scholarly journals Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1413
Author(s):  
Igor V. Kanatchikov
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Scalar Function ◽  
Functional Derivative ◽  
Space Time ◽  
Spin Connection ◽  
Curved Space ◽  
Schrödinger Representation ◽  
Product Integral

The functional Schrödinger representation of a nonlinear scalar quantum field theory in curved space-time is shown to emerge as a singular limit from the formulation based on precanonical quantization. The previously established relationship between the functional Schrödinger representation and precanonical quantization is extended to arbitrary curved space-times. In the limiting case when the inverse of the ultraviolet parameter ϰ introduced by precanonical quantization is mapped to the infinitesimal invariant spatial volume element, the canonical functional derivative Schrödinger equation is derived from the manifestly covariant partial derivative precanonical Schrödinger equation. The Schrödinger wave functional is expressed as the trace of the multidimensional spatial product integral of Clifford-algebra-valued precanonical wave function or the product integral of a scalar function obtained from the precanonical wave function by a sequence of transformations. In non-static space-times, the transformations include a nonlocal transformation given by the time-ordered exponential of the zero-th component of spin-connection.

Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization

2019 ◽  
Vol 16 (02) ◽  
pp. 1950017 ◽  
Author(s):  
I. V. Kanatchikov
Keyword(s):  
Scalar Field ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Delta Function ◽  
Functional Derivative ◽  
Field Configuration ◽  
Space Time ◽  
Schrödinger Representation ◽  
Limiting Case ◽  
Static Space

The functional Schrödinger representation of a scalar field on an [Formula: see text]-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder–Weyl Hamiltonian theory. The functional Schrödinger representation emerges from the precanonical quantization when the ultraviolet parameter [Formula: see text] introduced by precanonical quantization is replaced by [Formula: see text], where [Formula: see text] is the time-like tangent space Dirac matrix and [Formula: see text] is an invariant spatial [Formula: see text]-dimensional Dirac’s delta function whose regularized value at [Formula: see text] is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schrödinger equation is derived from the manifestly covariant Dirac-like precanonical Schrödinger equation which is independent of a choice of a codimension-one foliation.

On the Vanishing of the Cosmological Vacuum Energy

1997 ◽  
Vol 12 (16) ◽  
pp. 1127-1130 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Vacuum Energy ◽  
Ground State Solution ◽  
Space Time ◽  
State Solution ◽  
Superstring Theory ◽  
The Universe

By demanding the existence of a globally invariant ground-state solution of the Wheeler–De Witt equation (Schrödinger equation) for the wave function of the Universe Ψ, obtained from the heterotic superstring theory, in the four-dimensional Friedmann space-time, we prove that the cosmological vacuum energy has to be zero.

The Schrödinger Equation and Negative Energies

2018 ◽  
Vol 73 (12) ◽  
pp. 1129-1135
Author(s):  
S.A. Bruce
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Schrödinger Equation ◽  
Discrete Symmetries ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Negative Energy ◽  
Space Time ◽  
Energy Eigenstates

AbstractIt is known that there is no room for anti-particles within the Schrödinger regime in quantum mechanics. In this article, we derive a (non-relativistic) Schrödinger-like wave equation for a spin-$1/2$ free particle in 3 + 1 space-time dimensions, which includes both positive- and negative-energy eigenstates. We show that, under minimal interactions, this equation is invariant under $\mathcal{P}\mathcal{T}$ and 𝒞 discrete symmetries. An immediate consequence of this is that the particle exhibits Zitterbewegung (‘trembling motion’), which arises from the interference of positive- and negative-energy wave function components.

scholarly journals Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

2019 ◽  
Author(s):  
Igor V. Kanatchikov
Keyword(s):  
Scalar Field ◽  
Delta Function ◽  
Functional Derivative ◽  
Field Configuration ◽  
Space Time ◽  
Wave Functions ◽  
Curved Space ◽  
Nonlinear Scalar Field ◽  
Product Integral ◽  
Limiting Case

A relationship between the functional Schr\"odinger representation and the precanonical quantization of a nonlinear scalar field theory is extended to arbitrary curved space-times. The canonical functional derivative Schr\"odinger equation is derived from the manifestly covariant precanonical Schr\"odinger equation in a singular limiting case when the ultraviolet parameter $\varkappa$ introduced by precanonical quantization is identified with the invariant delta-function at equal spatial points. In the same limiting case, the Schr\"odinger wave functional is expressed as the trace of the multidimensional product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration. Thus the standard QFT in curved space-time in functional Schr\"odinger representation emerges from the precanonical formulation of quantum fields as a singular limiting case.

Solusi Persamaan Schrödinger Berbasis Panjang Minimal untuk Potensial Coulomb Termodifikasi

2018 ◽  
Vol 2 (2) ◽  
pp. 43-47
Author(s):  
A. Suparmi, C. Cari, Ina Nurhidayati
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Minimal Length ◽  
Energy Spectra ◽  
Atomic Particle ◽  
Approximate Analytical Solutions ◽  
Radial Schrödinger Equation

Abstrak – Persamaan Schrödinger adalah salah satu topik penelitian yang yang paling sering diteliti dalam mekanika kuantum. Pada jurnal ini persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Fungsi gelombang dan spektrum energi yang dihasilkan menunjukkan kharakteristik atau tingkah laku dari partikel sub atom. Dengan menggunakan metode pendekatan hipergeometri, diperoleh solusi analitis untuk bagian radial persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Hasil yang diperoleh menunjukkan terjadi peningkatan energi yang sebanding dengan meningkatnya parameter panjang minimal dan parameter potensial Coulomb Termodifikasi. Kata kunci: persamaan Schrödinger, panjang minimal, fungsi gelombang, energi, potensial Coulomb Termodifikasi Abstract – The Schrödinger equation is the most popular topic research at quantum mechanics. The  Schrödinger equation based on the concept of minimal length formalism has been obtained for modified Coulomb potential. The wave function and energy spectra were used to describe the characteristic of sub-atomic particle. By using hypergeometry method, we obtained the approximate analytical solutions of the radial Schrödinger equation based on the concept of minimal length formalism for the modified Coulomb potential. The wave function and energy spectra was solved. The result showed that the value of energy increased by the increasing both of minimal length parameter and the potential parameter. Key words: Schrödinger equation, minimal length formalism (MLF), wave function, energy spectra, Modified Coulomb potential

scholarly journals Geometrization of the Schrödinger equation: Application of the Maupertuis Principle to quantum mechanics

2014 ◽  
Vol 11 (08) ◽  
pp. 1450066 ◽  
Author(s):  
Antonia Karamatskou ◽  
Hagen Kleinert
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Particle Density ◽  
Free Particle ◽  
Beltrami Operator ◽  
Classical Particle ◽  
Geometric Form ◽  
Curved Space ◽  
Maupertuis Principle ◽  
Semiclassical Expansion

In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.

Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

2021 ◽  
Author(s):  
Annie Millet ◽  
Svetlana Roudenko ◽  
Kai Yang
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Random Variable ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Deterministic Setting

Abstract We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.

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