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In the present paper, the system of 11 equations for massive Stueckelberg particle is studied in presence of the external uniform electric field. We apply covariant formalism according to the general tetrad approach by Tetrode-Weyl-Fock-Ivanenko specified for cylindrical coordinates. After separating the variables, we derive the system of the first-order differential equations in partial derivatives with respect to coordinates (r, z). To resolve this system, we apply the Fedorov- Gronskiy method, thereby we consider the 11-dimensional spin operator and find on this base three projective operators, which permit us to expand the complete wave function in the sum of three parts. Besides, according to the general method, dependence of each projective constituent on the variable r should be determined by only one function. Also, in accordance with the general method we impose the first-order constraints which permit us to transform all differential equations in partial derivatives with respect to coordinates (r, z) into the system of 11 first-order ordinary differential equations in the variable z. The last system is solved in terms of confluent hypergeometric functions. In total, four independent types of solutions have been constructed, in contrast to the case of the ordinary spin 1 particle described by Daffin- Kemmer equation when only three types of solutions are possible.
Stueckelberg particle, tetrad formalism, cylindrical symmetry, external electric field, separation of the variables, differential equations in partial derivatives, exact solutions
1. The basic equation
The initial Stueckelberg system [1–5] of equations for a
massive particle in presence of external electromagnetic
fields is
−DaΨa − μΨ = 0,
DaΨ + DbΨab − μΨa = 0,
DaΨb − DbΨa − μΨab = 0,
where Da = ∂a + ieAa. As the wave function, we will use
the 11-dimensional column
Φ(x) = (Ψ;Ψ0,Ψ1,Ψ2,Ψ3;
Ψ01,Ψ02,Ψ03,Ψ23,Ψ31,Ψ12)t = (H,H1,H2)t,
where t denotes transposition. The above system can be presented
in the block form
DaGaH1 + μH = 0,
ΔaDaH + KaDaH2 − μH1 = 0, (1)
DaLaH1 − μH2 = 0,
or differently
(−DaΓa − μ)Φ = 0,
Γa =
0 −Ga 0
Δa 0 Ka
0 La 0
, Φ =
H
H1
H2
. (2)
All blocks were defined in [3–5]. This matrix equation for
Stueckelberg particle can be extended to the Riemannian
space-time in accordance with the known tetrad procedure
[
Γα(x)
(
∂
∂xα +
Σ
α
(x) − ieAα
)
− μ
]
Ψ(x) = 0.
(3)
Local matrices Γα(x) are determined through the tetrads
Γα(x) = eα(
a)(x)Γa =
=
0 −Gaeα(
a) 0
Δaeα(
a) 0 Kaeα(
a)
0 Laeα(
a) 0
. (4)
The connection Σα(x) is defined by the formulas
jab =
0 0 0
0 jab
1 0
0 0 jab
2
,
Σα(x) =
1
2
jabeβ
(a)(x)e(b)β;α(x), (5)
Σ1(x) =
1
2
jab
(1)eβa
(x)e(b)β;α(x),
Σ2(x) =
1
2
jab
(2)eβa
(x)e(b)β;α(x),
where jab
(1) and jab
(2) designate generators for vector Ψk(x)
and antisymmetric tensor Ψ[mn](x), respectively. Equation
(3) may be presented with the use of the Ricci rotation
coefficients
[
Γc
(
eα(
c)
∂
∂xα +
1
2
jabγabc − ieAc
)
− μ
]
Ψ(x) = 0.
(6)
In more detailed form, Eq. (6) reads
−Gc
(
eα(
c)∂α + jab
(1)
1
2
γabc − ieAc
)
H1 − μH = 0,
Δc (
eα(
c)∂α − ieAc
)
H+
+Kc
(
eα(
c)∂α + jab
(2)
1
2
γabc − ieAc
)
H2 − μH1 = 0,
(7)
Lc
(
eα(
c)∂α + jab
(1)
1
2
γabc − ieAc
)
H1 − μH2 = 0.
Let us consider the Stueckelberg equation in presence of the
uniform electric field. In cylindrical coordinates and corresponding
diagonal tetrad
xα = (t, r, ϕ, z),
dS2 = dt2 − dr2 − r2dϕ2 − dz2, A0 = −Ez,
the above equation takes the form (let eE ⇒ E):
[
Γ0
(
∂
∂t
+ iEz
)
+ Γ1 ∂
∂r
+
+Γ2 ∂ϕ + j12
r
+ Γ3 ∂
∂z
− μ
]
Ψ = 0. (8)
In block form, it reads
[
−G0
(
∂
∂t
+ iEz
)
− G1 ∂
∂r
−
−G2 1
r
(
∂
∂ϕ
+ j12
1
)
− G3 ∂
∂z
]
H1 − μH = 0,
[
Δ0
(
∂
∂t
+ iEz
)
+ Δ1 ∂
∂r
+ Δ2 1
r
∂ϕ + Δ3 ∂
∂z
]
H+
+
[
K0
(
∂
∂t
+ iEz
)
+ K1 ∂
∂r
+ (9)
+K2 ∂ϕ + j12
2
r
+ K3 ∂
∂z
]
H2 = μH1,
[
L0 ∂
∂t
+ L1 ∂
∂r
+ L2 ∂ϕ + j12
1
r
+ L3 ∂
∂z
]
H1 = μH2.
In the following, it will be convenient to apply the cyclic basis,
in which the third projection of the spin is diagonal (see
details in [3–5]).
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39
2. Separation of the variables
We apply the following substitution for the wave function
in cyclic basis
¯Ψ
= e−iϵteimϕ
¯H
(r, z)
¯H
1(r, z)
¯H
2(r, z)
, ¯H = h(r, z),
¯H
1 =
h0(r, z)
h1(r, z)
h2(r, z)
h3(r, z)
, ¯H2 =
(
Ei(r, z)
Bi(r, z)
)
. (10)
Then Eqs. (9) read
[
+i(ϵ − Ez)G0 − G1 d
dr
−
−G2 1
r
(
im + j12
1
)
− d
dz
G3
]
H1 = μH,
[
−i(ϵ − Ez)mΔ0 + Δ1 d
dr
+
im
r
Δ2 +
d
dz
Δ3
]
H+
+
[
−i(ϵ − Ez) ¯K 0 + K1 d
dr
+
+K2 im + j12
2
r
+
d
dz
K3
]
H2 = μH1,
[
−i(ϵ − Ez)L0 + L1 d
dr
+
+L2 im + j12
1
r
+
d
dz
L3
]
H1 = μH2.
After simple calculation, we obtain the system of 11 equations.
With the use of the shortening notations
am =
d
dr
+
m
r
, am+1 =
d
dr
+
m + 1
r
,
bm =
d
dr
− m
r
, bm−1 =
d
dr
− m − 1
r
,
(11)
it reads
i(ϵ − Ez)h0 +
d
dz
h2 − bm−1h1 + am+1h3 = μh,
−i(ϵ − Ez)h − d
dz
E2 + bm−1E1 − am+1E3 = μh0,
−amh + am+1B2 − d
dz
B3 + i(ϵ − Ez)E1 = μh1,
d
dz
h + i(ϵ − Ez)E2 − am+1B1 − bm−1B3 = μh2,
bmh + bmB2 +
d
dz
B1 + i(ϵ − Ez)E3 = μh3,
amh0 − i(ϵ − Ez)h1 = μE1,
− d
dz
h0 − i(ϵ − Ez)h2 = μE2,
−bmh0 − i(ϵ − Ez)h3 = μE3,
−bmh2 +
d
dz
h3 = μB1,
bm−1h1 + am+1h3 = μB2, − d
dz
h1 − amh2 = μB3.
3. The Fedorov-Gronskiy method
To resolve the last system, we will implement the
Fedorov-Gronskiy method [6]. To this end, let us consider the
11-dimensional spin operator Y = −i ¯ J12. We readily verify
that it satisfies the minimal equation Y (Y −1)(Y +1) = 0.
This permits us to introduce three projective operators
P1 =
1
2
Y (Y − 1), P2 =
1
2
Y (Y + 1),
P3 = 1 − Y 2, P0 + P+1 + P−1 = 1.
(12)
Therefore, the complete wave function may be decomposed
into the sum of three parts
Ψ = Ψ0 + Ψ+1 + Ψ−1,
Ψσ = PσΨ, σ = 0, +1,−1.
(13)
We can readily find an explicit formula of them. Besides, according
to the Fedorov-Gronskiy method, dependence of each
projective constituent on the variable r should be determined
by only one function
Ψ1(r, z) =
(
0, 0, h1(z), 0, 0,E1(z),
0, 0, 0, 0,B3(z)
)t
f1(r),
Ψ2(r, z) =
(
0, 0, 0, 0, h3(z), 0, 0,E3(z),
B1(z), 0, 0
)t
f2(r),
Ψ3(r, z) =
(
h1(z), h0(z), 0, h2(z),
0, 0,E2(z), 0, 0,B2(z), 0
)t
f3(r). (14)
Acting by projective operators on the above system of 11
equations Pi(A11×11Ψ) = 0, we get three subsystems. Besides,
in accordance with the general method, we should impose
the first-order constraints which permit us to transform
all differential equations in partial derivatives with respect
to coordinates (r, z) into the system of ordinary differential
equations of the variable z
P1
−amf3(r)h(z) + amf3(r)B2(z) − f1(r)
d
dz
B3(z)+
+i(ϵ − Ez)f1(r)E1(z) = μf1(r)h1(z) ⇒
amf3(r) = C1f1(r),
amf3(r)h0(z) − i(ϵ − Ez)f1(r)h1(z) =
= μf1(r)E1(z) ⇒ amf3(r) = C1f1(r),
−f1(r)
d
dz
h1(z) − amf3(r)h2(z) =
= μf1(r)B3(z) ⇒ amf3(r) = C1f1(r);
P2
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bmf3(r)h(z) + bmf3(r)B2(z) + f2(r)
d
dz
B1(z)+
+i(ϵ − Ez)f2(r)E3(z) = μf2(r)h3(z) ⇒
bmf3(r) = C2f2(r),
−bmf3(r)h0(z) − i(ϵ − Ez)f2(r)h3(z) =
= μf2(r)E3(z) ⇒ bmf3(r) = C2f2(r),
−bmf3(r)h2(z) + f2(r)
d
dz
h3(z) =
= μf2(r)B1(z) ⇒ bmf3(r) = C2f2(r);
P3
−i(ϵ − Ez)f3(r)h0(z) − f3(r)
d
dz
h2(z)+
+bm−1f1(r)h1(z) − bm−1f1(r)h3(z) =
= μf3(r)h(z) ⇒ bm−1f1(r) = C3f3(r),
−i(ϵ − Ez)f3(r)h(z) − f3(r)
d
dz
E2(z)+
bm−1f1(r)E1(z) − am+1f2(r)E3(z) = μf3(r)h0(z)
⇒ bm−1f1(r) = C3f3(r), am+1f2(r) = C4f3(r),
f3(r)
d
dz
h(z) + i(ϵ − Ez)f3(r)E2(z)−
−am+1f2(r)B1(z) − bm−1f1(r)B3(z) = μf3(r)h2(z)
⇒ bm−1f1(r) = C3f3(r), am+1f2(r) = C4f3(r),
−f3(r)
d
dz
h0(z) − iϵf3(r)h2(z) = μf3(r)E2(z),
bm−1f1(r)h1(z) + am+1f2(r)h3(z) = μf3(r)B2(z)
⇒ bm−1f1(r) = C3f3(r), am+1f2(r) = C4f3(r).
Thus, we get the following system
−C1h + C1B2 − d
dz
B3 + i(ϵ − Ez)E1 = μh1,
C1h0 −i(ϵ−Ez)h1 = μE1, − d
dz
h1 −C1h2 = μB3,
C2h + C2B2 +
d
dz
B1 + i(ϵ − Ez)E3 = μh3,
−C2h0 − i(ϵ − Ez)h3 = μE3,
−C2h2 +
d
dz
h3 = μB1,
−i(ϵ − Ez)h0 − d
dz
h2 + C3h1 − C3h3 = μh,
−i(ϵ − Ez)h − d
dz
E2 + C3E1 − C4E3 = μh0,
d
dz
h + i(ϵ − Ez)E2 − C4B1 − C3B3 = μh2,
− d
dz
h0 −i(ϵ−Ez)h2 = μE2, C3h1 +C4h3 = μB2,
and the constraints
bm−1f1(r) = C3f3(r), amf3(r) = C1f1(r),
am+1f2(r) = C4f3(r), bmf3(r) = C2f2(r).
(15)
Eqs. (15) transform into equations for separate functions
bm−1amf3(r) = C1C3f3(r),
ambm−1f1(r) = C1C3f1(r),
am+1bmf3(r) = C2C4f3(r),
(16)
bmam+1f2(r) = C2C4f2(r).
Evidently, within each pair we can assumeC3 = C1,C4 = C2.
Therefore, the above differential conditions and the secondorder
equations take on the form
bm−1f1(r) = C1f3(r), amf3(r) = C1f1(r),
am+1f2(r) = C2f3(r), bmf3(r) = C2f2(r);
(17)
[bm−1am − C2
1 ]f3(r) = 0,
[ambm−1 − C2
1 ]f1(r) = 0,
f3(r) = 0, [bmam+1 − C2
2 ]f2(r) = 0. (18)
Explicitly, Eqs. (18) are red as
(
d2
dr2 +
1
r
d
dr
− m2
r2
− C2
1
)
f3(r) = 0,
(
d2
dr2 +
1
r
d
dr
− (m − 1)2
r2
− C2
1
)
f1(r) = 0,
(
d2
dr2 +
1
r
d
dr
− m2
r2
− C2
2
)
f3(r) = 0,
(
d2
dr2 +
1
r
d
dr
− (m + 1)2
r2
− C2
2
)
f2(r) = 0.
So we get the following constraint C2
3 = C2
2 = C2
1 = C2,
and only three different equations
1
(
d2
dr2 +
1
r
d
dr
− (m − 1)2
r2
− C2
)
f1(r) = 0,
2
(
d2
dr2 +
1
r
d
dr
− (m + 1)2
r2
− C2
)
f2(r) = 0, (19)
3
(
d2
dr2 +
1
r
d
dr
− m2
r2
− C2
)
f3(r) = 0.
They are solved in Bessel functions. More details on the parameter
C2 are given later. The meaning of parameter C2
may be understood if we turn to the Klein-Fock-Gordon equation
in cylindrical coordinates in presence of the uniform electric
field
[
d2
dz2 + (ϵ − Ez)2 +
d2
dr2 +
1
r
d
dz
− m2
r2
− μ2
]
×
×eiϵteimϕR(r)Z(z) = 0.
The variables are separated as follows
[
d2
dz2 + (ϵ − Ez)2 − μ2 + λ
]
Z(z) = 0,
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41
[
d2
dz2 +
1
r
d
dz
− m2
r2
− λ
]
R(r) = 0,
so C2 = λ is the separation constant associated with the
cylindrical coordinate system (see (19)).
4. Solving the equations in the variable z
Below we will take into account the identitiesC1 = C2 =
C3 = C. We should solve the system of equations in the variable
z:
−Ch + CB2 − d
dz
B3 + i(ϵ − Ez)E1 = μh1,
Ch0 − i(ϵ − Ez)h1 = μE1, − d
dz
h1 − Ch2 = μB3,
Ch + CB2 +
d
dz
B1 + i(ϵ − Ez)E3 = μh3,
−Ch0 − i(ϵ − Ez)h3 = μE3, −Ch2 +
d
dz
h3 = μB1,
−i(ϵ − Ez)h0 − d
dz
h2 + Ch1 − Ch3 = μh, (20)
−i(ϵ − Ez)h − d
dz
E2 + CE1 − CE3 = μh0,
d
dz
h + i(ϵ − Ez)E2 − CB1 − CB3 = μh2,
− d
dz
h0 − i(ϵ − Ez)h2 = μE2, Ch1 + Ch3 = μB2.
First, we resolve the subsystem of 6 equations
−i(ϵ − Ez)h − d
dz
E2 + CE1 − CE3 = μh0,
d
dz
h + i(ϵ − Ez)E2 − CB1 − CB3 = μh2,
Ch0 − i(ϵ − Ez)h1 = μE1,
−Ch0 − i(ϵ − Ez)h3 = μE3, (21)
−Ch2 +
d
dz
h3 = μB1, − d
dz
h1 − Ch2 = μB3;
as algebraic one with respect to the variables
h0, h2,E1,E3,B1,B3. This results in (let dz =
d
dz
)
h0 =
dzE2μ − i(Ch1 − Ch3 + hμ)(Ez − ϵ)
2C2 − μ2 ,
h2 =
−dz(Ch1 − Ch3 + hμ) + iE2μ(Ez − ϵ)
2C2 − μ2 ,
E1 =
1
2C2μ − μ3
(
CdzE2μ + i(h1(C − μ)(C + μ)+
+C(Ch3 − hμ))(Ez − ϵ)
)
, (22)
E3 =
1
μ3 − 2C2μ
(
CdzE2μ − i
(
C2h1 + Chμ+
+h3(C − μ)(C + μ)
)
(Ez − ϵ)
)
,
B1 =
1
μ3 − 2C2μ
(
−dz
(
C2h1 + Chμ+
+h3(C − μ)(C + μ)
)
+ iCE2μ(Ez − ϵ)
)
,
B3 =
1
μ3 − 2C2μ
(
dz (h1(C − μ)(C + μ)+
+C(Ch3 − hμ)) + iCE2μ(Ez − ϵ)
)
.
Now substitute these expressions into remaining 5 equations
−i(ϵ − Ez)h0 − d
dz
h2 + Ch1 − Ch3 = μh,
− d
dz
h0 − i(ϵ − Ez)h2 = μE2,
Ch1 + Ch3 = μB2, (23)
−Ch + CB2 − d
dz
B3 + i(ϵ − Ez)E1 = μh1,
Ch + CB2 +
d
dz
B1 + i(ϵ − Ez)E3 = μh3.
As a result, we obtain
1
d2z
hμ
2C2 − μ2 +
Cd2z
h1
2C2 − μ2
− Cd2z
h3
2C2 − μ2+
+μ
(
(ϵ − Ez)2
2C2 − μ2
− 1
)
h +
(
C(ϵ − Ez)2
2C2 − μ2 + C
)
h1+
+Ch3
(
(ϵ − Ez)2
μ2 − 2C2
− 1
)
= 0;
2
C2d2z
h3
2C2μ − μ2 +
d2z
h1(μ2 − C2)
μ3 − 2C2μ
+
+
Cd2z
h
μ2 − 2C2 + B2C − C2h3(ϵ − Ez)2
μ3 − 2C2μ
+
+
(
−(μ2 − C2)(ϵ − Ez)2
2C2μ − μ3
− μ
)
h1+
+C
(
(ϵ − Ez)2
μ2 − 2C2
− 1
)
h = 0;
3
C2d2z
h1
2C2μ − μ3 +
d2z
h3(μ2 − C2)
μ3 − 2C2μ
+
+
Cd2z
h
2C2 − μ2 + B2C +
C2h1(ϵ − Ez)2
2C2μ − μ3 +
+h3
(
−(μ2 − C2)(ϵ − Ez)2
2C2μ − μ3
− μ
)
+
+
(
C(ϵ − Ez)2
2C2 − μ2 + C
)
h = 0;
4
[
d2
dz2 + (ϵ − Ez)2 − μ2 − 2C2
]
E2 = 0;
5 − μB2 + Ch1 + Ch3 = 0.
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With the use of equation 5, from equations 2 and 3 we can
eliminate the variable B2. In this way we obtain the system
of 3 equations for variables h, h1, h3
1 μ2(d2z
− 2C2 + μ2 + (ϵ − Ez)2)h+
+Cμ(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h1−
−Cμ(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h3 = 0, (24)
2 Cμ(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h+
+(μ2 − C2)(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h1−
−C2(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h3 = 0, (25)
3 Cμ(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h+
+C2(d2z
+ 2C2 − μ2 + (ϵ − Ez)2)h1−
−(μ2 −C2)(d2z
+2C2 −μ2 +(ϵ−Ez)2)h3 = 0. (26)
The structure of these equations may be presented shortly as
follows
1 A1h′′ + B1h + C1h′′
1+
+D1h1 +M1h′′
3 + N1h3 = 0,
2 A2h′′ + B2h + C2h′′
1+
+D2h1 +M2h′′
3 + N2h3 = 0,
3 A3h′′ + B3h + C3h′′
1+
D3h1 +M3h′′
3 + N3h3 = 0. (27)
We will combine these equations in three different ways.
The first variant is
(aA1 + bA2 + cA3)−1h′′ + (aB1 + bB2 + cB3)h+
+(aC1 + bC2 + cC3)−0h′′
1+
+(aD1 + bD2 + cD3)h1+
+(aM1 + bM2 + cM3)−0h′′
3+
+(aN1 + bN2 + cN3)h3 = 0.
This results in
h′′ + (aB1 + bB2 + cB3)h + (aD1 + bD2 + cD3)h1+
+(aN1 + bN2 + cN3)h3 = 0,
where a, b, c obey the linear system
aA1 + bA2 + cA3 = 1,
aC1 + bC2 + cC3 = 0, (28)
aM1 + bM2 + cM3 = 0;
its solution is
a =
1
μ2 − 2C2 , b =
C
2C2μ − μ3 , c =
C
2C2μ − μ3 .
The second variant is
(aA1 + bA2 + cA3)−0h′′ + (aB1 + bB2 + cB3)h+
+(aC1 + bC2 + cC3)−1h′′
1+
+(aD1 + bD2 + cD3)h1+
+(aM1 + bM2 + cM3)−0h′′
3+
+(aN1 + bN2 + cN3)h3 = 0.
This results in
h′′
1 + (aB1 + bB2 + cB3)h + (aD1 + bD2 + cD3)h1+
+(aN1 + bN2 + cN3)h3 = 0,
where a, b, c obey the linear system
aA1 + bA2 + cA3 = 0,
aC1 + bC2 + cC3 = 1, (29)
aM1 + bM2 + cM3 = 0;
its solution is
a =
C
2C2μ − μ3 , b =
1
μ2 − 2C2 , c = 0.
The third variant is
(aA1 + bA2 + cA3)−0h′′ + (aB1 + bB2 + cB3)h+
+(aC1 + bC2 + cC3)−0h′′
1+
+(aD1 + bD2 + cD3)h1+
+(aM1 + bM2 + cM3)−1h′′
3+
+(aN1 + bN2 + cN3)h3 = 0.
This results in
h′′
3 + (aB1 + bB2 + cB3)h + (aD1 + bD2 + cD3)h1+
+(aN1 + bN2 + cN3)h3 = 0,
where a, b, c obey the linear system
aA1 + bA2 + cA3 = 0,
aC1 + bC2 + cC3 = 0, (30)
aM1 + bM2 + cM3 = 1;
its solution is
a =
C
μ3 − 2C2μ
, b = 0, c =
1
2C2 − μ2 .
So after this transformation we get three second-order separate
equations
d2
dz2 h + (2C2 + μ2 + (ϵ − Ez)2)h = 0,
d2
dz2 h1 + (2C2 − μ2 + (ϵ − Ez)2)h1 − 2Chμ = 0,
d2
dz2 h3 +(2C2 −μ2 +(ϵ−Ez)2)h3 +2Chμ = 0. (31)
Let us introduce new variables
H = h1 + h3, G = h1 − h3. (32)
Then instead of (31) we can obtain one separate equation
[
d2
dz2 + 2C2 − μ2 + (ϵ − Ez)2
]
H = 0 (33)
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43
and one subsystem
[
d2
dz2 + 2C2 + μ+ + (ϵ − Ez)2
]
h = 0,
[
d2
dz2 + 2C2 + μ2 + (ϵ − Ez)2
]
G−
−2μ2G − 4μCh = 0.
The last subsystem can be presented in the matrix form
D
(
h
G
)
= 2μ
(
0 0
2C μ
)(
h
G
)
,
DΨ = 2μAΨ. (34)
Let us find transformation which diagonalizes the mixing matrix
A
¯Ψ
= SΨ, D¯Ψ = 2μ(SAS−1)¯Ψ, ¯Ψ =
(
¯h
¯G
)
.
For transformation matrix S we derive the following equations
SA = A¯S, A¯ =
(
λ1 0
0 λ2
)
,
(
s11 s12
s21 s22
)(
0 0
2C μ
)
=
(
λ1 0
0 λ2
)(
s11 s12
s21 s22
)
,
whence it follows
s122C = λ1s11, s12μ = λ1s12,
(
λ1 −2C
0 (λ1 − μ)
)(
s11
s12
)
= 0,
s222C = λ2s21, s22μ = λ2s22,
(
λ2 −2C
0 (λ2 − μ)
)(
s21
s22
)
= 0.
The first row is specified by relations λ1 = 0, s12 =
0, s11 = 1; the second row is specified as λ2 = μ, s22 =
1, s21 = 2C/μ. Thus, the needed transformation matrix S
is
S =
(
1 0
2C
μ 1
)
, S−1 =
(
1 0
−2C
μ 1
)
. (35)
Therefore, we derive three separate equations:
[
d2
dz2 + 2C2 + μ2 + (ϵ − Ez)2
]
¯h= 0, (
36)
[
d2
dz2 + 2C2 − μ2 + (ϵ − Ez)2
]
¯G
= 0, (37)
[
d2
dz2 + 2C2 − μ2 + (ϵ − Ez)2
]
¯H
= 0, (38)
where
¯h
= h, ¯H = H = h1 + h3,
G = h1 − h3, ¯G =
2C
μ
h + h1 − h3.
Besides we should remember the existence of the fourth independent
equation for the variable E2:
[
d2
dz2
− 2C2 − μ2 + (ϵ − Ez)2
]
E2 = 0. (39)
So, in total, four independent types of solutions exist for
Stueckelberg particle in the external uniform electric field,
in contrast to the ordinary spin 1 particle described by the
Daffin-Kemmer equation when only three independent solutions
are possible. All four equations (36)–(39) have the same
mathematical structure. In the papers [7, 8], solutions for
equation of the form (39) were constructed in terms of the
confluent hypergeometric functions.
The authors declare no conflict of interest.
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7. Ivashkevich, A. V. Vector particle with anomalous magnetic moment in external uniform electric field / A. V. Ivashkevich, Ya. A. Voinova // Proceedings of the Komi Science Centre of the Ural Branch of the Russian Academy of Sciences. Series “Physical and Mathematical Sciences”. – 2022. – № 5 (57). – P. 51–59.
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