(1.1)
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 σ1,
σ2 and σ 3, and Set 2 with operators τ1,
τ 2 and τ 3 respectively. They can be
represented as vector operators σ and τ, where
each vector is composed of three components, i.e. σ = [σ1,
σ2, σ 3] and τ = [τ1,
τ2, τ 3]. The following operation rules for a
pair of operators σ and τ are derivable from the
matrix expressions (1.1) and (1.2):
(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 τ. We know that the Dirac equation
generally is expressed as
(1.5)
Where the operators γμ satisfy anticommutive
relations:
(1.6)
Our representation is to replace γμ 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 τ in
the equation. We have seen that not only σ but also τ belong
to the inherent properties of the particles described by the equation.
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 Sμν
as that
(1.13)
Hence the Lorentz transformation operators Sμν and Λ
are distinct from the original expressions, and are perfect. Obviously the
vector
(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 τ. The Maxwell equations containing the matrices (1.1) and
(1.2) make the relations between Dirac equation and the equations of
electromagnetic field more close.
We know that the tensor Fμν of
electromagnetic field is represented by electric field E and
magnetic field B as follows
(2.1)
We
decompose Fμν into
the sum of six matrices, in which every matrix includes only one component of E
or B and the other elements are zero, then replace the six
matrices by using components of σ+τ and σ-τ.
The tensor finally is written in the form:
(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 Gμν as
follows to match Fμν, such that we can constitute a couple
of equation (2.7) from equations (2.3) and (2.6).
(2.8)
By making the analogous analysis of establishing expression (2.2), the
matrix Gμν also appears in such a form that contains σ
and τ.
(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 Aμ are
transformed according to
(2.12)
We still define transformation matrices Sκλ and
operator Λ similar to (1.12):
(2.13)
We require that the Lorenz transformation of Aμ 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 Fμν will be
performed by Λ and Λ-1 as follows
(2.17)
Calculating all the elements E’i and B’j
of F’μν on the basis of the transformation formula and
neglecting the terms of order εμκ ελν, we obtain
(2.18)
The transformation of Gμν is assumed just the same
as the tensor Fμν :
(2.19)
We also calculate all the elements E’i
and B’j of G’μν from
(2.19), neglecting the terms of order εμκ ελν
,
the
results coincide with (2.18) completely. It proved that formula (2.19) is
correct, so Gμν is a tensor under Lorenz transformation.
Equation (2.10) is a covariant form of Maxwell equations (2.3) and (2.6).
According to the
formulas (2.16), we concretely write each matrix of Sκλ as
follows
(2.20)
comparing the above matrices with (1.1) and (1.2), we finally arrive at
the formulas of Sμν 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[S23, S31, S12] and
another one is the vector –
i[S14, S24, S34].
3.
Runge-Lenz vector
We know that there exists Runge-Lenz vector[5] when there
exist Coulomb field. Runge-Lenz vector a
satisfy
(3.1)
Here l is angular momentum.
It is easy to prove that
the vector
– i[S14, S24, S34 ]
is Runge-lenz vector. For the Dirac equaton we have
(3.2)
Replace l withσ/2
and a
with στ3/2 in
formulas (3.1), we obtain
(3.3)
For Maxwell equations we have
(3.4)
Replace l with ( σ+τ)/2 and a
with ( σ-τ)/2 in formulas (3.1), we obtain
(3.5)
(3.3) and (3.5) coincide with (3.1), thus the vector –
i[S14, S24, S34 ] is Runge-lenz vector.
We have known that the vector – i[S23,
S31, S12] 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 – i[S14, S24, S34] should
be corresponding to the inherent electric property of particles.
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 C0, Cij, Ck
and Cl are arbitrary constants. It is easy to prove the above
formula because the traces of matrices σi
andτj
and
their productsσiτj
are
all equal to zero:
(4.5)
Hence we may construct a complete system of
4x4 matrices on the basis of the matrices of the operator σ and τ.
(2): Writing magnet field B
and electric field E of Maxwell
equations with potentials A and
φ.
The
magnetic field B and the
electric field E generally are
expressed by potentials A and φ
(4.6)
Here A is a vector
potential and φ is a quantity called
the scalar potential. We can rewrite (4.6) as
(4.7)
Where j=1,2,3. We assume
E4=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, 2nd
Ed, Oxford (1957)
[2] L. I. Schiff, Quantum Mechanics. 3rd Ed.
McGraw-Hill Book Company (1963)
[3] L.D.Landau and E.M.Lifshits, Quantum Mechanics. Pergamon Press (1977)
[4] Albert Messiah, Quantum Mechanics. 2nd 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