Abstract

The spin of electrons is studied (Spintronics) and applied to, for example, quantum computing. In this article, utilizing UMFT (Universal Mathematical Field Theory), we establish both Spin-Electromagnetics (Spin-EM) from the Coulomb’s law and the spin of electric source, and Spin-gravity from the Newton’s law and the spin of the gravity-g-charge (mass). Spin-EM predicts several new effects including the Spin-Lorentz-type force, and Spin-Lagrangian-Lorentz-type force. When experimentally confirmed, the Spin-force would be a NEW FORCE. The UMFT has been established and utilized to derive the Maxwell electromagnetics (Maxwell-EM) and Maxwell-type gravity (Gravito-EM). UMFT shows that mathematical identities lead to physical dualities, such as the duality between Maxwell-EM, Gravito-EM, Spin-EM, and Spin-gravity. The well-known-concepts, effects, and phenomena of Maxwell-EM may be directly converted to that of Spin-EM and Spin-Gravity. The Maxwell-EM, Gravito-EM, Spin-EM, and Spin-gravity are all derived from UMFT and thus, they are dual to each other, and can be unified in the frame of UMFT.

1. Introduction

Historically, Maxwell equations were established based on series experiments, except the introduction of the displacement current. The Coulomb’s law states that a rest e-charge induces a static vector electric field E. The created electric field is determined by only one quantity, e-charge. A heuristic phenomenon is that when an e-charge uniformly moves, it creates not only the electric field but a magnetic field. The magnetic field is determined by three quantities, e-charge, electric field and velocity. In classical EM theory, there are two main states of motion of an e-charge: (1) static and (2) uniform motion.

Now we ask a question: Does the spin of a charge create new field/force.

In Section 2, we review Universal Mathematical Field Theory (UMFT). To mathematically derive field equations of a spin-charge, we review Universal Mathematical Field Theory first.

In Section 3, Spin-Electromagnetics (Spin-EM) is established by utilizing UMFT, Coulomb’s law, e-charge and spin of the e-charge. Spin-EM provides classical foundations of variety phenomena of spintronics, and predicts new phenomena due to the spin of an e-charge.

In Section 4, Spin-Gravity is established by utilizing UMFT, Newton’s law, g-charge (gravitational mass) and spin of the g-charge. Spin-gravity predicts new phenomena.

Table 1 shows the comparison of the fields created by an e-charge and a g-charge, respectively.

Spin-EM predicts several new effects, the Spin-Lorentz-type force, and Spin-Lagrangian-Lorentz-type force. When experimentally confirmed, the Spin-forces would be NEW FORCES.

Spin-EM may be applied to different areas, for example, plasma/Tokamak.

The Maxwell-EM, Spin-EM, and Spin-gravity are all derived from UMFT and thus, they are dual to each other, and can be unified in the frame of UMFT.

2. Universal Mathematical Field Theory (UMFT)

There are three states of motion of a charge: (1) stationary; (2) uniform motion; (3) Spin. Utilizing UMFT, we have shown that the velocity of e-charge and g-charge produces Maxwell-EM ¹ and Gravito-EM ², respectively. UMFT provides the mathematic origin of dualities between different physic fields derived from it, such as the duality between Maxwell-EM and Gravito-EM.

In this article, we establish the Spin-EM and Spin-gravity. Duality is a powerful tool to find intrinsic similarities between apparently different phenomena, and predict new effects. “It turns out that most of the important concepts and theories of physics can be unified and understood by their common attribute of duality” (Damian P Hampshire).

Note: Spin-EM needs to be further tested experimentally.

To establish UMFT, we utilize mathematical identities that connecting divergences of either a vector or an axial vector with curls of induced axial vectors. For this aim, the following vector analysis identity is the most significant foundation of UMFT,

(2.1)

(2.2)

Equation (2.1) indicates that the combination of gradient and divergence of two arbitrary vectors including axial vector, and T, induces inevitably an axial vector . One of two terms, either or , represents fundamental inverse-square laws, and introduce a “charge”. Now there are three quantities, , T and “charge” in Equation (2.1).

It is useful to write Equation (2.1) in a different but equivalent form. By using another mathematical identity,

Equation (2.1) can be rewritten as an identity,

(2.3)

Based on mathematical vector identities, we establish self-consistent UMFT (Equation (2.1) to Equation (2.3) that universally describes physical fields induced by the different motions of a source.

We propose UMFT for spin.

(1). Maxwell-type equations.

The combination of the inverse-square law and the spin S of charges must induce axial vector fields. Let

(2.4)

Substituting Equation (2.4) into Equation (2.1), Equation (2.2) and Equation (2.3), respectively, we obtain Maxwell-type equations,

(2.5)

(2.6)

(2.7)

Note: physical quantities R representing are not limited to spin.

(2). Poynting theorem.

Let both “R” and “T” are fields, for example, , Equation (2.2) gives “Poynting theorem”,

(2.8)

Equation (2.8) validates Equation (2.2).

In Section 3, we utilize Equation (2.5) to Equation (2.8) to derive the Maxwell-type equations for the fields induced by the spin of e-charges, referred it as Spin-EM.

3. Spin-EM

UMFT is the math theory and thus can be applied to suitable physical situations. Applying UMFT, we have re-derived Maxwell equations, which justifies UMFT and shows that the experiments-based Maxwell equations have their mathematical origin. In Section 3, we establish Spin-Electromagnetics (referred as Spin-EM). The Spin-EM is powerful and fruitful in the perspective of fundamental physics: (1) predicts new forces, the Lorentz-type-spin-force and Lagrangian-Lorentz-type-spin-force, which are spin related force; (2) UMFT shows that mathematical identities lead to physical dualities including duality between Maxwell-EM and Spin-EM.

By applying UMFT to the combination of spin and EM, we establish Spin-EM below.

First let’s consider a spin-e-charge, such as an electron, characterized by electric charge and spin. The e-charge produces a Coulomb field, .

We propose that the spin of the e-charge induces axial vector fields:

(1). a static spin-e-charge with spin produces an electric field and a spin-magnetic field,

(3.1a)

(2) a spin-e-charge moving with velocity v produces an electric field, a magnetic field, a spin-magnetic field (polar vector), and a spin-electric field (Axial vector), where,

(3.1b)

(3.2)

Subscript “s” indicates the quantity related to spin. The and are type-2 dual to the B and E fields, respectively. The definitions, Equation (3.1) and Equation (3.2), predict a phenomenon: by interacting with the electric field E and the magnetic field B, the spin of e-charge produces a spin-magnetic field and a spin-electric field, respectively.

Note: (1) the definitions of and are conceptually different from that of Rashba-induced-Spin-EM.

(2) we do not consider the higher-order-terms of and.

Spin-EM can be derived from the UMFT formular, Equation (2.5) to Equation (2.8). To applying Equation (2.5) to Equation (2.8) to the spin-e-charge, let; is the spin of the spin-e-charge.

*) Spin-EM equations

Let (the electric field) and (the magnetic field), respectively. Then substituting Equation (3.1) and Equation (3.2) into Equation (2.5) to Equation (2.7) respectively, we obtain:

(3.3a)

(3.3b)

(3.4)

(3.5a)

(3.5b)

(3.6)

Note: it is interesting that in Maxwell-EM, we have .

Equation (3.3) and Equation (3.4) are dual to Equation (3.5) and Equation (3.6), respectively, under the transformation:

Substituting Maxwell equations into Equation (3.3) to Equation (3.6), we have Spin-EM equations:

(3.7a)

(3.7b)

(3.8)

(3.9a)

(3.9b)

(3.10)

Equation (3.7) to Equation (3.10) are the complete set of Spin-EM equations for constant spin.

Equation (3.7) and Equation (3.9) are the Ampere-type equation of spin and Faraday-type equations of spin, respectively.

Note: the spin generates the Spin-EM field, which is conceptually different from that of Rashba-induced-Spin-EM.

*) Poynting-type theorem of Spin-EM

Equation (2.2), shows the Poynting-type theorem of Spin-EM, Equation (3.11):

(*0) Let and , Equation (2.2) gives the standard Poynting theorem:

Which shows that Equation (2.2) is applicable to Poynting theorem.

(*1) Let and, Equation (2.2) gives the spin-Poynting theorem (a):

(3.11a)

(*2) Let and , Equation (2.2) gives the spin-Poynting theorem (b):

(3.11b)

(*3) Let and , Equation (2.2) gives the spin-Poynting theorem (c):

(3.11c)

Table 2 summarizes the Spin-EM equations including the Poynting-type theorem.

By analogy to “electric current”, let’s define the “spin-magnetic-current ” and the “spin-electric-current”, which induce the spin-magnetic field and the spin-electric field respectively, as

(3.12)

(3.13)

Equation (3.7b) and Equation (3.9b) become respectively,

(3.14)

(3.15)

By analogy to EM, defining spin-scalar-potential, , spin-vector-potential, , and gauge condition, as,

(3.16)

(3.17)

(3.18)

Under the gauge transformation,

(3.19)

the spin-electric and spin-magnetic fields, and, are invariant.

Substituting Equation (3.16) to Equation (3.18) into Equation (3.14) and Equation (3.15) respectively, we have spin waves:

(3.20)

Remark: The propagation speed of spin wave is to be determined.

Let’s study the similarity and difference between Classical-Spin-EM (C-Spin-EM) and Quantum-Spin-EM (Q-Spin-EM). To convert to quantum theory, we need to introduce the concept of phase,

(3.22)

(3.23)

We define the Spin-magnetic flux,

(3.24)

then have

(3.25)

Defining as the phase, . Spin-electric field and Spin-magnetic field of C-Sin-EM have the same form as that of Spin motive force and Berry curvature of Q-Spin-EM.

However, the fundamental differences between the C-Spin-EM and Q-Spin-EM are the definitions of spin-electric field and spin-magnetic field, as well the field equations.

For a non-relativistic non-spinning e-charge in the EM field, the regular Lagrangian and Hamiltonian are respectively,

(3.26)

For a rotating mass, the Lagrangian contains its rotation energy, .

We define the rotation energy of a spinning e-charge as,

(3.27)

where the spin angular momentum. Where is the coefficient.

By the duality between Extended EM and Spin-EM, let’s introduce Lagrangian,

(3.28)

Taking into account the interaction between the velocity and spin-vector-potential, and between the spin and vector potential, we obtain

(3.29)

Where to are coupling coefficients. Hereafter we ignore those coefficients.

The total Lagrangian of a spinning e-particle in EM fields and Spin-EM fields is

(3.30)

In the derivation of Spin-EM, we have replaced the velocity v by spin in UMFT. Now we use spin as a “generalized velocity”, substituting it into Hamiltonian,

(3.31)

we obtain the Hamiltonian for spinning e-particles,

(3.32)

which describes dynamics of spinning e-particles in both Extended EM and Spin-EM fields. Where the is a conjugate momentum corresponding to the spin,

(3.33)

For the situation, both uniform magnetic field B and uniform spin-magnetic field are in z-direction, the vector potential A and spin-vector-potential have similar form,

(3.34)

(3.35)

Equation (3.35) shows the relation between the spin-vector potential and spin-magnetic field induced by spin, which has the same form as that the vector potential A induced by magnetic field B, Eq, (3.34).

Substituting both Equation (3.34) and Equation (3.35) into Equation (3.32), we obtain four different forms,

(3.36a)

With Hamiltonian, Spin-EM can be converted to its quantum version. The Hamiltonian not only provides classical counterparts/origins of several quantum phenomena, but also predicts several classical effects that may be converted to quantum effects.

With the Hamiltonian, the following effects/phenomena have been predicted.

The first term on the right-hand side of Equation (3.36a) is the Zeeman effect,

(3.37)

Applying the definition, , the second term of Equation (3.36a) becomes

(3.38)

We refer Equation (3.38) as Spin-Zeeman effect (or Extended-Rashba-SOC (Comparing with the regular

Remark: The p represents a linear motion; represents an orbiting motion; thus, the term, , represents indeed a Spin-Orbit-coupling. Actually, when Rashba SOC is applied to several situations, the momentum p is replaced by angular momentum L, which indicates that Rashba SOC is not exactly the Spin-Orbit-coupling.

The third term of Equation (3.36a) shows the spin-angular momentum coupling to magnetic field .

(3.39)

The fourth term of Equation (3.36a) shows the Spin-Angular-Momentum, , coupling to Spin-magnetic field .

(3.40)

Equation (3.36b),

(3.40b)

shows that: the angular momentum L of a spin-particle couples to both the magnetic B field and spin-magnetic field. The spin-angular momentum of the spin-particle couples to both the magnetic B field and spin-magnetic field.

3.8.6. Total Angular Momentum Coupling to Magnetic Field and Spin-Magnetic Field B_s

Equation (3.36c)

(3.36c)

shows that: The total-angular momentum of an orbiting-spin-particle couples to both the magnetic B field and spin-magnetic field B_s.

3.8.7. Total Angular Momentum Coupling to Total magnetic Field (B+B_s)

Equation (3.36d)

(3.36d)

shows that: The total-angular momentum of an orbiting-spin-particle couples to total of magnetic B field and spin-magnetic field B_s,

We have predicted the effects of Lorentz Force on Spin:

(1) Classical Origin of Aharonov–Casher effect

(2). Spin-Stark Effect

(3). Magnetic-Rashba-type SOC.

Based on the type-2 duality between Extended EM and Spin-EM, we postulate that, beside the dualities between fields and field equations, there is a type-2 duality between Lorentz force and a spin related force, denoted as spin-Lorentz-type force, i.e., under the transformation, , , , Lorentz force, (abbreviate), converts to Spin-Lorentz-type force, (abbreviate), and vice versa,

(3.37)

We refer “” as spin-electric force, “” as spin-magnetic force.

We have predicted the effects of Spin-Lorentz-type Force:

(1) Dual-Hall Effect/Topological Insulator;

(2) Extended-Hall Effect/Topological Insulator

(3) Extended-Hall Effect Having Zero Longitudinal Resistivity

(4) Extended-Hall effect Contributing to GMR/TMR Effect,

(5) Extended-Hall Effect Contributing to Spin Hall Effect

(6) Extended-Hall effect Contributing to Anomalous-Hall Effect

(7) Temperature Dependence of Extended-Hall Effect;

(8) Magnetic Aharonov-Casher-type Effect

(9) Spin-Stark Effect;

(10) Spin-Potential-Coupling Contributing to Aharonov–Bohm Effect

Starting with Spin-Lagrangian’s equation, , where is,

(3.38)

The total Lagrangian-Lorentz-type Force

(3.39)

The “a” is a coefficient, such that and a have the unit of force. Here after, absorbing “a” into.

We have predicted the effects of Spin-Lagrangian-Lorentz-type Force

(1) Spin-Larmor-type Precession

(2) spin-magnetic force is the mechanism of Rashba Effect;

(3) Spin-magnetic-Rashba-type SOC.

Combining UMFT, Coulomb’s law and spin, we derived Spin-EM in the perspective of fundamental physics.

Spin-EM is powerful and fruitful, and achieves the following:

(1). Derives spin wave.

(2). Derives Spin-Lorentz-type force and Lagrangian-Lorentz-type force, which, for 3D model, cause Dual-Hall Effect, Extended-Hall effect, Lagrangian-Hall-type Effect, and Temperature Dependence of Extended-Hall effect; also cause Extended Rashba SOC.

(3). Extended-Hall effect contributes universally to zero longitudinal Hall coefficient/resistivity, GMR/TMR, Anomalous Hall effect, Spin Hall effect, and topological insulator.

(4). Predicts Spin-Potential-Coupling-Induces force that contributes to Aharonov–Bohm Effect.

(5). Provides counterparts of Aharonov-Bohm effect; Aharonov-Casher effect; Stark effec; Larmor Precession.

(6). Proposes several experiments to test proposed effects, such as, definition of spin-electric and spin-magnetic fields, Spin-Aharonov–Bohm effect, Dual-Hall Effect/Topological Insulator, whether of GMR/TMR, Spin-Potential-Coupling-Induced force, Zeeman effect/Extended-Rashba SOC.

We argue that the powerfulness and fruitfulness are evidences supporting Spin-EM. The Spin-EM predicts experimental results, which need to be tested.

The mathematical identities lead to the physical dualities. UMFT provides mathematical origins of physical dualities between Maxwell-EM, Gravito-EM and Spin-EM.

4. Spin-Gravity

We have proposed Spin-EM in the Section 3. In this Section we show that the spin of a gravitational mass/charge (g-charge) produces Spin-gravity fields.

We have shown that a mass (g-charge, , produces a gravito-electric-field g, , while a moving g-charge (velocity v) produces a gravito-magnetic-field,.

Now let us consider a spinning g-charge, for example a neutron, characterized by both g-charge and spin. In an external gravito-electric-field g and an external gravito-magnetic field , the spin produces the spin-gravito-electric-field and the spin-gravito-magnetic field :

(4.1)

(4.2)

The definitions, Equation (4.1) and Equation (4.2), state that: by interacting with a gravitation field g and a gravito-magnetic field respectively, the spin of a g-charge produces a spin-gravito-magnetic field and a spin-gravitation field, respectively. Subscript “s” indicates that the quantity is related to spin. The and are type-2 dual to the and g fields, respectively. The and are “Second order axial vector field” (SAF) and “Third order axial vector field” (TAF) respectively.

Assuming that the spin of a g-charge is constant. Let the “T” is the external gravito-electric-field g and the external gravito-magnetic field respectively. Equation (2.5) to Equation (2.7) give respectively,

(4.3a)

(4.3b)

(4.4a)

(4.4b)

(4.5a)

(4.5b)

Spin-gravity in this form has the same form as that of Spin-EM.

Equation (2.2), shows the Poynting-type theorem of Spin-gravity:

(*0) Let and, Equation (2.2) gives the normal Poynting theorem:

.(4.6a)

(*1) Let and, Equation (2.2) gives the spin-Poynting theorem (a):

(4.6b)

(*2) Let and, Equation (2.2) gives the spin-Poynting theorem (b):

(4.6c)

(*3) Let and, Equation (2.2) gives the spin-Poynting theorem (c):

(4.6d)

Note: the spin generates the spin-magnetic field, which is conceptually different from that of Rashba-induced-Spin-EM.

By analog to the Lorentz force, we propose the Lorentz-type gravitational force on a moving spin-g-charge

(4.7)

We apply UMFT to derive the physical field equations for describe different physical forces, including the known EM force, the new Gravito-EM force, the new Spin-EM force, and the new Spin-Gravity force. We suggest that those physical forces are the physical dual to each other. UMFT provides mathematical origins of physical dualities between Maxwell EM, Gravito-EM, Spin-EM, and Spin-gravity. They are all derived from UMFT and thus, they have the same symmetry, the U(1) symmetry, and can be unified in the frame of UMFT.

References