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

arXiv:2207.12825

A Flow Equation Approach Striving Towards an Energy-Separating Hamiltonian Unitary Equivalent to the Dirac Hamiltonian with Coupling to Electromagnetic Fields

Nils Schopohl, Nadine Suzan Cetin

Published Jul 26, 2022• 0 citations
arXivHTMLPDF

Abstract

The Dirac Hamiltonian $H^{\left(D\right)}$ for relativistic charged fermions minimally coupled to (possibly time-dependent) electromagnetic fields is transformed with a purpose-built flow equation method, so that the result of that transformation is unitary equivalent to $H^{\left(D\right)}$ and granted to strive towards a limiting value $H^{\left(NW\right)}$ commuting with the Dirac $β$-matrix. Upon expansion of $H^{\left(NW\right)}$ to order $\frac{v^2}{c^2}$ the nonrelativistic Hamiltonian $H^{\left(SP\right)}$ of Schrödinger-Pauli quantum mechanics emerges as the leading order term adding to the rest energy $mc^2$. All the relativistic corrections to $H^{\left(SP\right)}$ are explicitly taken into account in the guise of a Magnus type series expansion, the series coefficients generated to order $\left(\frac{v^{2}}{c^{2}}\right)^{n}$ for $n\geq2$ comprising partial sums of iterated commutators only. In the special case of static fields the equivalence of the flow equation method with the well known energy-separating unitary transformation of Eriksen is established on the basis of an exact solution of a reverse flow equation transforming the $β$-matrix into the energy-sign operator associated with $H^{\left(D\right)}$. That way the identity $H^{\left(NW\right)}=β\sqrt{H^{\left(NW\right)}H^{\left(NW\right)}}$ is established implying $H^{\left(NW\right)}$ being determined unambiguously.

DOI
10.48550/arxiv.2207.12825

Related papers

Hamiltonian Flow Equations for a Dirac Particle in an External Potential

Alexander B. Bylev, Hans J. Pirner

We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width

•Jan 1, 1998•12 citations

Dirac Theory as a Relativistic Flow

Asher Yahalom A

In previous papers we have shown how Schrödinger's equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields. This program was applied also to Pauli's equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. Here we repeat the analysis for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.

arXiv:2408.10216•Jul 22, 2024•0 citations

Gauge-invariant non-relativistic limit of an electron in a time-dependent electromagnetic field

Donald H. Kobe, Kuo‐Ho Yang

The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.

Journal of Physics A Mathematical and General•Oct 1, 1980•34 citations

Dirac Theory as Relativistic Flow Theory

Asher Yahalom A

In previous papers we have shown how Schr\"{o}dinger's equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields. This program was applied also to Pauli's equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. Here we repeat the analysis for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.

•Jul 23, 2024•0 citations

Exact correspondence between classical and Dirac-Pauli spinors in the weak-field limit of static and homogeneous electromagnetic fields

Dah-Wei Chiou, Tsung-Wei Chen

It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of $1/c$ in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.

arXiv:1504.03453•Apr 14, 2015•1 citation

Exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian by Kutzelnigg's method

Dah-Wei Chiou, Tsung-Wei Chen

We apply Kutzelnigg's method for the Foldy-Wouthuysen (FW) transformation of the Dirac-Pauli Hamiltonian. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, the long-held speculation is rigorously proven, by mathematical induction on the orders of $1/c$ in the power series, that the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.

•May 18, 2014•0 citations
The Dirac Hamiltonian H(D)H^{\left(D\right)}H(D) for relativistic charged fermions minimally coupled to (possibly time-dependent) electromagnetic fields is transformed with a purpose-built flow equation method, so that the result of that transformation is unitary equivalent to H(D)H^{\left(D\right)}H(D) and granted to strive towards a limiting value H(NW)H^{\left(NW\right)}H(NW) commuting with the Dirac βββ-matrix. Upon expansion of H(NW)H^{\left(NW\right)}H(NW) to order v2c2\frac{v^2}{c^2}c2v2​ the nonrelativistic Hamiltonian H(SP)H^{\left(SP\right)}H(SP) of Schrödinger-Pauli quantum mechanics emerges as the leading order term adding to the rest energy mc2mc^2mc2. All the relativistic corrections to H(SP)H^{\left(SP\right)}H(SP) are explicitly taken into account in the guise of a Magnus type series expansion, the series coefficients generated to order (v2c2)n\left(\frac{v^{2}}{c^{2}}\right)^{n}(c2v2​)n for n≥2n\geq2n≥2 comprising partial sums of iterated commutators only. In the special case of static fields the equivalence of the flow equation method with the well known energy-separating unitary transformation of Eriksen is established on the basis of an exact solution of a reverse flow equation transforming the βββ-matrix into the energy-sign operator associated with H(D)H^{\left(D\right)}H(D). That way the identity H(NW)=βH(NW)H(NW)H^{\left(NW\right)}=β\sqrt{H^{\left(NW\right)}H^{\left(NW\right)}}H(NW)=βH(NW)H(NW)​ is established implying H(NW)H^{\left(NW\right)}H(NW) being determined unambiguously.
We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width
In previous papers we have shown how Schrödinger's equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields. This program was applied also to Pauli's equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. Here we repeat the analysis for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.
The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.
In previous papers we have shown how Schr\"{o}dinger's equation which includes an electromagnetic field interaction can be deduced from a fluid dynamical Lagrangian of a charged potential flow that interacts with an electromagnetic field. The quantum behaviour was derived from Fisher information terms which were added to the classical Lagrangian. It was thus shown that a quantum mechanical system is drived by information and not only electromagnetic fields. This program was applied also to Pauli's equations by removing the restriction of potential flow and using the Clebsch formalism. Although the analysis was quite successful there were still terms that did not admit interpretation, some of them can be easily traced to the relativistic Dirac theory. Here we repeat the analysis for a relativistic flow, pointing to a new approach for deriving relativistic quantum mechanics.
It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of 1/c1/c1/c in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.
We apply Kutzelnigg's method for the Foldy-Wouthuysen (FW) transformation of the Dirac-Pauli Hamiltonian. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, the long-held speculation is rigorously proven, by mathematical induction on the orders of 1/c1/c1/c in the power series, that the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.