Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions

Abstract

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell
electrodynamics in vacuum is investigated within the matrix formalism. The
matrix form of electrodynamics includes three real 4 \times 4 matrices. Within
the squaring procedure we construct four formal solutions of the Maxwell
equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem
of separating physical electromagnetic waves in the linear space
\lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3}
is investigated, several particular cases, plane waves and cylindrical waves,
are considered in detail.

Full text

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arXiv:0906.1434v1 [math-ph] 8 Jun 2009
V.V. Kisel, E.M. Ovsiyuk, V.M. Red’kov, N.G. Tokarevskaya
Maxwell equations in matrix form, squaring procedure,
separating the variables, and structure of electromagnetic solutions
Belorussian State Pedagogical University, Minsk
Institute of Physics, National Academy of Sciences of Belarus
Belorussian Economical University, Minsk
The Riemann – Silberstein – Majorana – Oppenheimer approach to the Maxwell
electrodynamics in vacuum is investigated within the matrix formalism. The matrix
form of electrodynamics includes three real 4 × 4 matrices: (−i∂0+ αj∂j)Ψ(x) = 0,
where Ψ(x) = (0,E(x) + icB(x)). Within the squaring procedure we construct four
formal solutions of the Maxwell equations on the base of scalar Klein – Fock – Gordon
solutions: {Ψ0,Ψ1,Ψ2,Ψ3} = (i∂0+ α1∂1+ α2∂2+ α3∂3) Φ(x), where Φ(x) satisfies
equation ∂a∂aΦ(x) = 0. The problem of separating physical electromagnetic waves
in the linear space {λ0Ψ0+ λ1Ψ1+ λ2Ψ2+ λ3Ψ3} is investigated, several particular
cases, plane waves and cylindrical waves, are considered in detail.
1 Introduction
Special relativity arose from study of the Maxwell equations symmetry with respect to motion
of references frames: Lorentz [1], Poincar´ e [2], Einstein [3] Naturally, an analysis of the Maxwell
equations with respect to Lorentz transformations was the first objects of relativity theory:
Minkowski [4], Silberstein [5, 6], Marcolongo [8], Bateman [9], and Lanczos [10], Gordon [11],
Mandel’stam – Tamm [12, 13, 14].
After Dirac [15] discovery of the relativistic equation for a particle with spin 1/2 much work
was done to study spinor and vectors within the Lorentz group theory: M¨ oglich [16], Ivanenko –
Landau [17], Neumann [18], van der Waerden [19], Juvet [20]. As was shown any quantity which
transforms linearly under Lorentz transformations is a spinor. For that reason spinor quantities
are considered as fundamental in quantum field theory and basic equations for such quantities
should be written in a spinor form. A spinor formulation of Maxwell equations was studied by
Laporte and Uhlenbeck [21], also see Rumer [28]. In 1931, Majorana [23] and Oppenheimer
[22] proposed to consider the Maxwell theory of electromagnetism as the wave mechanics of the
photon. They introduced a complex 3-vector wave function satisfying the massless Dirac-like
equations. Before Majorana and Oppenheimer, the most crucial steps were made by Silberstein
[5], he showed the possibility to have formulated Maxwell equation in term of complex 3-vector
entities. Silberstein in his second paper [6] writes that the complex form of Maxwell equations
has been known before; he refers there to the second volume of the lecture notes on the differential
equations of mathematical physics by B. Riemann that were edited and published by H. Weber
in 1901 [7]. This not widely used fact is noted by Bialynicki-Birula [103]).
Maxwell equations in the matrix Dirac-like form considered during long time by many au-
thors: Luis de Broglie [24, 25, 30, 36], Petiau [26], Proca [27, 44], Duffin [29], Kemmer [31, 41, 61],
Bhabha [32], Belinfante [33, 34], Taub [35], Sakata – Taketani [37], Schr¨ odinger [39, 40], Heitler
[42], [45, 46], Mercier [47], Imaeda [48], Fujiwara [50], Ohmura [51], Borgardt [52, 59], Fe-
dorov [53], Kuohsien [54], Bludman [55], Good [56], Moses [57, 60, 76], Lomont [58], Bogush
– Fedorov [64], Sachs – Schwebel [66], Ellis [68], Oliver [70], Beckers – Pirotte [71], Casanova
[72], Carmeli [73], Bogush [74], Lord [75], Weingarten [77], Mignani – Recami – Baldo [78],
Newman [79], [80], [82], Edmonds [83], Silveira [86]; the interest to the Majorana – Oppen-
heimer formulation of electrodynamics has grown in recent years: Jena – Naik – Pradhan [87],
Venuri [88], Chow [89], Fushchich – Nikitin [90], Cook [92, 93], Giannetto [96], Y´ epez, Brito –
1

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Vargas [97], Kidd – Ardini – Anton [98], Recami [99], Krivsky – Simulik [101], Inagaki [102],
Bialynicki-Birula [103, 104, 128], Sipe [105], [106], Esposito [108], Dvoeglazov [109] (see a big
list of relevant references therein)-[111], Gersten [107], Kanatchikov [110], Gsponer [112], Ivezic
[113, 114, 115, 116, 117, 118, 119, 120, 121], Donev – Tashkova. [125, 126, 127].
Our treatment will be with a quite definite accent: the main attention is given to possibilities
given by the matrix approach for explicit constructing electromagnetic solutions of the Maxwell
equations. In vacuum case, the matrix form includes three real 4 × 4 matrices αj:
(−i∂0+ αj∂j)Ψ(x) = 0 ,
where Ψ(x) = (0,E(x) + icB(x)). With the use of squaring procedure one may construct four
formal solutions of the Maxwell equations on the base of scalar solution of the Klein – Fock –
Gordon equation:
{Ψ0,Ψ1,Ψ2,Ψ3} = (i∂0+ α1∂1+ α2∂2+ α3∂3) Φ(x) ,∂a∂aΦ(x) = 0 .
The problem of separating physical electromagnetic solutions in the linear space {λ0Ψ0+λ1Ψ1+
λ2Ψ2+ λ3Ψ3} is investigated. Several particular cases are considered in detail.
2 Complex matrix form of Maxwell theory in vacuum
Let us start with Maxwell equations in vacuum [38, 63, 82, 129] (with the use of usual notation
for current 4-vector ja= (ρ,J/c) , c2= 1/ǫ0µ0):
div cB = 0 ,rot E = −∂cB
∂ct
,
div E =ρ
ǫ0, rot cB =
j
ǫ0
+∂E
∂ct,
(1)
Let us introduce 3-dimensional complex vector ψk= Ek+ icBk, with the help of which
the above equations can be combined into (see Silberschtein [5, 6], Bateman [9], Majorana [23],
Oppenheimer [22], and many others)
∂1Ψ1+ ∂2Ψ0+ ∂3Ψ3= j0/ǫ0, −i∂0ψ1+ (∂2ψ3− ∂3ψ2) = i j1/ǫ0,
−i∂0ψ2+ (∂3ψ1− ∂1ψ3) = i j2/ǫ0, −i∂0ψ3+ (∂1ψ2− ∂2ψ1) = i j3/ǫ0. (2)
let x0 = ct, ∂0 = c∂t. These four relations can be rewritten in a matrix form using a 4-
dimensional column ψ with one additional zero-element [90, 130]:
(−i∂0+ αj∂j)Ψ = J , Ψ =
????????
0
ψ1
ψ2
ψ3
????????
,J =1
ǫ0
????????
j0
i j1
i j2
i j3
????????
,
α1=
????????
01
0
0
0
0
0
0
1
0
0−1
0
0
−1
0
????????
,α2=
????????
α2α3= −α3α2= α1,
0
0
0
0
0
1
0
0
0
0
1
0
0
−1
0−1
????????
,α3=
????????
0
0
0
0
0
1
0
01
0
0
0
−1
0
0−1
????????
(α1)2= −I,(α2)2= −I,(α2)2= −I,
α1α2= −α2α1= α3,α3α1= −α1α3= α2. (3)
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3Method to construct electromagnetic solutions from scalar
ones
The above matrix form of Maxwell theory :
(−i∂0+ αj∂j)Ψ = 0 ,Ψ =
????????
0
ψ1
ψ2
ψ3
????????
. (4)
permits us to develop a simple method of finding solutions of Maxwell equations on the base of
known solutions of the scalar massless equation by Klein – Fock – Gordon. Indeed, in virtue of
the above commutative relations we have an operator identity
(−i∂0+ α1∂1+ α2∂2+ α3∂3) (−i∂0− α1∂1− α2∂2− α3∂3) = (−∂2
0+ ∂2
1+ ∂2
2+ ∂3
3) .
Therefore, taking any special scalar solution (−∂2
construct four solutions of the Maxwell equation:
0+∂2
1+∂2
2+∂2
3) Φ(x) = 0 , one can immediately
(−i∂0+ α1∂1+ α2∂2+ α3∂3) Ψa= 0 ,
where Ψaare columns of the matrix
(i∂0+ α1∂1+ α2∂2+ α3∂3) Φ(x) =
∂1Φ∂2Φ
i∂0Φ−∂3Φ
∂3Φ i∂0Φ
−∂2Φ∂1Φ
=
????????
i∂0Φ
−∂1Φ
−∂2Φ
−∂3Φ
∂3Φ
∂2Φ
−∂1Φ
i∂0Φ
????????
= {Ψ0,Ψ1,Ψ2,Ψ3} .(5)
Thus, we have four formal solutions of the free Maxwell equations (let Fa(x) = ∂aΦ(x)):
{Ψ0,Ψ1,Ψ2,Ψ3} =
????????
iF0
−F1
−F2
−F3
F1
iF0
F3
−F2
F2
F3
F2
−F3
iF0
F1
−F1
iF0
????????
.(6)
4 Electromagnetic plane waves from scalar ones
Let us specify this method for an elementary example, starting from a scalar plane wave prop-
agating along the axis z:
Φ = Asin(ωt − kz) = Asin(k0x0− k3x3) = Acosϕ . (7)
The recipe (5) gives
{Ψa} = A
????????
ik0
00−k3
0
0
ik0
−k3
k3
ik0
0
0
k3
00 ik0
????????
cosϕ . (8)
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Because the left and the right columns have non-vanishing zero-component, they cannot repre-
sent any real solutions of the Maxwell equations. However, two remaining seem to be suitable
ones (the factor Acosϕ is omitted):
ΨI=
????????
0
ik0
−k3
0
????????
=
????????
0
E1+ icB1
E2+ icB2
E3+ ciB3
????????
,ΨII= A
????????
0
k3
ik0
0
????????
=
????????
0
E1+ icB1
E2+ icB2
E3+ ciB3
????????
.
Thus, we have two wave solutions:
IE2= −k3A cosϕ ,B1=k0
cA cosϕ ,
IIE1= k3A cosϕ ,B2=k0
cA cosϕ . (9)
In both cases, the ratio E/B equals to the speed of light:
E
B=
k3
k0/c= c .
Besides, both waves have are clockwise polarized:
I(E × B) = +e3
k3k0
c
k3k0
c
A2cos2ϕ ,
II(E × B) = +e3
A2cos2ϕ . (10)
Two waves are linearly independent and orthogonal to each other:
EIEII= 0 ,
BIBII= 0 .
In the same manner we can solve a more general problem of constructing plane wave solutions
with arbitrary wave vector k. Let us start with a scalar wave
Φ = A sin(k0x0− kixi) = A cosϕ .(11)
The matrix of solutions (5) will take the form
{Ψ0,Ψ1,Ψ2,Ψ3} = A
????????
ik0
k1
k2
k3
−k1
ik0
−k3
+k2
−k2
+k3
ik0
−k1
−k3
−k2
+k1
ik0
????????
cosϕ .(12)
We have four formal solutions Ψa, but they cannot be regarded as physical because each of them
has a non-vanishing zero-component. However, we can use linearity of the Maxwell equation and
combine elementary columns in (12) with any coefficients. In this way we are able to construct
physical solutions. For shortness let us omit the factor A sinϕ and operate only with the
columns of the matrix
{Ψ0,Ψ1,Ψ2,Ψ3} =
????????
ik0
k1
k2
k3
−k1
ik0
−k3
+k2
−k2
+k3
ik0
−k1
−k3
−k2
+k1
ik0
????????
.
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Taking into account properties of the wave vector
k2
0− k2= 0=⇒
k = k0n , n2= 1 ,(13)
previous matrix can be rewritten as follows (the common factor k0is omitted)
{Ψ0,Ψ1,Ψ2,Ψ3} = k0
????????
i−n1
i
−n3
+n2
−n2
+n3
i
−n1
−n3
−n2
+n1
i
n1
n2
n3
????????
∼
????????
i−n1
i
−n3
+n2
−n2
+n3
i
−n1
−n3
−n2
+n1
i
n1
n2
n3
????????
.
First, with the help of the column (0) let us produce a zero at the first component of the columns
(1) – (2) – (3):
{Ψ0,Ψ1,Ψ2,Ψ3} ∼
????????
i
n1
n2
n3
0
−in1n1+ i
−in1n2− n3
−in1n3+ n2
0
−in2n1+ n3
−in2n2+ i
−in2n3− n1
0
−in3n1− n2
−in3n2+ n1
−in3n3+ i
????????
.
Noting that the column (3) is a linear combination of the columns (1) and (2):
i n2(1) − i n1(2) = (3) ;
in other words, solution (3) is a linear combination of (1) and (2). Therefore from the above it
follows (multiplying the columns (1) and (2) by imaginary i):
{Ψ0,Ψ1,Ψ2,Ψ3} ∼
????????
i
n1
n2
n3
0
n2
n1n2− in3
n1n3+ in2
0
n2n1+ in3
n2n2− 1
n2n3− in1
0
0
0
0
1− 1
????????
.(14)
Thus, two physical Maxwell solutions are
ΨI=
????????
0
n2
1− 1
n1n2− in3
n1n3+ in2
????????
=
????????
0
E1+ icB1
E2+ icB2
E3+ ciB3
????????
,ΨII=
????????
0
n2n1+ in3
n2n2− 1
n2n3− in1
????????
=
????????
0
E1+ icB1
E2+ icB2
E3+ ciB3
????????
, (15)
or differently (the factor A cosϕ is omitted)
I
E = (n2
1− 1, n1n2, n1n3) ,
E = (n2n1, n2n2− 1, n2n3) ,
cB = (0, −n3, n2) ;
cB = (n3, 0, −n1) .II(16)
It is the matter of simple calculation to verify the identity for amplitudes: cB = E. Also, both
these waves are clockwise polarized:
EI× BI=
??????
n2n1
e1
e2
e3
n2
1− 1n1n2
−n3
n1n3
0n2
??????
??????
= (1 − n2
1) (n1e1+ n2e2+ n3e1) ∼ k ,
EII× BII=
??????
e1
e2
e3
n2n2− 1n2n3
−n1
n3
0
= (1 − n2
2) (n1e1+ n2e2+ n3e1) ∼ k .(17)
Besides, they are independent solutions (but not orthogonal ones)
EIEII= −n1n2,
BIBII= −n1n2.(18)
5