Elastic scattering and the path integral

Efimov, G. V.

doi:10.1007/s11232-014-0172-z

Cited by 9 publications

(13 citation statements)

References 8 publications

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“…It is only in the combination of G.V. Efimov's scientific works which ended with publications [85][86][87][88], the nontrivial physical intuition with rigorous mathematical closure, that every point of nonlocal QFT methods described and discussed in this paper is able to take a worthy space in description of nature.…”

Section: Discussionmentioning

confidence: 95%

S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

et al. 2019

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Nonlocal quantum theory of one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of S-matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by evolution of Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the S-matrix in terms of abstract functional integral over primary scalar field, the S form of a grand canonical partition function is found. And, by expression of S-matrix in terms of the partition function, the representation for S in terms of basis functions is obtained. Derivations are given for discrete case where basis functions are Hermite functions, and for continuous case where basis functions are trigonometric functions. The obtained expressions for the S-matrix are investigated within the framework of variational principle based on Jensen inequality. And, by the latter, the majorant of S (more precisely, of − ln S) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both the polynomial theory ϕ 4 (with suggestions for ϕ 6 ) and the nonpolynomial sine-Gordon theory. A new definition of the S-matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and illustrated numerically, with plots of variational functions q and corresponding majorants for the S-matrices of the theory. For simplicity of numerical calculation: the D = 1 case is considered, and propagator for the free theory G is in the form of Gaussian function typically in the Virton-Quark model, although the obtained analytical inferences are not limited to these particular choices in principle. The formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process. Recently, the discovery of Higgs boson, the last SM element in energy domain, where existence is most natural, occurred: Notably, by the remarkable event confirming validity of SM, a quantum-trivial local quantum theory of scalar field is, after all, not quantum trivial if the SM is a sector of a non-Abelian gauge theory; analogous to QED event.Groundbreaking of Supersymmetric Non-Abelian QFTs as well as Integrable QFTs [1-9] in earnest search for quantum theory of gravity will naturally complement superstring theory. Under such sophistication for superstring theory, which is almost surely strongest pick for fundamental theory of nature, the ultimate truth, it will not be impossible to view all QFTs as effective (low-energy) theory given by renormalizable and nonrenormalizable QFTs, respectively. In other words, every field theory will be a limit in superstring theory. Hypothetically, bosonic string...

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Section: Discussionmentioning

confidence: 95%

S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

et al. 2019

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“…The definition of the GFs is given by the expression (13). Let us note that ∀n > 2 linear and closed equations for S (n) are obtained and no n, n + 2 coupling takes place.…”

Section: Hamilton-jacobi and Continuity Functional Equations Hierarchiesmentioning

confidence: 99%

“…The functional (path) integral formalism underlies many modern theories from various fields of science [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. For this reason, calculation of functional integrals is an important mathematical problem [15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

et al. 2020

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We consider the functional Hamilton–Jacobi (HJ) equation, which is the central equation of the holographic renormalization group (HRG), functional Schrödinger equation, and generalized Wilson–Polchinski (WP) equation, which is the central equation of the functional renormalization group (FRG). These equations are formulated in D-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a “holographic” scalar field Λ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant-field configurations of Λ. For all the functional equations above a rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. An advantage of the HJ hierarchy compared to Schrödinger or WP hierarchies is that the HJ hierarchy splits into independent equations. Using the integration formula, the functional (arbitrary configuration of Λ) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant-field configurations of Λ, this solution is studied in detail. A separable solution for a two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schrödinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schrödinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of Λ are obtained. In context of the continuity equation and open quantum field systems, an optical potential is briefly discussed. The mode coarse-graining growth functional for the WP action (WP functional) is analyzed. Based on this analysis, an approximation scheme is proposed for the generalized WP equation. With an optimized (Litim) regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically. For Λ=0 these solutions are monotonic in each of the momentum variables.

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“…The interaction potential of two nucleons through the exchange of scalar particles is called the Yukawa potential [16]. It has form:   where V 0 is the magnitude coupling constant,  is another scaling constant which is related to R-the effective size where the potential is nonrezo as…”

Section: Cross-section With the Yukawa Potentialmentioning

confidence: 99%

“…If the collision parameters are small compared to the wavelength of the scattering particle [16]   …”

Section: Applying the Modified Perturbation Theory To High Energy Scamentioning

confidence: 99%

Applying the Modified Perturbation Theory to High Energy Scattering in the Quasi-Potential Approach

Han¹,

NhuXuan²,

ToanThang³

2017

JPSA

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Asymptotic behavior of the scattering amplitude for two scalar particles at high energy and fixed momentum transfer is reconsidered in quantum field theory. In the framework of the quasi-potential approach and the modified perturbation theory, a systematic scheme for finding the leading eikonal scattering amplitudes and its corrections are developed and constructed. The differential cross section is also discussed.

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Elastic scattering and the path integral

Cited by 9 publications

References 8 publications

S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

Applying the Modified Perturbation Theory to High Energy Scattering in the Quasi-Potential Approach

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