ZHANG Xia-fu

**Pacc
Code: **0370

**0. Introduction
**

We suggest a special representation of the Dirac equation，which can be derived by a unitary transformation from the acquainted representation of the Dirac equation. The feature of the representation is that, besides spin angular momentum, another intrinsic spin is included in the equation. The representation of the Dirac equation contains two intrinsic spins, thus it involves more symmetries.

Corresponding to the mentioned representation of the Dirac equation, we also rewrite Maxwell equations with the matrices that exist in the Dirac equation. As a result Maxwell equations show that they also contain two intrinsic spins, and are related to the Dirac equation with the common spin matrices.

**1.
The Dirac equation containing two intrinsic spins
**

**
**When
we make mention of spin we always remember 2×2 Pauli matrices. As we show in the following, there exist two sets of
three* *4×4 matrices, the properties of which are just the same as the 2×2
Pauli matrices. We will write the Dirac equation with them instead of Pauli
matrices.

Set
1 of the three 4×4
matrices is written as:

(1.1)

Set 2 of the three 4×4
matrices is written as:

(1.2)

All
the matrices are Hermitiean. We denote Set 1 with operators *σ*_{1},*
σ*_{2} and* σ*_{ 3}, and Set 2 with operators *τ*_{1},
*τ*_{ 2} and* τ*_{ 3} respectively. They can be
represented as vector operators ** σ** and

(1.3)

Specifically, the matrices of Set 1 commute with the matrices of Set 2,
that is

(1.4)

The form of the Dirac equation can be improved by
means of ** σ** and

*
* * *(1.5)

Where the operators *γ*_{μ} satisfy anticommutive
relations:

(1.6)

Our representation is to replace *γ*_{μ} in the above
equation by

(1.7)

Considering rules (1.3) and (1.4), we can verify that expressions (1.7)
obey anticommutive relations shown in expression (1.6):

(1.8)

The matrices of expressions (1.7) can be deduced from the conventional
four Dirac matrices by means of a unitary transformation (see Appendix (1)).
Substituting (1.7) into (1.5), consequently, a special representation of the
Dirac equation is founded

(1.9)

Where contains a pair of vector operator ** σ **and

Under
Lorentz transformations

(1.10)

Relativistic invariance of Eq. (1.5) requires that

(1.11)

For infinitesimal transformation, we have acquainted the following
formulas

(1.12)

Corresponding to the representation of the Dirac equation given by us,
in terms of rule (1.3) and (1.4) we can establish the expressions of *S _{μν}*
as that

(1.13)

Hence the Lorentz transformation operators* S _{μν }*and

(1.14)

is just the spin angular momentum of the particles of spin *S **=1/2*.

**2.
Representing Maxwell Equations with The Above
Matrices
**

We
can also write Maxwell equations with the matrices (1.1) and (1.2) of ** σ**
and

We know that the tensor *F _{μν }*of
electromagnetic field is represented by electric field

** **(2.1)

We
decompose *F _{μν }*into
the sum of six matrices, in which every matrix includes only one component of

(2.2)

Maxwell equations that are used to describe electromagnetic field are,
in Caussian units,

(2.3)

(2.4)

(2.5)

(2.6)

The covariant form of Lorenz transformation of equations (2.4) and (2.5)
can be written as

(2.7)

It is apparent that equation (2.7) contains the matrices of the
intrinsic spins when we substitute expression (2.2) in the above equation.

We also define a matrix *G _{μν }*as
follows to match

(2.8)

By making the analogous analysis of establishing expression (2.2), the
matrix *G _{μν }*also appears in such a form that contains

(2.9)

Base on the expressions (2.8) and (2.9), the Maxwell equations (2.3) and
(2.6) can be simply taken in the form:

(2.10)

The pair of equations (2.7) and (2.10) is the complete Maxwell equations
in which intrinsic spins are contained. The pair of the equations shows us to be
more symmetries than the original forms. The equation (2.10) is only formally
different but equivalent to the following covariant form of equations (2.3) and
(2.6):

(2.11)

Under
infinitesimal Lorentz transformation, four-vector *A _{μ }*are
transformed according to

(2.12)

We still define transformation matrices *S _{κλ }*and
operator Λ similar to (1.12):

(2.13)

We require that the Lorenz transformation of *A _{μ }*will
be performed according to

(2.14)

Comparing the above formula with (2.12), we have

(2.15)

From equations (2.13) and (2.15) we derive

(2.16)

The Lorenz transformation of tensor *F _{μν }*will be
performed by Λ and Λ

(2.17)

Calculating all the elements *E ^{’}_{i}* and

(2.18)

The transformation of *G _{μν }*is assumed just the same
as the tensor

(2.19)

We also calculate all the elements *E ^{’}_{i}*
and

According to the
formulas (2.16), we concretely write each matrix of *S _{κλ}* as
follows

(2.20)

comparing the above matrices with (1.1) and (1.2), we finally arrive at
the formulas of *S _{μν}* of electromagnetic field.

(2.21)

It is clear for electromagnetic field the vector

(2.22)^{
}

is just the spin angular momentum of photon of spin *S *=1. The
spin vector is contained in the forms (2.7) and (2.10) of Maxwell equations.

Equation
(1.9), equations (2.7) and (2.10) disclose that, the Dirac equation and Maxwell
equations bring two intrinsic spins with themselves. One is the vector *–
i[S _{23}, S_{31}, S_{12}] *and
another one is the vector

**3.
Runge-Lenz vector
**

**
**We know that there exists Runge-Lenz vector^{[5]} when there
exist Coulomb field.** **Runge-Lenz vector

(3.1)

Here ** l **is angular momentum.

It is easy to prove that
the vector_{
}*– i[S _{14}, S_{24}, S_{34 }]*
is Runge-lenz vector. For the Dirac equaton we have

(3.2)

Replace ** l **with

(3.3)

For Maxwell equations we have

(3.4)

Replace ** l** with (

(3.5)

(3.3) and (3.5) coincide with (3.1), thus the vector_{ }*–
i[S _{14}, S_{24}, S_{34 }] *is Runge-lenz vector.

We have known that the vector *– i[S _{23},
S_{31}, S_{12}] *is
corresponding to the inherent magnetic property of particles and is called spin
angular momentum. From (2.2) and (4.7) (see appendix (2)) we can also see that the
vector

**4.Appendix
**

(1):
Transformation from ordinary representation of the Dirac equation to the above
representation.

For Dirac matrices

(4.1)

there exist unitary matrix:

(4.2)

The Dirac matrices (3.1) can be transformed into the forms of expression
(1.7) by the unitary matrix *U*:

(4.3)

In addition, any 4x4
matrices *A* can be expressed by

(4.4)

where *C _{0}, C_{ij, }C_{k}
and C_{l}* are arbitrary constants. It is easy to prove the above
formula because the traces of matrices

(4.5)

Hence we may construct a complete system of
4x4 matrices on the basis of the matrices of the operator ** σ** and

(2): Writing magnet field ** B**
and electric field

The
magnetic field ** B** and the
electric field

(4.6)

Here ** A **is a vector
potential and

(4.7)

Where *j=1,2,3.* We assume*
E _{4}=0*, such that from (4.7) we get

(4.8)

It
is just Lorenz condition of gauge transformation. In (4.7) we can see that the
magnetic field is only associated with the spin angular momentum and the
electric field is only associated with the Runge-Lenz vector.

**References**:

[1] P. A. M. Dirac, *The Principles of Quantum Mechanics*, 2^{nd}
Ed, Oxford (1957)

[2] L. I. Schiff, *Quantum Mechanics. *3^{rd} Ed.
McGraw-Hill Book Company (1963)

[3] L.D.Landau and E.M.Lifshits, *Quantum
Mechanics.* Pergamon Press (1977)

[4] Albert Messiah, *Quantum
Mechanics.* 2^{nd} Ed, Unabridged Dover (1999)

[5] Thomas F.Jordan, *Quantum Mechanics in Simple Matrix Form. *New York (1986)

作者：张侠辅（昆明理工大学理学院物理系退休教师）

地址：云南昆明，昆明理工大学莲华校区教工宿舍121幢103号，邮编650093