Search Results (31)

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume. Full article

Using a Dyson–Schwinger approach, we perform an analysis of the non-trivial ground state of thermal $SU\left(N\right)$ Yang–Mills theory in the non-perturbative regime where chiral symmetry is dynamically broken by a mass gap. Basic thermodynamic observables such as energy density and pressure are derived analytically, using Jacobi elliptic functions. The results are compared with the lattice results. Good agreement is found at low temperatures, providing a viable scenario for a gas of massive glue states populating higher levels of the spectrum of the theory. At high temperatures, a scenario without glue states consistent with a massive scalar field is observed, showing an interesting agreement with lattice data. The possibility is discussed that the results derived in this analysis open up a novel pathway beyond lattice to precision studies of phase transitions with false vacuum and cosmological relics that depend on the equations of state in strong coupled gauge theories of the type of Quantum Chromodynamics (QCD). Full article

In the framework of the quantum theory of many-particle systems, we study the compatibility of approximated non-equilibrium Green’s functions (NEGFs) and of approximated solutions of the Dyson equation with a modified continuity equation of the form ${\partial }_t⟨\rho ⟩+(1-\gamma )\nabla ·⟨\mathbf{J}⟩=0$. A continuity equation of this kind allows the e.m. coupling of the system in the extended Aharonov–Bohm electrodynamics, but not in Maxwell electrodynamics. Focusing on the case of molecular junctions simulated numerically with the Density Functional Theory (DFT), we further discuss the re-definition of local current density proposed by Wang et al., which also turns out to be compatible with the extended Aharonov–Bohm electrodynamics. Full article

The gauge covariance of the quark gap equation is compared for the case of three different quark–gluon vertices: the bare vertex, a Ball–Chiu-like vertex constrained by the corresponding Slavnov–Taylor identity, and a full vertex including the transverse components derived from transverse Slavnov–Taylor identities. The covariance properties are verified with the chiral quark condensate and the pion decay constant in the chiral limit. Full article

For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is $L^2\left(\mathbb{R}\right)$. Then, an alternative operator for generating the time evolution that is manifestly unitary is suggested, regardless of the choice of Hamiltonian. The new construction involves an additional positive operator that normalizes the wave function locally and allows us to preserve unitarity, even when dealing with infinite dimensional or non-normed spaces. Our considerations show that Schrödinger and Liouville equations are, in fact, two sides of the same coin and together they provide a unified description for unbounded quantum systems. Full article

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function. Full article

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups ${\left(e^{t(T+K_e)}\right)}_{t\ge 0}$ and ${\left(e^{t(T+K_c+K_e)}\right)}_{t\ge 0}$, generated by the operators $T+K_e$ and $T+K_c+K_e$, respectively, have the same essential type. According to Proposition 1, it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that $R_2\left(t\right)$ is compact on $L^2(\mathsf{\Omega }\times V,dxdv)$, and, using an interpolation argument (see Proposition 2), we establish the compactness of $R_2\left(t\right)$ on $L^p(\mathsf{\Omega }\times V,dxdv)$-spaces for $1<p<+\infty$. To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem 1, is new for periodic boundary conditions. Full article

We discuss local magnetic field quenches using perturbative methods of finite time path field theory (FTPFT) in the following spin chains: Ising and XY in a transverse magnetic field. Their common characteristics are: (i) they are integrable via mapping to a second quantized noninteracting fermion problem; and (ii) when the ground state is nondegenerate (true for finite chains except in special cases), it can be represented as a vacuum of Bogoliubov fermions. By switching on a local magnetic field perturbation at finite time, the problem becomes nonintegrable and must be approached via numeric or perturbative methods. Using the formalism of FTPFT based on Wigner transforms (WTs) of projected functions, we show how to: (i) calculate the basic “bubble” diagram in the Loschmidt echo (LE) of a quenched chain to any order in the perturbation; and (ii) resum the generalized Schwinger–Dyson equation for the fermion two-point retarded functions in the “bubble” diagram, hence achieving the resummation of perturbative expansion of LE for a wide range of perturbation strengths under certain analyticity assumptions. Limitations of the assumptions and possible generalizations beyond it and also for other spin chains are further discussed. Full article

A strategy is developed for writing the time-dependent Schrödinger Equation (TDSE), and more generally the Dyson Series, as a convolution equation using recursive Fourier transforms, thereby decoupling the second-order integral from the first without using the time ordering operator. The energy distribution is calculated for a number of standard perturbation theory examples at first- and second-order. Possible applications include characterization of photonic spectra for bosonic sampling and four-wave mixing in quantum computation and Bardeen tunneling amplitude in quantum mechanics. Full article

We derive a Nambu–Jona-Lasinio (NJL) model from a non-local gauge theory and show that it has confining properties at low energies. In particular, we present an extended approach to non-local QCD and a complete revision of the technique of Bender, Milton and Savage applied to non-local theories, providing a set of Dyson–Schwinger equations in differential form. In the local case, we obtain closed-form solutions in the simplest case of the scalar field and extend it to the Yang–Mills field. In general, for non-local theories, we use a perturbative technique and a Fourier series and show how higher-order harmonics are heavily damped due to the presence of the non-local factor. The spectrum of the theory is analysed for the non-local Yang–Mills sector and found to be in agreement with the local results on the lattice in the limit of the non-locality mass parameter running to infinity. In the non-local case, we confine ourselves to a non-locality mass that is sufficiently large compared to the mass scale arising from the integration of the Dyson–Schwinger equations. Such a choice results in good agreement, in the proper limit, with the spectrum of the local theory. We derive a gap equation for the fermions in the theory that gives some indication of quark confinement in the non-local NJL case as well. Confinement seems to be a rather ubiquitous effect that removes some degrees of freedom in the original action, favouring the appearance of new observable states, as seen, e.g., for quantum chromodynamics at lower energies. Full article

The anomalous dimension ${\gamma }_m=1$ in the infrared region near the conformal edge in the broken phase of the large $N_f$ QCD has been shown by the ladder Schwinger–Dyson equation and also by the lattice simulation for $N_f=8$ and for $N_c=3$. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, ${\gamma }_m=1$, by comparing the pion matrix element of the trace of the energy-momentum tensor $⟨\pi \left(p_2\right)|(1+{\gamma }_m)·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=⟨\pi \left(p_2\right)|{\theta }_{\mu }^{\mu }|\pi \left(p_1\right)⟩=2M_{\pi }^2$ (up to trace anomaly) with the estimate of $⟨\pi \left(p_2\right)|2·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=2M_{\pi }^2$ through the Feynman–Hellmann theorem combined with an assumption $M_{\pi }^2\sim m_f$ characteristic of the broken phase. We show that this is not justified by the explicit evaluation of each matrix element based on the dilaton chiral perturbation theory (dChPT): $⟨\pi \left(p_2\right)|2·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=2M_{\pi }^2+[(1-{\gamma }_m)M_{\pi }^2·2/(1+{\gamma }_m)]=2M_{\pi }^2·2/(1+{\gamma }_m)\ne 2M_{\pi }^2$ in contradiction with his estimate, which is compared with $⟨\pi \left(p_2\right)|(1+{\gamma }_m)·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=(1+{\gamma }_m)M_{\pi }^2+\left[(1-{\gamma }_m)M_{\pi }^2\right]=2M_{\pi }^2$ (both up to trace anomaly), where the terms in $\left[\right]$ are from the $\sigma$ (pseudo-dilaton) pole contribution. Thus, there is no constraint on ${\gamma }_m$ when the $\sigma$ pole contribution is treated consistently for both. We further show that the Feynman–Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the $\sigma$ pole contribution is absent, and with $M_{\pi }^2\sim m_f^{2/(1+{\gamma }_m)}$ instead of $M_{\pi }^2\sim m_f$, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of $f_0\left(500\right)$ to $\pi \pi$ for $N_f=2$. It is shown to be consistent with the reality, when $f_0\left(500\right)$ is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance. Full article

We demonstrate that the Finite-Time-Path Field Theory is an adequate tool for calculating neutrino oscillations. We apply this theory using a mass-mixing Lagrangian which involves the correct Dirac spin and chirality structure and a Pontecorvo–Maki–Nakagawa–Sakata (PMNS)-like mixing matrix. The model is exactly solvable. The Dyson–Schwinger equations transform propagators of the input free (massless) flavor neutrinos into a linear combination of oscillating (massive) neutrinos. The results are consistent with the predictions of the PMNS matrix while allowing for extrapolation to early times. Full article

Dynamical chiral symmetry breaking (D$\chi$SB) in quantum chromo dynamics (QCD) for light quarks is an indispensable concept for understanding hadron physics, i.e., the spectrum and the structure of hadrons. In functional approaches to QCD, the respective role of the quark propagator has been evident since the seminal work of Nambu and Jona-Lasinio has been recast in terms of QCD. It not only highlights one of the most important aspects of D$\chi$SB, the dynamical generation of constituent quark masses, but also makes plausible that D$\chi$SB is a robustly occurring phenomenon in QCD. The latter impression, however, changes when higher n-point functions are taken into account. In particular, the quark–gluon vertex, i.e., the most elementary n-point function describing the full, non-perturbative quark–gluon interaction, plays a dichotomous role: It is subject to D$\chi$SB as signalled by its scalar and tensor components but it is also a driver of D$\chi$SB due to the infrared enhancement of most of its components. Herein, the relevant self-consistent mechanism is elucidated. It is pointed out that recently obtained results imply that, at least in the covariant gauge, D$\chi$SB in QCD is located close to the critical point and is thus a delicate effect. In addition, requiring a precise determination of QCD’s three-point functions, D$\chi$SB is established, in particular in view of earlier studies, by an intricate interplay of the self-consistently determined magnitude and momentum dependence of various tensorial components of the gluon–gluon and the quark–gluon interactions. Full article

In this paper, we consider the orthogonal polynomials with respect to the weight $w\left(x\right)=w(x;s):=x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]},x\in {\mathbb{R}}^{+},$ where $\lambda >0$, $N>0$ and $0\le s\le 1$. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients ${\alpha }_n\left(s\right)$ and ${\beta }_n\left(s\right)$. We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when $q:=n/N$ is fixed as $n\to \infty$. Full article

Internal solitary waves evolving with time in shallow water are known to affect sound propagation significantly. Unlike prior work studying the acoustic effects of individual internal-wave properties separately, this paper elucidates and evaluates the influence of a complete evolution process of internal waves on acoustic fields both theoretically and by the coupled ocean-acoustic simulation. Two evolving wave properties considered here are shape deformations including the variations of wave amplitudes and widths and packet dispersion manifested as the increasing wavelength (i.e., the distance between successive solitons). The acoustic modal intensity expressed by the Dyson series solution is reformulated to explicitly reveal the modulation effects induced by the deformation and dispersion of internal waves. Dispersion leads to modal interference and causes the intensity envelope to oscillate with the varying wavelength. Deformation modulates intensity in a non-oscillatory manner that is less predictable due to the complexity of amplitude and width variations. In the environment reconstructed from the field observations of internal waves in the South China Sea, the modal intensity simulated by the parabolic-equation model exhibits pronounced modulation effects, where the modal interference due to dispersion dominates the intensity-envelope shape, and deformation affects the extremum positions of envelopes. Full article