Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Zeitschrift für Naturforschung A

Normalization and Probability Densities for Wavefunctions of Quantum Fieldtheoretic Energy Equations

Harald Stumpf

Published Apr 1, 1989• 2 citations
Publisher

Abstract

Abstract Quantum fields can be characterized by the set of transition amplitudes {〈0|π(A)|a〉V̶|a,〉∈V} where π(A) is a representation of the field operator algebra in V. This set has to satisfy renormalized energy equations and the elements of this set are called wavefunctions. However, these wavefunctions are not identical with wavefunctions of conventional quantum theory in Fockspace. Thus a theoret­ical interpretation is needed. In the present paper, by means of some theorems a method of normalization and construction of probability densities for these wavefunctions is given, which differs from the method of derivation of the normalization condition for Bethe-Salpeter amplitudes. The method can be applied both to nonrelativistic and relativistic fields with positive definite or indefinite state spaces, provided the renormalized energy equations possess finite solutions.

DOI
10.1515/zna-1989-0402

Related papers

The probability operator in quantum theory and quantization of a system of free fields

Sujit Basu, V. V. Kuryshkin

A theoretical framework for quantization, defined by the normalized positive-definite probability operator establishing dynamical correspondence between classical and quantum Poisson brackets, is presented. The resulting quantum theory, unlike the conventional one, admits consistent probabilistic interpretation. It is shown that, in the nonrelativistic case, quantization based on the probability operator leads to the theory known as ‘‘quantum mechanics with a non-negative quantum distribution function.’’ A generalization of the proposed framework to the case of the relativistic theory of fields is attempted. Four auxiliary problems of constructing probability operators of one-dimensional field oscillators in Bose and Fermi algebras are formulated and solved. On the basis of these solutions it is concluded that spinor fields are not quantizable in the Bose algebra with the help of the probability operator (the analog of Pauli’s theorem). An equation for the probability operator of a system of free fields is derived from the principles of dynamical correspondence and translational invariance. The physical meaning of the operators corresponding to classical field amplitudes, such as annihilation and creation operators of field quanta with definite energy-momentum, is shown to emerge as a consequence of this equation. It is shown that the quantization of a system consisting only of tensor fields or only of spinor fields in the formalism of the probability operator leads to difficulties. It is shown further that these difficulties can be removed by considering quantization of a system containing both tensor and spinor fields. As an illustration, quantization of a system consisting of a massive vector field and a massive spinor field is considered and it is found that a noncontradictory quantization requires the mass of the vector particle to be less than that of the spinor particle. The probability operator thus acts as a mechanism of selection of the fields to be quantized already at the level of free fields.

Journal of Mathematical Physics•Apr 1, 1990•0 citations

Quantum field theory for classical fields

Christof Wetterich

For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.

quant-pharXiv:2603.05061•APS Open Science•Mar 6, 2026•0 citations

The indefinite metric in relativistic quantum mechanics

K. J. Le Couteur

Abstract There are many problems of relativistic quantum mechanics which cannot be treated by working with a positive definite metric in the vector space of the fully quantized system. In this paper it is shown that, independently of the signs of energy and charge in the c number theory, any relativistic wave equation can be quantized and interpreted in terms of positive energies by use of an indefinite metric of suitable signature. A powerful method of obtaining positive definite transition probabilities from an indefinite metric is introduced and the concept of ‘negative probability’ is entirely avoided.The transition probabilities are defined as the squared matrix elements of a unitary matrix T, which is obtained from the solution of the Schrodinger equation by means of a special transformation.These techniques make it possible to treat relativistic wave equations which describe particles with several mass and spin states and to give a satisfactory account of the transition between two such mass states, a problem which is otherwise quite intractable. The relationship of spin and statistics is discussed and it is concluded that, apart from the special cases of spin 0, ½, 1, there is no necessary connexion between spin and statistics.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 22, 1949•4 citations

The Schrodinger equation in terms of probability densities

Peter Rastall

The probability density of a non-relativistic particle is written as the sum of two non-negative density functions. The Schrodinger equation is equivalent to continuity equations (with source terms) for these functions. The equations are interpreted as describing two modes of the particle, which continually transform into each other.

Journal of Physics A Mathematical and General•Mar 7, 1994•0 citations

The Wavefunction in the Scretching–Schrödinger Equation, Nonrelativistic Quantum Mechanics, and Relativistic Quantum Mechanics

Daniel Scretching

This paper describes, explains, and compares the wavefunction in three related quantum frameworks: nonrelativistic quantum mechanics (NRQM), relativistic quantum mechanics (RQM), and the Scretching–Schrödinger Equation (SSE). In NRQM, the wavefunction is treated as a complex probability amplitude whose squared modulus gives the probability density for finding a particle in a specified region of space. For bound atomic systems, it is governed by the Schrödinger equation and is commonly labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms, which specify the principal energy level, orbital angular momentum, magnetic projection, and spin projection. In RQM, the wavefunction must be compatible with special relativity. For spin-0 particles, this leads to the Klein–Gordon scalar wavefunction, while for spin-12\frac{1}{2}21 particles, the relativistic wavefunction becomes the four-component Dirac spinor. These relativistic forms preserve probability-amplitude interpretation but require Lorentz-compatible energy–momentum relations, relativistic covariance, and, in the Dirac case, explicit treatment of spin, antimatter solutions, and relativistic current conservation. The SSE framework extends the ordinary Schrödinger wavefunction by imposing a deterministic spectroscopic closure condition through the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, the wavefunction retains the standard quantum-number structure of atomic quantum mechanics, including radial, angular, and spin components, but it is additionally constrained to reproduce measurable spectroscopic quantities such as oscillator strength, transition dipole moment, Einstein AAA and BBB coefficients, transition frequency, absorbance-linked optical quantities, and the SQC invariant Kν=2πe2ε0mec3.K_\nu=\frac{2\pi e^2}{\varepsilon_0 m_e c^3}.Kν=ε0mec32πe2. Thus, the SSE wavefunction is not merely a probability amplitude. It is a probability amplitude constrained by experimentally anchored spectroscopic closure. The referenced SSE formalism defines the SSE Hamiltonian as H^SSE=H^0+V^SQC,\hat{H}_{\mathrm{SSE}}=\hat{H}_0+\hat{V}_{\mathrm{SQC}},H^SSE=H^0+V^SQC, where H^0\hat{H}_0H^0 is the ordinary quantum Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic constraint operator. The SSE wavefunction may therefore be written schematically as ΨSSE(r,θ,ϕ,s)=RSSE(r) Ylm(θ,ϕ) χs [fconstraint],\Psi_{\mathrm{SSE}}(r,\theta,\phi,s) = R_{\mathrm{SSE}}(r)\,Y_l^m(\theta,\phi)\,\chi_s\,[f_{\mathrm{constraint}}],ΨSSE(r,θ,ϕ,s)=RSSE(r)Ylm(θ,ϕ)χs[fconstraint], where RSSE(r)R_{\mathrm{SSE}}(r)RSSE(r) is the radial component modified or constrained by SSE closure, Ylm(θ,ϕ)Y_l^m(\theta,\phi)Ylm(θ,ϕ) is the spherical harmonic angular component, χs\chi_sχs is the spin function, and [fconstraint][f_{\mathrm{constraint}}][fconstraint] represents the SQC/MSC spectroscopic constraint that forces agreement between the wavefunction matrix elements and measured or computed optical transition quantities. The central comparison developed in the paper is that NRQM describes the wavefunction as a probability amplitude, RQM generalizes the wavefunction into Lorentz-compatible scalar or spinor fields, and SSE adds a deterministic spectroscopic layer to the quantum wavefunction. In this sense, SSE does not discard Schrödinger or Dirac quantum mechanics; instead, it proposes an additional closure architecture in which allowed wavefunctions must satisfy both the usual quantum dynamical equation and the SQC/MSC spectroscopic consistency conditions.

•May 1, 2026•0 citations

Quantization as a universal probabilistic phenomenon

Luigi Accardi, Yun Gang Lu

We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field $X$ admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative $*$-algebra on which the quantum extension of the expected value relative to the probability distribution of the field $X$ induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).

Теория вероятностей и ее применения•Jan 1, 2025•0 citations
Abstract Quantum fields can be characterized by the set of transition amplitudes {〈0|π(A)|a〉V̶|a,〉∈V} where π(A) is a representation of the field operator algebra in V. This set has to satisfy renormalized energy equations and the elements of this set are called wavefunctions. However, these wavefunctions are not identical with wavefunctions of conventional quantum theory in Fockspace. Thus a theoret­ical interpretation is needed. In the present paper, by means of some theorems a method of normalization and construction of probability densities for these wavefunctions is given, which differs from the method of derivation of the normalization condition for Bethe-Salpeter amplitudes. The method can be applied both to nonrelativistic and relativistic fields with positive definite or indefinite state spaces, provided the renormalized energy equations possess finite solutions.
A theoretical framework for quantization, defined by the normalized positive-definite probability operator establishing dynamical correspondence between classical and quantum Poisson brackets, is presented. The resulting quantum theory, unlike the conventional one, admits consistent probabilistic interpretation. It is shown that, in the nonrelativistic case, quantization based on the probability operator leads to the theory known as ‘‘quantum mechanics with a non-negative quantum distribution function.’’ A generalization of the proposed framework to the case of the relativistic theory of fields is attempted. Four auxiliary problems of constructing probability operators of one-dimensional field oscillators in Bose and Fermi algebras are formulated and solved. On the basis of these solutions it is concluded that spinor fields are not quantizable in the Bose algebra with the help of the probability operator (the analog of Pauli’s theorem). An equation for the probability operator of a system of free fields is derived from the principles of dynamical correspondence and translational invariance. The physical meaning of the operators corresponding to classical field amplitudes, such as annihilation and creation operators of field quanta with definite energy-momentum, is shown to emerge as a consequence of this equation. It is shown that the quantization of a system consisting only of tensor fields or only of spinor fields in the formalism of the probability operator leads to difficulties. It is shown further that these difficulties can be removed by considering quantization of a system containing both tensor and spinor fields. As an illustration, quantization of a system consisting of a massive vector field and a massive spinor field is considered and it is found that a noncontradictory quantization requires the mass of the vector particle to be less than that of the spinor particle. The probability operator thus acts as a mechanism of selection of the fields to be quantized already at the level of free fields.
For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.
Abstract There are many problems of relativistic quantum mechanics which cannot be treated by working with a positive definite metric in the vector space of the fully quantized system. In this paper it is shown that, independently of the signs of energy and charge in the c number theory, any relativistic wave equation can be quantized and interpreted in terms of positive energies by use of an indefinite metric of suitable signature. A powerful method of obtaining positive definite transition probabilities from an indefinite metric is introduced and the concept of ‘negative probability’ is entirely avoided.The transition probabilities are defined as the squared matrix elements of a unitary matrix T, which is obtained from the solution of the Schrodinger equation by means of a special transformation.These techniques make it possible to treat relativistic wave equations which describe particles with several mass and spin states and to give a satisfactory account of the transition between two such mass states, a problem which is otherwise quite intractable. The relationship of spin and statistics is discussed and it is concluded that, apart from the special cases of spin 0, ½, 1, there is no necessary connexion between spin and statistics.
This paper describes, explains, and compares the wavefunction in three related quantum frameworks: nonrelativistic quantum mechanics (NRQM), relativistic quantum mechanics (RQM), and the Scretching–Schrödinger Equation (SSE). In NRQM, the wavefunction is treated as a complex probability amplitude whose squared modulus gives the probability density for finding a particle in a specified region of space. For bound atomic systems, it is governed by the Schrödinger equation and is commonly labeled by the quantum numbers nnn, lll, mlm_lml, and msm_sms, which specify the principal energy level, orbital angular momentum, magnetic projection, and spin projection. In RQM, the wavefunction must be compatible with special relativity. For spin-0 particles, this leads to the Klein–Gordon scalar wavefunction, while for spin-12\frac{1}{2}21 particles, the relativistic wavefunction becomes the four-component Dirac spinor. These relativistic forms preserve probability-amplitude interpretation but require Lorentz-compatible energy–momentum relations, relativistic covariance, and, in the Dirac case, explicit treatment of spin, antimatter solutions, and relativistic current conservation. The SSE framework extends the ordinary Schrödinger wavefunction by imposing a deterministic spectroscopic closure condition through the Scretching Quantum Chain (SQC) and Maxwell–Scretching Chain (MSC). In this formulation, the wavefunction retains the standard quantum-number structure of atomic quantum mechanics, including radial, angular, and spin components, but it is additionally constrained to reproduce measurable spectroscopic quantities such as oscillator strength, transition dipole moment, Einstein AAA and BBB coefficients, transition frequency, absorbance-linked optical quantities, and the SQC invariant Kν=2πe2ε0mec3.K_\nu=\frac{2\pi e^2}{\varepsilon_0 m_e c^3}.Kν=ε0mec32πe2. Thus, the SSE wavefunction is not merely a probability amplitude. It is a probability amplitude constrained by experimentally anchored spectroscopic closure. The referenced SSE formalism defines the SSE Hamiltonian as H^SSE=H^0+V^SQC,\hat{H}_{\mathrm{SSE}}=\hat{H}_0+\hat{V}_{\mathrm{SQC}},H^SSE=H^0+V^SQC, where H^0\hat{H}_0H^0 is the ordinary quantum Hamiltonian and V^SQC\hat{V}_{\mathrm{SQC}}V^SQC is the spectroscopic constraint operator. The SSE wavefunction may therefore be written schematically as ΨSSE(r,θ,ϕ,s)=RSSE(r) Ylm(θ,ϕ) χs [fconstraint],\Psi_{\mathrm{SSE}}(r,\theta,\phi,s) = R_{\mathrm{SSE}}(r)\,Y_l^m(\theta,\phi)\,\chi_s\,[f_{\mathrm{constraint}}],ΨSSE(r,θ,ϕ,s)=RSSE(r)Ylm(θ,ϕ)χs[fconstraint], where RSSE(r)R_{\mathrm{SSE}}(r)RSSE(r) is the radial component modified or constrained by SSE closure, Ylm(θ,ϕ)Y_l^m(\theta,\phi)Ylm(θ,ϕ) is the spherical harmonic angular component, χs\chi_sχs is the spin function, and [fconstraint][f_{\mathrm{constraint}}][fconstraint] represents the SQC/MSC spectroscopic constraint that forces agreement between the wavefunction matrix elements and measured or computed optical transition quantities. The central comparison developed in the paper is that NRQM describes the wavefunction as a probability amplitude, RQM generalizes the wavefunction into Lorentz-compatible scalar or spinor fields, and SSE adds a deterministic spectroscopic layer to the quantum wavefunction. In this sense, SSE does not discard Schrödinger or Dirac quantum mechanics; instead, it proposes an additional closure architecture in which allowed wavefunctions must satisfy both the usual quantum dynamical equation and the SQC/MSC spectroscopic consistency conditions.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).