Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

Dzhunushaliev, Vladimir; Zloshchastiev, Konstantin G.

doi:10.2478/s11534-012-0159-z

Cited by 24 publications

(14 citation statements)

References 52 publications

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“…For example, it was recently shown that the logarithmic term y S r ln ; | ( )|  of equation (1) dominates the standard Gross-Pitaevskii terms for Helium-4 at low temperatures in certain regimes [16]. A superfluid vacuum theory using a log SE appears in one formulation for the Higgs potential [17,18]. The log SE even has geophysical applications in magma transport [19].…”

Section: Introductionmentioning

confidence: 99%

Solution of the 3D logarithmic Schrödinger equation with a central potential

Shertzer

Scott

2020

J. Phys. Commun.

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Some form of the time-independent logarithmic Schrödinger equation (log SE) arises in almost every branch of physics. Nevertheless, little progress has been made in obtaining analytical or numerical solutions due to the nonlinearity of the logarithmic term in the Hamiltonian. Even for a central potential, the Hamiltonian does not commute with L 2 or L ; z the Hamiltonian is invariant under the parity operation only if the wave function is an eigenstate of the parity operator. We show that the solutions with well-defined parity can be expressed as a linear combination of eigenstates of L 2 and L , z where the parity restrictions on l and m determine the nodal structure of the wave function. The dominant contribution in the sum is designated as l˜and m; these serve as approximate quantum numbers. Using an iterative finite element approach, we also carry out fully converged numerical calculations in 1D, 2D and 3D for the special case of a Coulomb potential. The nodal structure of the wave functions and the expectation values á ñ L 2 and á ñ L z 2 are consistent with the analytical predictions. Values for the energy and expectations values are tabulated for the low-lying states. The methods developed for this problem are applicable across many areas of physics.

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Section: Introductionmentioning

confidence: 99%

Solution of the 3D logarithmic Schrödinger equation with a central potential

Shertzer

Scott

2020

J. Phys. Commun.

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“…A version of SVT favors a logarithmic Schrödinger equation over a Gross-Pitaevski equation [9,15]. In particular, the logarithmic model can account for the upside-down Mexican hat shape of the Higgs potential and its solution is claimed to be even more stable and energetically favorable than the model with a quartic (Higgslike) potential [9,16,17]. Although the Higgs boson reported at 125 Gev has been confirmed, the Higgs potential has not been firmly established.…”

Section: Introductionmentioning

confidence: 99%

Solution of the logarithmic Schrödinger equation with a Coulomb potential

Scott

Shertzer

2018

J. Phys. Commun.

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The nonlinear logarithmic Schrödinger equation (log SE) appears in many branches of fundamental physics, ranging from macroscopic superfluids to quantum gravity. We consider here a model problem, in which the log SE includes an attractive Coulomb interaction. We derive an analytical solution for the ground state energy and wave function as a function of the strength of the logarithmic interaction. We develop an iterative finite element method to solve the Coulombic log SE for the spherically symmetric states. The ground state results agree with the exact solution to better than one part in 10 10 . The excited states (n>1) are converged to better than one part in 10 8. We also construct a remarkably simple variational wave function, consisting of a sum of Gaussons with n free parameters. One can obtain an approximation to the energy and wave function that is in good agreement with the finite element results. Although the Coulomb problem is interesting in its own right, the iterative finite element method and the variational Gausson basis approach can be applied to any central force Hamiltonian.

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“…One of the best known examples is provided by [16,17] in which the toy models are considered in the form of the nonlinear logarithmic Schrödinger Equation (5) with the wave-function solutions ψ ∈ L 2 ( d ) studied in an interval of time t ∈ (t 0 , t 1 ). This equation, along with its relativistic analogue, finds multiple applications in the physics of quantum fields and particles [49][50][51][52][53][54][55], extensions of quantum mechanics [16,56], optics and transport or diffusion phenomena [57][58][59][60], nuclear physics [61,62], the theory of dissipative systems and quantum information [63][64][65][66][67][68], the theory of superfluidity [69][70][71][72] and the effective models of the physical vacuum and classical and quantum gravity [73][74][75][76], where one can utilize the well-known fluid/gravity analogy between inviscid fluids and pseudo-Riemannian manifolds [77][78][79][80][81]. The relativistic analogue of Equation (5) is obtained by replacing the derivative part with the d'Alembert operator and is not considered here.…”

Section: Broader Context In Physicsmentioning

confidence: 99%

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

2017

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Schrödinger equations with non-Hermitian, but P T -symmetric quantum potentials V(x) found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an "anomalous" refraction index. This turned attention to the nonlinear Schrödinger equations in which the interaction term becomes state-dependent, V(x) → W(ψ(x), x). Here, the state-dependence in W(ψ(x), x) is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction W(ψ(x), x) is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by P T -symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schrödinger equations.

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Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

Cited by 24 publications

References 52 publications

Solution of the 3D logarithmic Schrödinger equation with a central potential

Solution of the 3D logarithmic Schrödinger equation with a central potential

Solution of the logarithmic Schrödinger equation with a Coulomb potential

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

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