elements, a different, relatively simpler approach to this problem, with respect to conventional quantum-mechanical treatments, is obtained.

Path-integral approach to Debye–Waller factors in EXAFS, EELS and XPD for cubic and quartic anharmonic potentials

In this paper thermal effects in extended X-ray absorption fine structure (EXAFS) and X-ray photoelectron diffraction (XPD) due to atomic vibration in cubic and quartic potentials are studied by use of Feynman's path-integral approach. This approach can be applied to strongly anharmonic systems where the cumulant analyses break down. It is closely related to the well known classical approach which is only valid at high temperature. The phase of the thermal factor plays an important role both in EXAFS and XPD analyses for the asymmetric potential with strong anharmonicity. At low temperature the cumulant expansion up to the second order for the thermal damping function agrees well with the self-consistent result, but up to higher orders should be taken into account for the phase function. At high temperature the result from self-consistent calculations shows the characteristic behaviour: the thermal damping function is negative in the high-k region for both strongly and weakly anharmonic systems. The cumulant approximation cannot reproduce this behaviour. For the strongly anharmonic systems the quantum result shows qualitatively different behaviour from the classical approximation at low temperature: the former does not show the negative values even in the high-k region, while the latter shows the phase inversion in the amplitude.

On the use of Feynman–Hibbs effective potentials to calculate quantum mechanical free energies of activation

The use of Feynman–Hibbs effective potentials to estimate quantum mechanical free energies of activation is shown to be a well defined approximation to a more accurate version of quantum activated rate theory. The potential pitfalls of employing such an approximation are also discussed.

Path-integral approach to resonant tunneling.

Resonant tunneling is studied by use of Feynman's path integrals. The semiclassical Green's function is obtained for a one-dimensional double-barrier structure. This approach leads to a clearer and deeper understanding of the physical process of resonant tunneling. It is shown that sequential tunneling never gives rise to negative differential resistance.

From single-particle states to n-particle states.

With the use of Feynman's laws for combining quantum-mechanical amplitudes, expressions for the n-particle states in terms of the single-particle states (for noninteracting particles) are derived rather than postulated. The usual direct-product form for distinguishable particles and the symmetric (antisymmetric) combination of direct products for identical particles emerge as the only forms consistent with the quantum-mechanical laws for compounding amplitudes.

Fourier transform of a two-center product of exponential-type orbitals. Application to one- and two-electron multicenter integrals

A new formula for the Fourier transform (FT) of a two-center RBF (reduced Bessel function) charge distribution permitting partial-wave analysis is derived with the use of Feynman's identity. This formula is valid for all quantum numbers. It is also independent of the orientation of the coordinate axes. A new representation for the two-center overlap integral (to which the kinetic energy, two-center attraction and Coulomb repulsion integrals can be readily reduced) and for the three-center attraction integral is obtained with the help of the FT. It is stable for all values of the orbital exponents. The method developed by Graovac et al. for computing repulsion integrals for $s$ states is generalized to include all states. Numerical test values of several one- and two-electron integrals are also reported. A strategy which should enhance the efficiency of computation by making maximal use of the FT's already computed is suggested.

Theory of Quantum-Mechanical Effects at the Liquid-Gas Critical Point

A theory is developed, with use of Feynman's effective-potential method, for the quantum-mechanical (QM) effects on the critical properties of Lennard-Jones fluids. The theory (i) provides a good description of the QM effects on the critical temperature, pressure, and volume; (ii) predicts no QM effects on critical exponents; (iii) correctly describes the QM corrections to critical amplitudes; and (iv) introduces QM corrections to the ''correction-to-scaling'' confluent singular terms obtained via classical renormalization-group theory.

A quantum-statistical Monte Carlo method; path integrals with boundary conditions

A new Monte Carlo method for problems in quantum-statistical mechanics is described. The method is based on the use of iterated short-time Green’s functions, for which ’’image’’ approximations are used. It is similar to the use of Feynman or Wiener path integrals but with a modification to take account of hard-core boundary conditions. It is applied to two one-dimensional test problems: that of a single particle in a hard-walled box and that of two hard particles in a hard-walled box. For these test problems, the results are in excellent agreement with exact quantum-mechanical results both at high temperatures (near the classical limit) and at very low temperatures such that essentially only the ground state is occupied. Generalizations to three-dimensional systems, to many-body systems, and to more realistic potentials are discussed briefly.

High-energy cosmic-ray muons at ground level and below ground level

Sea-level muon spectrum predictions for the vertical and greatly inclined directions have been derived from recently developed rigorous calculations by Maeda and also by Cantrell by means of both energy-independent and energy-dependent hadron-nucleus inelastic crosssections. Most recently observed ground-level muon spectra are found to be adequately described as arising from decay process of pions and kaons produced in hadron collisions. Significant differences that can be distinguished by the existing cosmic-ray muon measurements are not expected to arise between muon spectra derived from conventional models and those from the use of Feynman's predictions about the scaling behaviour of very-high-energy hadron collisions. Included in the analysis there are some conclusions from previous works to show that there is a lack of agreement among them and with the present results.

Recalculation of the unitary single planar dual loop in the critical dimension of space time

Abstract The single planar dual-loop amplitudes are recalculated in the critical dimension of space time paying particular attention to the unitary property by ensuring that the only states propagating internally in the loop are those needed to factorize tree diagram residues in a positive definite way. The two new technical features which make this possible are 1. (i) our newly discovered physical state projection operator valid if the space-time dimension takes the critical value; 2. (ii) the use of Feynman's tree theorem whereby it is sufficient to have the above projection operator on the mass shell. The final result agrees with a previous conjecture in that it differs from the original calculations with unprojected propagators by two inverse power of a certain partition function. The results apply to both the ordinary dual model and the Neveu-Schwarz model.

Path-integral theory of an electron gas in a random potential

An ideal gas of non-degenerate electrons of mass m in a Gaussian random potential is investigated. The potential is characterized by two parameters: y (whose square is the variance of the potential energy) and L (correlation length). Relevance to the case of a polycrystalline non-degenerate semiconductor is suggested. The autocorrelation function of the potential is taken Gaussian, W(r' — r") = exp [ — ( r' — r") 2 / L 2 ]. The averaged canonical density matrix (C B (r, r0) > is calculated by the use of Feynman’s path-integral formulation. Using the replacement W ( r - r" ) 2 ~ 1 — (r' —r") 2 /- L 2 (after discussing conditions under which it is possible), we derive an explicit formula for . A characteristic frequency W G = (n/L) (2/ mk B T)% is found and interpreted as being due to itinerant oscillators. A characteristic mass mG = coth (^/3hojG), ft = l/kBT is found; the ratio (mG —m)/m is interpreted as a measure of localization of the electrons in the random potential under consideration. A formula for the energy-level density is discussed with respect to conditions of applicability of the quasi-classical approximation. The function is a simple expression if the condition {hr//L J (2m)}» kBT is satisfied; moreover, if rj kBT, then the ‘quantal distribution function’ (averaged Wigner function) is shown to equal the Maxwell distribution function f(Ek) = f (0) exp (— f 3Ek) where Ek is the kinetic energy, Ek = h 2 k 2 /2m, and f (0) is practically independent of ?. Additional conditions of validity of the Boltzmann-equation theory are thus found.

Asymptotic Expansion of the Bardeen-Cooper-Schrieffer Partition Function by Means of the Functional Method

The canonical operator exp [−β(H − μN)] associated with the Bardeen-Cooper-Schrieffer (BCS) model Hamiltonian of superconductivity is represented as a functional integral by the use of Feynman's ordering parameter. General properties of the partition function in this representation are discussed. Taking the inverse volume of the system as an expansion parameter, it is possible to calculate the thermodynamic potential including terms independent of the volume. This yields a new proof that the BCS variational value is asymptotically exact. The behavior of the canonical operator for large volume is described and related to the state of free quasiparticles. A study of the terms of the thermodynamic potential which are of smaller order in the volume in the low-temperature limit, shows that the ground state energy is nondegenerate and belongs to a number eigenstate.