Materials Science and Nanotechnology

Review Article - Materials Science and Nanotechnology (2018) Volume 2, Issue 1

Unitary relativistic quantum theory.

Eliade Stefanescu*

Advanced Studies in Physics Centre of the Romanian Academy, Academy of Romanian Scientists Bucharest, Romania

Corresponding Author:
Eliade Stefanescu
Advanced Studies in Physics Centre of the Romanian, Academy
Academy of Romanian Scientists Bucharest, Romania
Tel: +4021-3188106/3138
E-mail: [email protected]

Accepted date: February 05, 2018

Citation: Stefanescu E. Unitary relativistic quantum theory. Mater Sci Nanotechnol. 2018;2(1):17-28.

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Abstract

We consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrödinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation.

Keywords

Quantum particle, Wave packet, Group velocity, Lagrangian, Hamiltonian, Maxwell equations, Spin, metric tensor, Covariant derivation.

Introduction

We perceive the physical world as a collection of objects, in a three-dimensional space of coordinates defined by a position vector equation, and in time, defined by a scalar t [1-5]. For a matter object we define an inertial property called mass, M0 and a dynamic quantity as the product of the mass with the velocity, equation, called momentum, equation. We also define a conservative quantity, called energy,

equation

as a sum of the kinetic energy equation, which depends on momentum, and the potential energy equation, which depends on the coordinates [6-8]. From the conservation condition:

equation

we obtain the dynamic equations called Hamilton equations,

equation

while the energy function of coordinates and momenta,

equation

is called Hamiltonian. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time, with a single quantum constant equation [9-11].

equation

In this case, one can define a momentum operator

equation

and a Hamiltonian operator,

equation

With these operators one obtains eigenvalue equations, for momentum,

equation

and for energy, as a Schrödinger equation:

equation

However, when the group velocities are calculated for the wave packets (1.5), which with (1.1) are of the form

equation

we obtain an erroneous equation, contradictory to the corresponding Hamilton equation (1.3):

equation

- a minus sign is missing [12-14]. We get back the minus sign only if instead the Hamiltonian equation we consider the Lagrangian

equation

with group velocities in agreement with the Hamilton equations (1.3):

equation

However, this description is still unrealistic, having an infinite spectrum of waves, as a function of the wave velocity equation. A finite spectrum is obtained when the relativistic Lagrangian equation

equation

From (1.15), we notice that the invariance of the space- time interval of the Theory of Relativity is equivalent to the invariance of the time dependent phase variation of a quantum particle, and define a Relativistic Quantum Principle: The scalar tim-dependent phase variation of a quantum particle wave is an invariant for an arbitrary change of coordinates. On this basis, in section 2 we obtain the relativistic transform of the space- time coordinates, and the relativistic dynamics of the particle waves. In section 3, we consider a quantum particle in a field described by a scalar potential conjugated to time, and a vector potential conjugated to coordinates. We obtain the Lagrange equation as the velocity of the particle waves, the Lorentz force, and the Maxwell equations. In section 4, based on the Relativistic Quantum Principle, we obtain the dynamics of a quantum particle in electromagnetic field. In section 5, from the invariance of the time dependent phase of a quantum particle at an arbitrary change of coordinates, we obtain the relativistic transform of the electromagnetic field. In section 6, we obtain a relativistic equation for the quantum particle waves, show that the solution of the Schrödinger equation is only the amplitude of the wave function of a quantum particle, which also includes a rapidly varying factor depending on the particle rest mass, and derive the spin of a quantum particle wave. In section 7, we consider the inversion of two particles as a double rotation of these particles, and obtain the spin-statistic relation. In section 8, we consider a coordinate deformation according to the Theory of General Relativity, and obtain the dynamics of the particle waves in gravitational field. In section 9, we consider the particle waves as a distribution of matter described by a density and a velocity field, and find that these waves are perpendicular to the geodesic tracks. We derive an invariant for the matter density of a quantum particle, and an equation of conservation of this matter. Section 10 is for conclusions.

Relativistic Kinematics and Dynamics

A realistic particle has a finite spectrum as a function of the wave propagation velocity. A finite spectrum is obtained for a relativistic Lagrangian. According to (1.15), the invariance of the scalar time dependent phase variation equation of a quantum particle wave function is equivalent to the invariance of the space-time interval ds. For such a particle, we obtain the momentum,

equation

as functions of the particle velocity equation and the cut-off velocity c. The invariance of the space-time interval means that a change of coordinates is in fact a rotation of the space-time coordinates (Figure 1).

Figure 1: The rotation of the space-time system of coordinates for a velocityequation of the system equation over the system equation.

By a well-known calculation, the relativistic transform of the coordinate intervals is obtained for the quantum particle waves. Thus, from the coordinate transform,

equation

with the expressions of the rotation angle trigonometric functions,

equation

for the coordinates of a quantum particle wave, we obtain the transform:

equation

which, in this case does not refer to the coordinates of some classical particle as in the conventional Theory of Relativity, but to the coordinates of the quantum particle waves.

Electromagnetic Field

When a particle with a charge e is placed in a field of a vector potential equation conjugated to the coordinates, and a scalar potential equation conjugated to time, for the two wave packets of this particle

equation

a time dependent phase variation arises, with terms proportional to the coordinate variations, and the time variation,

equation

with a mechanical component depending on the particle mass and velocity, and an electromagnetic component as the product of the particle charge with the vector potential. For a quantum particle in a field we consider again the relativistic quantum principle: The time dependent phase variation of a quantum particle is the same in any system of coordinates.

We are in agreement with the Aharonov-Bohm effect [15]: the time-dependent phase of a quantum particle equation

equation

in a magnetic field includes a term proportional to the space integral of the vector potential of this field:

equation

This effect has been experimentally put out into evidence (Figure 2).

Figure 2: Experimental evidence of the Aharonov-Bohm effect: between the two electron beams, a phase difference arises due to the vector potential equation [4].

Besides this variation, we consider a time-dependent phase variation with a term proportional to the scalar potential and time, and the invariance of this phase. From equation (3.3), we obtain the electric force:

equation

while from a particle wave velocity in the momentum space we obtain the Lagrange equation:

equation

Taking into account that the curl of the gradient is null, from (3.8) with (3.9) we obtain the Faraday-Maxwell law of the electromagnetic induction:

equation

Taking into account that the divergence of the curl is null, from (3.9) we obtain the Gauss-Maxwell law for the magnetic induction flow:

equation

With the Gauge condition

equation

we obtain the Gauss-Maxwell law for the electric field flow:

equation

with the Laplacian of the scalar potential as source of this divergence, considered as a ratio of the charge density equation and the dimensional constant ε0 called electric permittivity. Considering a current density under the action of the electric field,

equation

with the vector formula

equation

we obtain the time derivative of the electric field of the form:

equation

For the relation (3.10) of the two vector fields, asserting that a time variation of a magnetic field determines a curl of the electric field, we consider a symmetric relation, namely that a time variation of the electric field determines a curl of the magnetic field:

equation

From (3.16) and (3.17), the Ampère-Maxwell law of the magnetic circuit is obtained. In this way we obtain the whole system of Maxwell equations:

equation

This force acts on the particle wave functions (3.1), with the time-dependent phase coefficient:

equation

as a function of the two potentials of the electromagnetic field:

equation

It is interesting that by the hypothetical relation (3.17), we obtain a field propagating in waves with a velocity equation in agreement with the electromagnetic theory. For the physical consistency, this velocity takes the limit value of the velocity of the quantum particle waves (Figure 3):

Figure 3: Wave packet of a quantum particle, with a limit velocity c equal to the electromagnetic field velocity.

equation

Otherwise, an electromagnetic field not interacting with a quantum particle, or a quantum particle not interacting with any electromagnetic field could exist, which is contradictory to our basic hypothesis of the particle-field interaction.

Dynamics of a Quantum Particle in Electromagnetic Field

We consider the space-time interval,

equation

and the phase variation (3.2),

equation

as functions of the coordinate four-vector

equation

and the field four-vector

equation

For the wave velocity in the space of the momentum (3.3), from (3.1) we obtain a wave/group velocity of the form of the Lagrange equation:

equation

which means that the time-dependent phase of a quantum particle wave function is of the form of the action of the Lagrangian (3.20),

equation

equation

for the interaction of a particle wave with an electromagnetic field.

Relativistic Transform of a Field Interacting with a Quantum Particle

For a coordinate four-vector transform,

equation

with the transformation matrix,

equation

in the least action equation (4.7), we obtain an invariant including a mechanical term and an electromagnetic term:

equation

Since the mechanical term is invariant to any change of coordinates,

equation

the electromagnetic term must be also an invariant,

equation

On this basis, we find the transform for a change of coordinates of a field interacting with a quantum particle

equation

Particle Wave Function and Spin

From the Lagrange equation obtained as the group velocity (3.5) with the momentum expression (3.3),

equation

we obtain the Hamilton equation

equation

At the same time, with the definition expression (6.2), the wave packets (3.1) take a form

equation

as solutions of a wave equation

equation

which, compared to the Schrödinger equation includes a term depending on the velocity.

On the other hand, from (3.1) we notice that, in a classical approximation,

equation

the Schrödinger wave function equation, we usually use in our studies, in fact is not the particle wave function, bat only an amplitude of a wave function including a rapidly varying factor,

equation

We consider an electromagnetic field with a constant scalar potential for a stationary electric field, and a time-dependent vector potential for a coherent radiation field. From equation (6.2) with (3.3) and (3.20), we notice that although the vector potential equation depends on time, the Hamiltonian does not:

equation

With this Hamiltonian we, obtain the total energy as a function of the mechanical energy and the potential energy in electric field:

equation

With the mechanical energy as a function of the canonical momentum (3.3),

equation

we obtain the canonical form of the Hamiltonian

equation

According to Dirac’s well-known spin theory, we consider the Schrödinger equation

equation

with the Hamiltonian

equation

which depends on Dirac’s spin operators

equation

as functions of Pauli’s spin operators,

equation

These operators satisfy the commutation relations:

equation

With the notation

equation

the Schrödinger equation (6.13) with the Hamiltonian (6.14) takes a form:

equation

with a wave function which can be written as a vector with two components, or a vector with four components,

equation

We obtain the two-dimensional Schrödinger-Dirac equation,

equation

with two coupled Schrödinger-Dirac equations for the two components equation and equation

equation

By eliminating the coupling terms, the two Schrödinger- Dirac equations take non-linear forms, which, in a non- relativistic approximation, small velocity, small electric potential, become linear:

equation

In the non-relativistic approximation, we get Schrödinger- Dirac equations

equation

The Hamiltonian of these equations includes a kinetic term depending on the momentum and the Pauli spin operators,

equation

Thus, in the kinetic term of the Hamiltonian, besides the mechanical term, proportional to the square of the mechanical momentum, we obtain a magnetic potential, proportional to the magnetic field, as of a proper rotation, called spin:

equation

We obtain Schrödinger-Dirac equations with a Hamiltonian including the kinetic energy, the electric potential energy, and a potential energy in magnetic field, due to the particle spin:

equation

in the direction of the magnetic field. Taking into account the commutation relation

equation

of the Hamiltonian

equation

with the total angular momentum, which includes the orbital angular momentum l3 and the spin angular momentum s3, we obtain a relation between the spin angular momentum and the orbital angular momentum which is known:

equation

These equations have a solution of the form

s3 = sα1α2 (6.40)

with the coefficient

equation

Thus, we obtain the spin angular momentum in the direction of the magnetic field

equation

From (6.34) and (6.41) we obtain the gyromagnetic ratio

equation

In this way, the spin is obtained from the relativistic quantum principle, in the framework of a unitary relativistic quantum theory.

Spin-Statistic Relation

We consider a system of two particles, in the states equation and equation, with the position vectors equation and equation (Figure 4).

Figure 4: Two-particle system – a particle inversion equivalent to a double rotation with the angle π.

For a two particle wave function

equation

On the other hand, we notice that an inversion is equivalent with a double rotation with the angle π (Figure 4),

equation

For a wave function rotation with a differential angle equation,

equation

we find a rotation operator depending on the spin operator equation,

equation

which, for a rotation with an arbitrary angle equation generates equation

For α =π we obtain

equation

With this expression, from (7.4) and (7.5) we obtain the inversion eigenvalues as a function of spin,

equation

Quantum Particle in Gravitational Field

We consider the wave functions (1.16) in a system of curvilinear coordinates equation,

equation

with the time-space differential interval ds,

equation

and use Dirac’s formalism of General Theory of Relativity [16]. In (8.1), we used the notation:

equation

For a non-relativistic case, small particles compared with the non-uniformities of any gravitational field, small velocities, we consider:

equation

In this way, for a quantum particle we obtain wave packets with linear phases in space and time,

equation

For this wave packet we obtain a self-consistent expression of the wave velocity,

equation

and the acceleration of a particle on a geodesic as a function of the Christoffel symbol equation of the second kind,

equation

We introduce the Christoffel symbol of the first kind, and consider the expression of this symbol as a function of the metric tensor derivatives:

equation

For the non-relativistic case, considered here, we neglect the spatial coordinate derivatives compared to the time derivative, and take into account a stationary state, which means that the derivatives with time disappear. We obtain a particle accelerations proportional to the derivatives of the metric tensor element g00:

equation

which means that this matrix element behaves as a potential. Considering the Newtonian potential V in this matric element,

equation

With the Schwarzschild solution for the contravariant metric elements conjugated to the spatial coordinates, we obtain the three components of the acceleration,

equation

We notice that, for a rather small distance r, we obtain a correction of the Newtonian force, increasing the gravitational attraction.

Quantum Particle as a Distribution of Matter

We notice that, in this framework, a particle is conceived as a distribution of matter with a matter velocity field:

equation

From the relativistic quantum principle of the space-time interval invariance,

equation

we obtain that the covariant derivative of velocity vector is perpendicular to this vector,

equation

If, besides the acceleration on a geodesic track, we consider an additional component equation,

equation

we find that this component is perpendicular to the velocity vector:

equation

This means that any additional acceleration to the acceleration on a geodesic is perpendicular to the wave velocity, which is in agreement with wave propagation (Figure 5).

Figure 5: Plane wave – any additional particle acceleration Aμ to the acceleration on a geodesic is perpendicular to the velocity x.μ.

In particle propagation on a geodesic, other matter propagations are allowed only in perpendicular directions on the particle velocity.

We consider a quantum particle as a normalized matter distribution – a normalized integral of the matter density, in Cartesian coordinates,

equation

on a space-time volume equation, in two systems of coordinates.

equation

and remark that the elements of the Jacobian,

equation

are also elements of a tensor transform, which, for the metric tensor is:

equation

Calculating the determinants,

equation

we obtain the Jacobian as a function of the determinants of the metric tensor in the two systems of coordinates,

equation

Introducing this expression of the Jacobian in the integral equation (9.9), we obtain an invariant for the matter density:

equation

We consider a matter flow density equation, as the product of the matter density with the velocity

equation

and the matter conservation as a null covariant divergence

equation

From the general expression of the Christoffel symbol of the second kind as a function of the metric tensor,

equation

From (9.18) and (9.19), we obtain the Christoffel symbol of the second kind in the expression (9.16),

equation

which takes the form of a divergence of the product of the matter flow density with the square root of the metric determinant,

equation

Integrating this expression on a volume V of spatial coordinates,

equation

and separating the time derivative from the coordinate derivatives, we find a conservation law, as a matter variation in a volume V by the flow of this matter through the surface ΣV of the volume V:

equation

In the nonrelativistic case, equation, weak gravitational field, we obtain

equation

while the conservation equation (9.23) takes a more explicit form, of density and matter flow,

equation

Conclusion

We discovered that a packet of Schrödinger wave functions, representing a quantum particle, is not in agreement with the Hamilton equations which describe the dynamics of such a particles – such an agreement is obtained only when instead of the Hamiltonian we consider the Lagrangian function. For a realistic particle, which must have a finite spectrum, we considered the relativistic Lagrangian, with a cut-off velocity c. We defined a Relativistic Quantum Principle: The time-dependent phase of a quantum particle is invariant to any change of coordinates. We obtained a wave equation for a quantum particle, depending on velocity, while the conventional Schrödinger equation describes only the amplitude of the particle wave-function - it includes a rapidly varying factor, with a phase proportional to the particle rest mass. On this basis, the relativistic kinematics and dynamics, the electromagnetic field equations, the particle spin, and the Schwarzschild-Newtonian dynamics in a gravitational field have been obtained. A quantum particle has been described by a wave function representing a distribution of conservative matter in motion, according to the General Theory of Relativity.

References