Abstract Quantum mechanics describes multi-electronic systems by using probabilistic theories. All of these theories provide a probabilistic conception of quantum mechanics, with an astonishing accuracy (for instance the Helium atom ground state is computed with more than 20 digits of accuracy) but unfortunately seems to be a bit “disconnected” with macro-scale physics. Indeed, the real observable world is for us a 3D+time Minkowski space but quantum mechanics is expressed in multi-dimensional Hilbert space: which is not intuitively physical but seems to be purely mathematical. In this article we try to construct a relativistic equation that gives an approximately correct description of atomic physics in the spacetime (i.e the “real space”) for 2 electrons systems: i.e a 2-electrons Dirac equation. The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a matter-wave. For this we start from the Hartree equation, which gives already a spacetime description of each electron wave function. Then, we switch to Hartree-Fock representation, which gives birth to the exchange potential, that can be also expressed in spacetime. By extending the concept of exchange potential to the relativistic 4-exchange: we obtain a relativistic 2-electrons spacetime Dirac-Hartree-Fock (STDHF) equation (i.e in Minkowski space). That equation takes into account relativistic effects but not correlation effects, which is the difficulty of this study. Then we will show the equivalence of the 2-electrons STDHF equation with the Hartree-Fock equation in non-relativistic, stationary and scalar regime. Electron correlation issues will be discussed as well.
Abstract Quantum mechanics describes multi-electronic systems by using probabilistic theories. All of these theories provide a probabilistic conception of quantum mechanics, with an astonishing accuracy (for instance the Helium atom ground state is computed with more than 20 digits of accuracy) but unfortunately seems to be a bit “disconnected” with macro-scale physics. Indeed, the real observable world is for us a 3D+time Minkowski space but quantum mechanics is expressed in multi-dimensional Hilbert space: which is not intuitively physical but seems to be purely mathematical. In this article we try to construct a relativistic equation that gives an approximately correct description of atomic physics in the spacetime (i.e the “real space”) for 2 electrons systems: i.e a 2-electrons Dirac equation. The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a matter-wave. For this we start from the Hartree equation, which gives already a spacetime description of each electron wave function. Then, we switch to Hartree-Fock representation, which gives birth to the exchange potential, that can be also expressed in spacetime. By extending the concept of exchange potential to the relativistic 4-exchange: we obtain a relativistic 2-electrons spacetime Dirac-Hartree-Fock (STDHF) equation (i.e in Minkowski space). That equation takes into account relativistic effects but not correlation effects, which is the difficulty of this study. Then we will show the equivalence of the 2-electrons STDHF equation with the Hartree-Fock equation in non-relativistic, stationary and scalar regime. Electron correlation issues will be discussed as well.
Quantum mechanics describes multi-electronic systems by using probabilistic theories (non-relativistic many-body Hamiltonian, Bethe-Salpeter equation, Breit equation, density functional theory and so on).All of these theories are approximations and provide a probabilistic conception of quantum mechanics, which prevents it to be compatible with general relativity (or "macro scale" physics).Indeed, the real observable world (or Minkowski space) is for us a 3D+time space but quantum mechanics is expressed in multi-dimensional Hilbert space, which is not physical but purely mathematical.In this article we try to make a relativistic equation that gives a correct description of atomic physics in the real space for N ≥ 2 electrons systems: i.e a N-electrons Dirac equation.The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a matter-wave.For this we start from the Hartree equation, which gives already a 3D space description of each electron wave function.Then we jump into Hartree-Fock theory, that takes into account the electron exchange and can be also expressed in 3D, thanks to Slater-Condon rules.By extending the concept of exchange potential to a relativistic 4-exchange potential, and doing the same for the electron-electron repulsion, kinetic and rest mass energy: we obtain a relativistic N electrons Dirac equation, in 3D+time space that takes into account relativistic and correlation effects.The non-relativistic, stationary, uncorrelated and scalar approximation will show the reduction of the N-electrons Dirac equation into Hartree-Fock equation.
Quantum mechanics describes multi-electronic system by using non-relativistic probabilistic theories (non-relativistic many-body Hamiltonian, Bethe-Salpeter equation, Breit equation, density functional theory and so on).All of these theories are approximations and provide a probabilistic conception of quantum mechanics, which prevents it to be compatible with general relativity (or "macro scale" physics).Indeed, the real observable world (or Minkowski space) is for us a 3D+time space but quantum mechanics is expressed in multi-dimensional Hilbert space, which is not physical but purely mathematical.In this article we try to make a relativistic equation that gives a correct description of atomic physics in the real three-dimensional space for N ≥ 2 electrons systems: i.e a N-electrons Dirac equation.The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a mini oscillating cloud or matter-wave.For this we start from the Hartree equation, which gives already a 3D space description of each electron wave function.Then we jump into Hartree-Fock theory, that takes into account the electron exchange and can be also expressed in 3D, thanks to Slater-Condon rules.By extending the concept of exchange potential to a relativistic 4-exchange potential, and doing the same for the electron-electron repulsion potential, we have obtained a relativistic N electrons Dirac equation, in 3D+time space, including new relativistic interactions.The non-relativistic equivalence to Hartree-Fock equations will be demonstrated, which shows that the equation is already correct to describe light atoms.
A relativistic configuration space (many-time) presentation of quantum electrodynamics is developed, which, being equivalent to the Tomonaga-Schwinger formalism, may be expected to have the advantage of its direct applicability to bound-state problems of multi-electron systems. The main purpose of this paper is therefore to get an evidently relativistically invariant description of the interactions between individual electrons, in such a way, however, as to avoid those divergence difficulties which may be accounted for by the use of electron plane waves for building up the actual state of the system. The order of the presentation is as follows: a general relativistically invariant proof of equivalence of the Tomonaga-Schwinger formalism and the many-time formalism of Dirac, Fock, and Podolski given in Sec. I will provide a necessary starting point for further generalization in the form of Eq. (53). This equation, when eventually solve in Sec. II, gives rise to a formal extension of the multi-electron wave function concept into regions for which the latter remained hitherto undefined. In Sec. III a method of elimination of virtual processes is outlined and invariant expressions for the multi-electron interactions are derived. All explicit derivations in this paper are carried out to terms of the order of e2. An example, a relativistic two-electron wave equation, is given. This equation accounts for the Coulomb and Breit interactions.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed derivatives on coordinates. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a birth or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Schrödinger equation the wave function is represented by a single complex quantity, in the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a birth or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics.