The simple solution of relativistic wave equations for charged particles in constant electric field and pair production
We obtained the generalization of the simple solution of Dirac equations for the electron in constant electric field on the case of the Kemmer-Duffin-Proca equations for the charged vector boson with arbitrary gyromagnetic ratio g moving in constant electric field. The pair production of spin 1/2 an...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production / D.J. Cirilo Lombardo, Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 182-184. — Бібліогр.: 8 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859666664349499392 |
|---|---|
| author | D.J. Cirilo Lombardo Stepanovsky, Yu.P. |
| author_facet | D.J. Cirilo Lombardo Stepanovsky, Yu.P. |
| citation_txt | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production / D.J. Cirilo Lombardo, Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 182-184. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | We obtained the generalization of the simple solution of Dirac equations for the electron in constant electric field on the case of the Kemmer-Duffin-Proca equations for the charged vector boson with arbitrary gyromagnetic ratio g moving in constant electric field. The pair production of spin 1/2 and spin 1 particles is discussed.
|
| first_indexed | 2025-11-30T11:27:38Z |
| format | Article |
| fulltext |
THE SIMPLE SOLUTION OF RELATIVISTIC WAVE EQUATIONS
FOR CHARGED PARTICLES IN CONSTANT ELECTRIC FIELD
AND PAIR PRODUCTION
D.J. Cirilo Lombardoa,, Yu.P. Stepanovskyb
aBuenos Aires University, Buenos Aires, Argentina
e-mail: diegoc@iafe.uba.ar
bNational Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: yustep@kipt.kharkov.ua
We obtained the generalization of the simple solution of Dirac equations for the electron in constant electric field
on the case of the Kemmer-Duffin-Proca equations for the charged vector boson with arbitrary gyromagnetic ratio g
moving in constant electric field. The pair production of spin 1/2 and spin 1 particles is discussed.
PACS: 03.65.Pm, 12.20.-m
1. INTRODUCTION
There is a well known sad confession of A.Einstein:
“All the fifty years of conscious brooding have brought
me no closer to the answer to the question "what are
light quanta?" Of course, today, every rascal thinks he
knows the answer, but he is deluding himself.”
A.I. Akhieser has taught his pupils do not change
into such rascals and to distinguish understanding from
a lack of understanding or from misunderstanding.
A.I. Akhieser himself never was afraid of showing that
he did not understand something and always took great
pains to comprehend incomprehensible, and he disliked
deeply those “human individuals who are able to fool us
for a while into believing that they possess some
understanding, when it finally emerges that indeed they
possess none whatever!”, [1].
A.I. Akhieser was passionately fond of science. He
was not the man, who took to science "out of a joyful
sense of superior intellectual power" or the man who
"had offered the products of his brain on the altar for
purely utilitarian purposes". The driving force of
scientific research of A.I. Akhieser was "the devoted
striving to comprehend a portion, be it ever so tiny, of
the Reason that manifests itself in nature." (All
quotations are taken from A. Einstein, who was the
most favourite hero to A.I. Akhieser). A.I. Akhieser had
wide interests in all theoretical physics, but quantum
electrodynamics was the subject of permanent fixed
attention of him. One of us (Yu. S.) has had frequently
the lucky opportunities to discuss with A.I. Akhieser a
lot of problems of quantum electrodynamics, such, for
instance, as relativistic wave equations for arbitrary spin
particles, the equivalence of different forms of
relativistic wave equations, the description of states of
polarizations of arbitrary spin particles, the different
exact solutions of Dirac equation for electron, the
production of particles and antiparticles by constant
electric fields, black hole evaporation, and others. This
report is the feeble echo of those discussions.
2. SIMPLE SOLUTION
OF DIRAC EQUATIONS
Let us consider the Dirac equations for the electron
moving in an electric field E with components (0, 0, E ).
Taking the no vanishing component of vector potential
A3 = (z-t)/2, scalar potential ϕ = – E(z-t)/2, and using the
notations e = | e |, z-t = ς, z+t = η, we obtain
( ) ( ) ( )
.0]
2
)2(
2
[
4321
434343
=ψ+γγ−−γ
∂
∂+γ
∂
∂+
γ−γζ+γ+γ
ζ∂
∂+γ−γ
η∂
∂
m
m
eEg
yx
ieEiii
(1)
(Other notations are as in the book [2], g is gyro-
magnetic factor for the electron). Put ψ in the form
( )ζΦ=ψ +ηΠ yipie 2 . Then the equations (1) become
( ) ( ) ( )
.0]
2
)2(
2
[
4322
434343
=Φ+γγ−−γ+
γ−γζ+γ+γ
ζ∂
∂+γ−γΠ
m
m
eEgip
ieEiiii
(2)
If we introduce the new variable
)2/(/2 ζ+Π=ξ eEeE and put on ψ the
polarization condition Φ±=Φγγ 15i the Eqs. (2)
reduce to
,0]~~
22
[ 243
4343 =Φ+γγ+ξ
γ−γ+
ξ
γ+γ mpii
d
di
(3)
where
.2/~,2/]2/)2([~
22 eEmmeEmeEgpp =−−±=
Let us introduce now two bispinors that are determined
by the conditions
+−++++
γ−γ
=±=γγ=γγ uiuuuiuui
2
;, 43
1534 (4)
and put ( ) ( ) ( ) ,−+ ξχ+ξφ=ξΦ uu introducing two
new scalar functions, φ and χ. From (3) we obtain the
182 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 182-184.
equations
,0)~~2( 23 =Ψ++ξ+
ξ
−+ mpiSiS
d
dS (5)
where
χ
φ
=Ψ and S+, S− and S3 are the usual spin
matrices for spin ½. We can rewrite (5) in the form
,0
~~
~~
2
2 =
χ
φ
−ξ
ξ
+
pimi
d
dpim
(6)
and finally we obtain the simple solution of the Dirac
equations (1)
( ) ( ) ( )
( ) .
2
2/)2(2
]28
22
2
22
2
2
2
2
[2
+
ξ
−−±−×
ξ=ψ
−+
−+−
+
−+ηΠ
uu
eE
meEgpimi
e m
eEg
m
pg
eE
pm
iyipi
(7)
Replacing in the solution (7) Π by Π=(p3−ε)/2, where
2
3
2
2
2 ppm ++=ε , we can verify after some
simple calculations that in the limit eE → 0, (ξ≠0) our
solution (7) passes into ordinary plane wave solution
,32 ue zipyipti ++ε−=ψ (8)
where u satisfies the equations
( ) .043322 =+ε γ−γ+γ umipip (9)
The solution (7) for g = 2 is well known [3,4]. It can
also be obtained as the generalization of the solution,
obtained in [5,6]. But our method of derivation of this
solution may be easily generalized on the case of the
more complex problem of the charged vector boson
with arbitrary g moving in constant electric field.
3. SIMPLE SOLUTION OF KEMMER-
DUFFIN-PROCA EQUATIONS
There are very many ways to describe vector boson.
The corresponding equations were invented by A. Proca
(1938), R.J. Duffin (1938), N. Kemmer (1939),
W. Pauli and M. Firz (1939), V, Bargman and
E.P. Wigner (1948), H.S. Green (1949) (see the book
[6] for review). All these equations are equivalent one to
another if the vector boson is free. But it is not the case
for vector boson interacting with electromagnetic field.
Let us consider the Kemmer-Duffin equations (that
are completely equivalent to the Proca equations) for the
negative charged vector boson with arbitrary
gyromagnetic ratio g in the form similar to (1),
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) .0}2])()[(
2
)1(][][
[
2
][][{
2
43
1
43
2
2
1
2
2
1
1
1
2
43
1
43
2
43
1
43
2
43
1
43
=ψ+γγ+γγ−−γ+γ
∂
∂+γ+γ
∂
∂+
γ−γ+γ−γζ+γ+γ+γ+γ
ζ∂
∂+γ−γ+γ−γ
η∂
∂
m
m
eEg
yx
iieEiiiii
(10)
Here ψ is symmetrical spin-tensor, ψαβ = ψβα, α,β = 1,2,3,4, and the Dirac matrices with superscript (1) and (2) act on
the first or second indices of ψαβ correspondingly. After transformations similar to the Dirac case we obtain instead
of (3)
( ) ( ) ( ) ( )
( ) ( ) ,0}~2~])()[(
]
22
[]
22
{[
2
2
43
1
43
2
43
1
43
2
43
1
43
=Φ+γγ+γγ+
ξ
γ−γ
+
γ−γ
+
ξ
γ+γ
+
γ+γ
mp
iii
d
dii
(11)
where
.2/~,2/]2/)1([~
22 eEmmeEmeEgpp =−−±=
Let us introduce now three quantities u+ , u0, and u−
that are determined by the conditions
( ) ( )
( ) ( )
( ) ( )
0
2
43
1
43
2
43
1
43
0
2,1
15
2,1
34
]
22
[
,]
22
[
;)(,)(
uiiu
u
ii
u
uuiuui
γ−γ
+
γ−γ
=
γ−γ
+
γ−γ
=
±=γγ=γγ
−
+
++++
(12)
and put
( ) ( ) ( ) ( ) ,0 −+ ξθ+ξχ+ξφ=ξΦ uuu
introducing three new scalar functions, φ, χ and θ. From
(11) we obtain the equations
,0)~2~2( 23 =Ψ++ξ+
ξ
−+ mpiSiS
d
dS (13)
where
θ
χ
φ
=Ψ and S+, S− and S3 are the usual spin
matrices for spin 1. We can rewrite (13) in the form
183
0
~~
2
10
2
1~
2
1
0
2
1~~
2
2
=
θ
χ
φ
−ξ
ξ
ξ
ξ
+
pmi
d
dmi
d
dpim
(14)
and finally we obtain the simple solution of the
Kemmer-Duffin equations (10)
( ) ( ) ( ) ( )
( )
( )
.
2/)1(
]
22
2/)1([
1
2
0
2
]28
31
2
22
2
2
2
2
[2
−−±−
ξ−+
ξ
+
ξ
−−±−
×
ξ
ξ
=ψ
−
+
−−+−
+
−+ηΠ
u
meEgpim
eEiu
u
m
eE
eE
meEgpimi
e m
eEgg
m
pg
eE
pm
iyipi
(15)
Of course, one should to find the solution for the third
state of polarization of vector boson that is determined
by condition
( ) ( ) 0])()[( 2
15
1
15 =ψγγ+γγi .
This solution must be similar to (15) but to obtain it one
needs to make somewhat more complicated calculations
than in considered case.
Replacing in the solution (15) Π by Π=(p3−ε)/2,
where 2
3
2
2
2 ppm ++=ε , we can verify after some
calculations that our solution (15) in the limit eE → 0,
(ξ≠0) also passes into ordinary plane wave solution
,32 ue zipyipti ++ε−=ψ (16)
where u satisfies the equations similar to (9)
( ) ( ) ( )( ) .02,1
4
2,1
33
2,1
22 =+ε γ−γ+γ umipip (17)
4. THE DEPENDENCE OF PAIR
PRODUCTION FROM POLARIZATIONS
The solutions (7) and (15) have the remarkable
features, they have branching point at ξ=0. At this point
the value of the wave function amplitude springs and
multiplies on the factor
( ) ( ) ]
8
2
2
2
2
[ 2
22
2
2
2
m
eEg
m
pg
eE
pm
e
−+−+π− (18)
for electron and on the factor
( ) ( ) ( ) ]
8
31
2
2
2
[ 2
2
2
2
2
m
eEgg
m
pg
eE
pm
e
−−+−+π− (19)
for vector boson. It is known that [4,5] that the squares
of these factors determine the probabilities of pair
production by constant electric field. We see that these
probabilities do not depend from the polarizations of the
particles only if g = 2. (It agrees with the results of [8]
where the case of charged vector boson with g = 2 is
considered).
This dependence from the polarizations is due to the
fact that if the classical magnetic dipole µ moves and
the direction of motion of magnetic dipole is orthogonal
to its orientation then the electric dipole d = µp/m arises.
This electric dipole interacts with electric field, and this
interaction depends from the magnetic dipole
orientation. We see that only (g-2) part of quantum
magnetic dipole manifests itself as the classical
magnetic dipole.
ACKNOWLEDGMENTS
We are very grateful to A.I. Nikishov who kindly
drew our attention on the fact that the simple solution of
Dirac equations for g = 2 is well known. We are very
obliged to G.N. Afanasiev whose great interest to the
moving magnetic dipole suggested our explanation of
the physical reason of dependence of pair production by
electric field from polarizations. One of us, (D. C.),
thanks deeply the people of the Institute for Theoretical
Physics for great hospitality during his visit in Kharkov
Institute of Physics and Technology.
REFERENCES
1. R. Penrose. Shadows of the mind: a search
for the missing science of consciousness. London:
“Vintage”, 1995, 458 p.
2. A.I. Akhiezer, V.B. Berestecky. Quantum
electrdynamics. Moscow: “Nauka”, 1981, 432 p.
3. N.B. Narozhny, A.I. Nikishov. The solutions
of Klein-Gordon and Dirac equations for the
particle in constant electric field and in the field of
the plane electromagnetic wave propagating along
the electric field // Theor. Mat. Fis. 1976, v. 26,
p. 16-34 (in Russian).
4. N.B. Narozhny, A.I. Nikishov. The particle in
constant electric field and in the field of the plane
electromagnetic wave propagating along the
electric field // Trudy FIAN. 1986, v. 168, p. 175-
199 (in Russian).
5. A. Jannussis. Die relativistische Bewegung
eines Electrons im äußeren homogenen Magnetfeld
// Z. Phys. 1966, v. 190, p. 99-109.
6. A.S. Bakai, Yu.P. Stepanovsky. Adiabatic
invariants. Kiev: “Naukova dumka”, 1981, 284 p.
(in Russian).
7. E.M. Corson. Introduction to tensors,
spinors, and relativistic wave equations. London
and Glasgow “Blackie & Son Limited”, 1953,
221 p.
8. A.I. Nikishov. Vector boson in constant
electromagnetic field. hep-th/0104019, 20 p.
184
OF DIRAC EQUATIONS
Let us consider the Dirac equations for the electron moving in an electric field E with components (0, 0, E ). Taking the no vanishing component of vector potential A3 = (z-t)/2, scalar potential = – E(z-t)/2, and using the notations e = | e |, z-t = , z+t = , we obtain
4. THE DEPENDENCE OF PAIR PRODUCTION FROM POLARIZATIONS
ACKNOWLEDGMENTS
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79481 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T11:27:38Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | D.J. Cirilo Lombardo Stepanovsky, Yu.P. 2015-04-02T15:30:35Z 2015-04-02T15:30:35Z 2001 The simple solution of relativistic wave equations for charged particles in constant electric field and pair production / D.J. Cirilo Lombardo, Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 182-184. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 03.65.Pm, 12.20.-m https://nasplib.isofts.kiev.ua/handle/123456789/79481 We obtained the generalization of the simple solution of Dirac equations for the electron in constant electric field on the case of the Kemmer-Duffin-Proca equations for the charged vector boson with arbitrary gyromagnetic ratio g moving in constant electric field. The pair production of spin 1/2 and spin 1 particles is discussed. We are very grateful to A.I. Nikishov who kindly drew our attention on the fact that the simple solution of Dirac equations for g = 2 is well known. We are very obliged to G.N. Afanasiev whose great interest to the moving magnetic dipole suggested our explanation of the physical reason of dependence of pair production by electric field from polarizations. One of us, (D. C.), thanks deeply the people of the Institute for Theoretical Physics for great hospitality during his visit in Kharkov Institute of Physics and Technology. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields The simple solution of relativistic wave equations for charged particles in constant electric field and pair production Простое решение релятивистских волновых уравнений для заряженных частиц в постоянном электрическом поле и рождение пар Article published earlier |
| spellingShingle | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production D.J. Cirilo Lombardo Stepanovsky, Yu.P. Electrodynamics of high energies in matter and strong fields |
| title | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production |
| title_alt | Простое решение релятивистских волновых уравнений для заряженных частиц в постоянном электрическом поле и рождение пар |
| title_full | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production |
| title_fullStr | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production |
| title_full_unstemmed | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production |
| title_short | The simple solution of relativistic wave equations for charged particles in constant electric field and pair production |
| title_sort | simple solution of relativistic wave equations for charged particles in constant electric field and pair production |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79481 |
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