Equations Of Quantum Field Theory Research Articles

Noether’s theorem and Ward-Takahashi identities from homotopy algebras

We derive the new identity in homotopy algebras which directly corresponds to the Schwinger-Dyson equations in quantum field theory. As an application, we derive the Ward-Takahashi identities. We demonstrate that the Ward-Takahashi identities are reproduced in several examples. In general, our formula contains divergence. We mediate this problem by introducing stubs known in the context of string field theory. With the regularization, we can calculate the anomaly such as axial U(1) anomaly in vector-like U(1) gauge theory.

Covariant tetraquark equations in quantum field theory

We derive general covariant coupled equations of QCD describing the tetraquark in terms of a mix of four-quark states $2q2\bar q$, and two-quark states $q\bar q$. The coupling of $2q2\bar q$ to $q\bar q$ states is achieved by a simple contraction of a four-quark $q\bar q$-irreducible Green function down to a two-quark $q\bar q$ Bethe-Salpeter kernel. The resulting tetraquark equations are expressed in an exact field theoretic form, and are in agreement with those obtained previously by consideration of disconnected interactions; however, despite being more general, they have been derived here in a much simpler and more transparent way.

RGE++: A C++ library to solve renormalisation group equations in quantum field theory

In recent years three-, four- and five-loop beta functions have been computed for various phenomenologically interesting models. However, most of these results have not been implemented in easy to use software packages. RGE++ bridges this gap by providing a flexible, template-based, C++ library to solve renormalisation group equations. Furthermore, we implement the available beta functions for the Standard Model, the minimal supersymmetric extension of the Standard Model and two-Higgs-doublet models, as well as right-handed neutrino extensions of the former two. Program summaryProgram Title:RGE++CPC Library link to program files:https://doi.org/10.17632/c9xmrzg46b.1Developer's repository link:www.github.com/Herren/RGEppLicensing provisions: MITProgramming language:C++Nature of problem: Values of coupling constants in dimensionally regularized quantum field theories depend on the chosen renormalization scale μ. The change of coupling constants w.r.t a change of μ is encoded in coupled first-order differential equations. RGE++ provides routines for solving these differential equations in the Standard Model and selected models beyond it. Furthermore, it provides templates for the implementation of additional models.Solution method:RGE++ is implemented as a set of C++ template classes. To numerically solve the differential equations, we employ Eigen3 [1] and odeint [2]. Furthermore, various additional C++ routines are implemented to extract physical quantities from Yukawa matrices.Additional comments including restrictions and unusual features: For non-physical choices of the coupling constants Landau poles might occur.

Symmetries of the Renormalization Group Equations

The determination of the renormalization group equations in quantum field theory is a very laborious task. For example in the Standard Model the full set of these equations is known only up to two loops, while only some partial results are obtained at higher orders. We argue that one can simplify the calculation of the renormalization group equations by using the symmetry of a system. The origin of the simplification lies in the use of the invariant polynomials of the symmetry group of the theory. We consider a quantum field theory with three scalar fields that is invariant under the action of the permutation group S 3. We show, using the theorem of Molien, that for such a model there is a significant reduction of the amount of work needed for the derivation of the renormalization group equations.

Comment on “the solitons and periodic travelling wave solutions for Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations in quantum field theory, Optik 168 (2018) 807-816”

In the recently published article, S.Saha Ray investigated Dodd-Bullough-Mikhailov (DBM) and Tzitzeica-Dodd-Bullough (TDB) equations via his new extended auxiliary equation method. And he claimed that new types of Jacobi elliptic function solutions had been obtained and meanwhile a new family of explicit travelling wave solutions were derived. In this comment, we show that all of exact traveling solutions presented by S.Saha Ray do not satisfy the new extended auxiliary equation method and therefore are incorrect. In addition, general traveling wave solutions of the two differential equation are presented by making full use of certain results published previously.

Poincaré-covariant analysis of heavy-quark baryons

We use a symmetry-preserving truncation of meson and baryon bound-state equations in quantum field theory in order to develop a unified description of systems constituted from light- and heavy-quarks. In particular, we compute the spectrum and leptonic decay constants of ground-state pseudoscalar- and vector-mesons: $q^\prime \bar q$, $Q^\prime \bar Q$, with $q^\prime,q=u,d,s$ and $Q^\prime,Q = c,b$; and the masses of $J^P=3/2^+$ baryons and their first positive-parity excitations, including those containing one or more heavy quarks. This Poincar\'e-covariant analysis predicts that such baryons have a complicated angular momentum structure. For instance, the ground states are all primarily $S$-wave in character, but each possesses $P$-, $D$- and $F$-wave components, with the $P$-wave fraction being large in the $qqq$ states; and the first positive-parity excitation in each channel has a large $D$-wave component, which grows with increasing current-quark mass, but also exhibits features consistent with a radial excitation. The configuration space extent of all such baryons decreases as the mass of the valence-quark constituents increases.

The solitons and periodic travelling wave solutions for Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations in quantum field theory

In this paper, new types of Jacobi elliptic function solutions of Dodd–Bullough–Mikhailov(DBM) and Tzitzeica–Dodd–Bullough(TDB) equations have been obtained using a new extended auxiliary equation method. A new family of explicit travelling wave solutions is derived. The solitary wave solutions and periodic solutions for these equations are formally derived from the Jacobi elliptic function solutions. The proposed method has been efficiently applied to solve the DBM and TDB equations.

Terminal chords in connected chord diagrams

Rooted connected chord diagrams form a nice class of combinatorial objects. Recently they were shown to index solutions to certain Dyson–Schwinger equations in quantum field theory. Key to this indexing role are certain special chords which are called terminal chords. Terminal chords provide a number of combinatorially interesting parameters on rooted connected chord diagrams which have not been studied previously. Understanding these parameters better has implications for quantum field theory. Specically, we show that the distributions of the number of terminal chords and the number of adjacent terminal chords are asymptotically Gaussian with logarithmic means, and we prove that the average index of the first terminal chord is 2n/3 . Furthermore, we obtain a method to determine any next-to ^i leading log expansion of the solution to these Dyson–Schwinger equations, and have asymptotic information about the coecients of the log expansions.

Nonlinearity of Quantum Mechanics and Solution of the Problem of Wave Function Collapse

The problem of the wave function collapse (a problem of measurement in quantum mechanics) is considered. It is shown that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The particle creation and annihilation processes, which are described based on quantum field theory, play a key role in the measurement processes. Superposition principle is not valid for the system of equations of quantum field theory for particles and fields, because this system is a non-linear. As a result of the creation (annihilation) of a particle, an additional uncertainty arises, which “smears” the interference pattern. The imposition of such a large number of uncertainties in the repetitive measurements leads to the classical behavior of particles. The decoherence theory also implies the creation and annihilation of particles, and this processes are the consequence of non-linearity of quantum mechanics. In this case, the term “collapse of the wave function” becomes a consequence of the other statements of quantum mechanics instead of a separate postulate of quantum mechanics.

Coupled S-States OF Relativistic Two-Particle Systems WITH Energy-Dependent Single Boson Exchange Potentials

Numerical solutions of the equations of quantum field theory describing bound s-states of systems of two particles for single boson exchange potentials depending on the energy of the system have been found. On the basis of these solutions, the transition frequencies of the system from the first excited state to the ground state and the decay widths of the bound state of two particles into two photons have been calculated. The obtained energy spectra and decay widths are compared with the experimentally measured values for positronium.

Numerical solution of relativistic problems on bound states of systems of two spinless particles

Numerical solutions of the equations of quantum field theory describing bound s-states of systems of two scalar particles have been found for the following three cases: a one-boson exchange potential, a superposition of such potentials, and also the Coulomb potential in the relativistic configuration representation. On the basis of these solutions, the elastic scattering form factors and decay constants for decay of a two-particle system into two photons have been obtained. It is shown that the number of zeros of the elastic form factors coincides with the number of zeros of the wave functions and is equal to the quantum number of the state of the system minus one. The obtained energy spectra and decay constants are compared with the experimentally measured values for positronium.

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.

A theory of anti-selfdual Lagrangians: stationary case

We develop the concept of anti-self dual Lagrangians that seems inherent to many problems in mathematical physics, Riemannian geometry, and differential equations. On one hand, they represent gradients of convex functions which usually drive dissipative systems, and on the other, their structure is rich enough to also cover – certain representations of – skew-symmetric operators which normally generate unitary flows. These Lagrangians provide variational formulations and resolutions for several non-potential boundary value problems many of which do not fit in the Euler–Lagrange theory. Solutions are minima of functionals of the form L ( u , Λ u ) where L is an anti-self dual Lagrangian and where Λ is a skew-adjoint operator. However, and just like the self (antiself) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to minimal solutions of our variational problems are not derived from the fact they are critical points of the associated functionals, but because they are also zeroes of the corresponding Lagrangians. To cite this article: N. Ghoussoub, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Unexpected Applications of Hill's Differential Equations in Quantum Field Theory and Cosmology

The effect of the exponential pair creation from vacuum by the external field periodic in time is discussed. Two prospective applications of this physical effect in quantum field theory and in inflationary cosmology are considered. Being a nontrivial example of a parametric resonance, the effect of exponential pair creation may serve as an illustration of the effectiveness of mathematics in physical theory.

The complex germ method in fock space. I. Wave packet type asymptotics

In this paper, we establish a new method of constructing approximate solutions to secondary-quantized equations, for instance, for many-particle Schrodinger and Liouville equations written in terms of the creation and annihilation operators, and also for equations of quantum field theory. The method is based on transformation of these equations to an infinite-dimensional Schrodinger equation, which is investigated by semiclassical methods. We use, and generalize to the infinite-dimensional case, the complex germ method, which yields wave packet type asymptotics in the Schrodinger representation. We find the corresponding asymptotics in the Fock space and show that the state vectors obtained are actually asymptotic solutions to secondary-quantized equations with an accuracy of O(ωM/2), M e N, with respect to the parameter e of the semiclassical expansion.

N-PARTICLE PROBLEM IN QUANTUM FIELD THEORY AND THE FUNCTIONAL LEGENDRE TRANSFORMS

The Bethe–Salpeter equations for the n-particle irreducible Green functions in quantum field theory are derived. Their kernels are expressed in terms of the functional Legendre transforms, and n-particle irreducibility of the perturbation theory graphs for the kernel of the n-particle equation is proved. The method for obtaining the Faddeev–Yakubovski equations in quantum field theory is demonstrated for the three-particle case.

Pair-interaction approximation in the equations of quantum field theory for a four-body system

It is shown that for the correct formulation of the pair-interaction approximation in the dynamical equations for a system of four and more particles in the various approaches of quantum field theory not only the sum of the pair-interaction potentials but also additional, “two-pair” kernels must be taken into account. Equations of two types with compact kernels that take into account these terms are obtained. The kernels of these equations are expressed in terms of the two-annd three-particle scattering amplitudes. The equations of the first type have a structure identical to their nonrelativistic analog, whereas the equations of the second type differ from the nonrelativistic equations even in the number of components. An advantage of these equations is that they can be generalized to the case of arbitrarily many particles.

On Series Expansions Giving Closed-Form Solutions of Korteweg–de Vries-Like Equations

Korteweg–deVries (KdV)-like equations with higher-degree nonlinearity are solved by a direct algebraic technique due to Hereman et al. [J. Phys. A, 19 (1986), pp. 607–628]. For two KdV-like equations, one with fifth-degree nonlinearity, the other a combined KdV and mKdV equation, for particular choices of the coefficients of the nonlinear terms, the kink and antikink solutions found by Dey are recovered. Furthermore, soliton solutions of the combined KdV and mKdV equation are found for all values of the coefficients. Closed-form solutions for the Calogero–Degasperis–Fokas modified mKdV equation are also developed. Applications of the solutions of these equations in quantum field theory, plasma physics, and solid-state physics are mentioned. The Hereman et al. method is illustrated and slightly extended and this direct series method is briefly compared to Hirota’s method.

Renormalizability of the time-dependent variational equations in quantum field theory.

The field-theoretic time-dependent variational approximation allows the study of dynamics by a formalism which reflects some of the nonlinearities of the full theory. We study renormalizability of the time-dependent variational equations which are obtained when a Gaussian trial wave functional is used. Renormalizability depends on the specified initial data. We develop reasonable criteria for limiting the initial data, and show that the time-dependent variational equations are made finite by the renormalization prescription used in the vacuum sector.

Bound states in quantum field theory and coherent states: A fresh look.

We consider bound-state equations in quantum field theory where the state explicitly includes radiation quanta as constituents with the number of such quanta not fixed. The fully interacting system is dealt with through equal-time commutators/anticommutators of field operators. The multiparticle channel for the radiation field is approximated through coherent-state representations.