PERTURBATIVE QED: THE SCHRÖDINGER PICTURE

The principle of least action is known to apply to many domains of Physics. Yet its justifications and derivations usually come from archaic classical principles like the Fermat or Maupertuis principles, or mathematical analytical frameworks used for classical mechanics like the Lagrange, Hamilton and D’Alembert formalisms. As is well known, it is widely applicable to other domains, without necessarily a clear justification for it, especially when applied to Physics where Lagrangians or Hamiltonians may not be defined or do not exist. In addition, the Feynman Path Integral formulation of Quantum Mechanics gave it a second life, and another way to justify its validity not only in classical, but also in quantum Physics. However, the conventional logic to reach these conclusions is at time circular, and the principle is applied without proof that it is applicable: it just works and everybody does it. It is this applicability that led us in the past to treat the Action path integral as God’s equation.The paper justifies the principle of least action and Feynman’s path integrals through different physical reasoning instead of the traditional mathematical formalism, and on that basis we confirm its validity in most situations, even if Hamiltonians or Lagrangians may be shaky to define, or the system may be dissipative. We also explain when, and why, sometimes, it may not conventionally apply. The derivation uncovers different links between Entropy and Action, which also opens a different way to look at the origin of the probability distribution of even just a single lone particle; especially when complemented with the W-type multi-fold hypothesis, and makes some considerations on Feynman diagrams, instantons, renormalons and resurgence. Also discontinuities in the paths of the path integrals imply that spacetime is discrete, and non-commutative, and supersymmetry in flat spacetime is unphysical. Surfaceology, as optimized computation of the Feynman diagram is shown to trivially predict that all gauge theories double copy dual to gravity share the same forbidden scatterings. Relying on a recently encountered paper, we show how the results of the multi-fold theory, in particular the multi-fold space time matter induction and scattering, built explicitly on the path integral formalism, its underlying principles, and the principle of least action, allows recovering almost everything in Today’s physics. It leaves out essentially only some of the coupling constants and mixing angles/parameters. This goes a long way towards the ambition of a Theory Of Everything (TOE), even if not presented as usually expected. It maybe the best that can be achieved considering Gödel's incompleteness theorems. Our justification of the principle of least action, and path integrals, and our work on multi-fold theory encounters relationships between Entropy and Action. We review these encounters, including how our early prediction that entanglement might be irreversible match recent published results that argue against the existence of a (second) Law of entanglement analogous to the second Law of Thermodynamics. Also, from our approach, we obtain a formulation of the evolution of entropy, including for irreversible systems, which matches Onsager and Prigogine’s famous models and principles.

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.

.The worldline approach to Quantum Field Theory (QFT) allows to efficiently compute several quantities, such as one-loop effective actions, scattering amplitudes and anomalies, which are linked to particle path integrals on the circle. A helpful tool in the worldline formalism on the circle are string-inspired (SI) Feynman rules, which correspond to a specific way of factoring out a zero mode. In flat space this is known to generate no difficulties. In curved space, it was shown how to correctly achieve the zero mode factorization by applying BRST techniques to fix a shift symmetry. Using special coordinate systems, such as Riemann Normal Coordinates, implies the appearance of a non-linear map --originally introduced by Friedan-- which must be taken care of in order to obtain the correct results. In particular, employing SI Feynman rules, the map introduces further interactions in the worldline path integrals. In the present paper, we compute in closed form Friedan’s map for RNC coordinates in maximally symmetric spaces, and test the path integral model by computing trace anomalies. Our findings match known results.

Probability theory deals with limit theorems, which consider limits (when time tends to infinity) of some functions (observables) on a sample space, or averages of these observables over an infinite time interval. But what is happening in a finite time or over finite time intervals? Such questions, important for virtually all applications, seem to be intractable mathematically (and generally sound unreasonable). For instance, equilibrium statistical mechanics deals with phase transitions (a number of equilibrium probability distributions/states) rather than time evolution, while nonequilibrium statistical mechanics is concerned with convergence of nonequilibrium states to equilibrium ones. Again, such processes occur on infinite time intervals. It turns out, however, that there are natural and reasonable questions about finite time dynamics of random and deterministic chaotic systems, which can be answered and, moreover, rigorously answered. This allows one to make predictions about a finite time evolution of such systems.

Lyapunov modal analysis and participation factors applied to small-signal stability of power systems

The question of operator ordering in the quantum Yang-Mills Hamiltonian is discussed for Coulomb gauge. Unconventional terms, first proposed by Schwinger, are found and their effect on the Coulomb gauge Feynman rules is analysed.

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time τ, on a compact space manifold Ω. We explain how to compute the evolution operator U(tf, ti) between the initial time ti and the final time tf = ti + τ, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite τ and compact Ω are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity U†(tf, ti)U(tf, ti) = 1. The dimensional regularization is extended to finite τ and compact Ω, and used to prove, under general assumptions, that renormalizability holds whenever it holds at τ = ∞, Ω = ℝ3. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator Uph(tf, ti) is unitary, but does not satisfy the more general identity Uph(t3, t2)Uph(t2, t1) = Uph(t3, t1). As a consequence, Uph(tf, ti) cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles.

Aims. We describe various expansion schemes that can be used to study gravitational clustering. Obtained from the equations of motion or their path-integral formulation, they provide several perturbative expansions that are organized in a different fashion or involve different partial resummations. We focus on the two-point and three-point correlation functions, but these methods also apply to all higher-order correlation and response functions.Methods. We present the general formalism, which holds for the gravitational dynamics and for any similar models, such as the Zeldovich dynamics, that obey similar hydrodynamical equations of motion with a quadratic nonlinearity. We give our explicit analytical results up to one-loop order for the simpler Zeldovich dynamics. For the gravitational dynamics, we compare our one-loop numerical results with numerical simulations.Results. We check that the standard perturbation theory is recovered from the path integral by expanding over Feynman's diagrams. However, the latter expansion is organized in a different fashion, and it contains some UV divergences that cancel out as we sum all diagrams of a given order. Resummation schemes modify the scaling of tree and one-loop diagrams, which exhibit the same scaling over the linear power spectrum (contrary to the standard expansion). However, they do not significantly improve over standard perturbation theory for the bispectrum, unless one uses accurate two-point functions (e.g. a fit to the nonlinear power spectrum from simulations). Extending the range of validity to smaller scales, to reach the range described by phenomenological models, seems to require at least two-loop diagrams.

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on $${\mathbb{R}^n}$$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in $${\hbar}$$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the $${t \to 0}$$ limit approaches the δ distribution. As such, our construction gives explicitly the full $${\hbar\to 0}$$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.

A dynamical picture of phylogenetic evolution is given in terms of Markov models on a state space, comprising joint probability distributions for character types of taxonomic classes. Phylogenetic branching is a process which augments the number of taxa under consideration, and hence the rank of the underlying joint probability state tensor. We point out the combinatorial necessity for a second-quantised, or Fock space setting, incorporating discrete counting labels for taxa and character types, to allow for a description in the number basis. Rate operators describing both time evolution without branching, and also phylogenetic branching events, are identified. A detailed development of these ideas is given, using standard transcriptions from the microscopic formulation of nonequilibrium reaction-diffusion or birth-death processes. These give the relations between stochastic rate matrices, the matrix elements of the corresponding evolution operators representing them, and the integral kernels needed to implement these as path integrals. The `free' theory (without branching) is solved, and the correct trilinear `interaction' terms (representing branching events) are presented. The full model is developed in perturbation theory via the derivation of explicit Feynman rules which establish that the probabilities (pattern frequencies of leaf colourations) arising as matrix elements of the time evolution operator are identical with those computed via the standard analysis. Simple examples (phylogenetic trees with 2 or 3 leaves), are discussed in detail. Further implications for the work are briefly considered including the role of time reparametrisation covariance.

Optimal control of stochastic prey–predator models

Based on Kleinert's variational perturbation (KP) theory [Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. (World Scientific, Singapore, 2004)], we present an analytic path-integral approach for computing the effective centroid potential. The approach enables the KP theory to be applied to any realistic systems beyond the first-order perturbation (i.e., the original Feynman-Kleinert [Phys. Rev. A 34, 5080 (1986)] variational method). Accurate values are obtained for several systems in which exact quantum results are known. Furthermore, the computed kinetic isotope effects for a series of proton transfer reactions, in which the potential energy surfaces are evaluated by density-functional theory, are in good accordance with experiments. We hope that our method could be used by non-path-integral experts or experimentalists as a "black box" for any given system.

Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.

The study of boson models and the problem of their solutions is as old as the free Bose gas model for which the celebrated phenomenon of Bose-Einstein condensation (BEC 1924) has been detected. This phenomenon puts in evidence a macroscopic, purely quantum phenomenon. Solving boson models means that we are interested in finding the ground and/or temperature states of the models, or that we are interested in deriving at least some of their properties from first quantum-mechanical principles. For a long time activities in this area belonged to the field of many-body physics, a field of high activity in theoretical physics. Green’s functions, series expansions, Feynman graphs and their summations, and much of numerics are the standard technologies. The ultimate aim is, as always, to understand the physical world from the point of view of its basic laws and constituents and, therefore, to derive as many exact results as possible having a large or universal range of validity for these systems.KeywordsCanonical Commutation RelationBoson SystemCanonical Commutation RelationGround State CondensationBogoliubov ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we use a new method and combining the partial ordering method to study the existence of the solutions for the first order nonlinear impulsive integro‐differential equations of Volterra type on finite interval in Banach spaces and for the first order nonlinear impulsive integro‐differential equations of Volterra type on infinite interval with infinite number impulsive times in Banach spaces. By introducing an interim space and using progressive estimation method, some restrictive conditions on impulsive terms, used before, such as, prior estimation, noncompactness measure estimations are deleted.