Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin
We consider the Vavilov-Cherenkov radiation by a particle with arbitrary spin using the relativistic quantum treatment of the one-photon Cherenkov emission. The probability of the one-photon emission by relativistic particle is calculated in the framework of quantum electrodynamics using the covaria...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2007 |
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| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin / G.N. Afanasiev, M.V. Lyubchenko, and Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 153-155. — Бібліогр.: 18 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860106457388679168 |
|---|---|
| author | Afanasiev, G.N. Lyubchenko, M.V. Stepanovsky, Yu.P. |
| author_facet | Afanasiev, G.N. Lyubchenko, M.V. Stepanovsky, Yu.P. |
| citation_txt | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin / G.N. Afanasiev, M.V. Lyubchenko, and Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 153-155. — Бібліогр.: 18 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | We consider the Vavilov-Cherenkov radiation by a particle with arbitrary spin using the relativistic quantum treatment of the one-photon Cherenkov emission. The probability of the one-photon emission by relativistic particle is calculated in the framework of quantum electrodynamics using the covariant parameterization of electromagnetic current for particle with arbitrary spin.
Розглянуте випромінювання Вавілова-Черенкова частинкою з довільним спіном. Iмовірність випромінювання одного фотона релятивістською частинкою обчислена в рамках квантової електродинаміки з використанням коваріантної параметризації матричних елементів електромагнітного струму частинки з довільним спіном.
Рассмотрено излучение Вавилова-Черенкова частицей с произвольным спином. Вероятность излучения одного фотона релятивистской частицей рассчитана в рамках квантовой электродинамики с использованием ковариантной параметризации матричных элементов электромагнитного тока частицы с произвольным спином.
|
| first_indexed | 2025-12-07T17:31:56Z |
| format | Article |
| fulltext |
QUANTUM THEORY OF THE VAVILOV-CHERENKOV RADIATION
FOR A PARTICLE WITH ARBITRARY SPIN
G.N. Afanasiev1, M.V. Lyubchenko2, and Yu.P. Stepanovsky3
1Joint Institute for Nuclear Research, Dubna, Russia;
e-mail: afanasev@thsun1.jinr.ru;
2V.N. Karazin National University, Kharkov, Ukraine;
e-mail: rosomasha@yandex.ru;
3National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: yustep@kipt.kharkov.ua
We consider the Vavilov-Cherenkov radiation by a particle with arbitrary spin using the relativistic quantum
treatment of the one-photon Cherenkov emission. The probability of the one-photon emission by relativistic particle
is calculated in the framework of quantum electrodynamics using the covariant parameterization of electromagnetic
current for particle with arbitrary spin.
PACS: 13.40.Gp, 41.60.-m
1. INTRODUCTION
The theory for the Vavilov-Cherenkov radiation
(VCR) of electrical and magnetic multipoles has a long
history. The VCR radiation of a magnetic dipole and of
electric and magnetic dipoles was first considered by
Ginzburg (1940) and Frank (1942) [1,2]. In 1952 ap-
peared two other publications by Ginzburg and Frank
on the same subject [3,4], and they were followed on by
their publications [5,6] in 1984. In detail this subject
was considered in the books by Ginzburg and Frank
[7,8] (Ginzburg regarded electric and magnetic dipoles
only, Frank considered VCR of arbitrary electrical and
magnetic multipoles). Note also that the exact electro-
magnetic fields and the VCR of electric, magnetic and
toroidal dipoles moving uniformly in unbounded non-
dispersive medium was considered in [9,10].
To calculate the probability of VCR for particle of
arbitrary spin S we take a relativistic expression for
electromagnetic current through “physical” form factors
and [11-13]. The expression for elec-
tromagnetic current mentioned above is based essen-
tially on using Chebyshev polynomials of a discrete
variable [14,15].
)( 2qQl )( 2qM l
2. THE COVARIANT PARAMETERIZATION
OF ELECTROMAGNETIC CURRENT
FOR PARTICLE WITH ARBITRARY SPIN
The electrical and magnetic multipole momenta are
proportional to completely symmetrical traceless “mul-
tipole” tensors ( ): liiiS ...21 Sl 2≤
( ) ( )
( ) ( )
;
!!2
!2!2
...2... 2121 ll iiiliii S
lS
lSlQQ −
= (1)
( ) ( )
( ) ( )
,
!!2
!2!2
...2... 2121 ll iiiliii S
lS
lSlMM −
= (2)
where Ql and Ml are equal to averaged quantities of
Qz…z and Mz…z in state with spin S projection on z axis.
Tensors may be unambiguously expressed
through spin operators
liiiS ...21
iS
,liklikki SiSSSS ε=− ( )ISSSS ii 1+= (3)
up to factor, that we fix by the next condition: the con-
struction represents i-power polyno-
mial of with the unit coefficient by ( , is
arbitrary vector.
ll iiiiii qqqS ...2121 ...
)qS( lqS ) q
It is known [13], that
( ) ( aSaaaS liiiiiil
l
ll ϕ=...2121 ...!
!2 ) , (4)
where ai is unit vector, are Chebyshev polynomi-
als.
( )xlϕ
Acting by the polynomial on the wave func-
tion of a particle with spin S and its projection m on
direction , for eigenvalue we obtain Chebyshev poly-
nomial
)( aSlϕ
a
(ml )ϕ of discrete variable m:
( ) ( ) ( ) ( ).,, mSvmmSvaS ll ϕϕ = (5)
Polynomials are different for different values of spin S,
i.e. by ( )mlϕ we always mean ( )Sml ,ϕ .
Note, that Chebyshev polynomials ( )mlϕ are not so
popular as famous Chebyshev polynomials T and
. In view of it let’s consider some properties of
polynomials
( )xn
( )xU n
( )mlϕ .
A notation ( )mlϕ was introduced by Chebyshev
[14], but in modern mathematical literature [15] instead
of them a little different polynomials are used:
( ) ( ).!
1 mmSt lll ϕ=+ (6)
For some purposes polynomials are more con-
venient,
( )mpl
( ) ( ) ( )
( ) ( ).!12
12!2
!
1 mmSp llS
llS
ll ϕ++
+−=+ (7)
Recurrence expressions for Chebyshev polynomials are:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p.153-155. 153
( ) ( ) ( ) ( )[ ] ( )12212 1
222
1 mlSlmmlm lll −+ −+−+= ϕϕϕ .
Orthogonality relations are:
( ) ( ) ;llll
S
Sn
mpmp ′′
−=
=∑ δ
( ) ( ) .
2
0
mmll
S
l
mpmp ′
=
=′∑ δ
The trace of two Chebyshev polynomials product is
( ) ( )[ ] ( ),baPbSpaSpSp lllll ′′ = δ (8) (Ql M
where
( ) ( )l
dx
d
ll xxP l
l
l 12
!2
1 −=
is a Legendre polynomial. The relation with Klebsch-
Gordan coefficients is
( ) ( ) ,0|1, lmSSmmSp mS
l −−= − (9)
where
jmjjmmjjjmmmjj 1212121212 || ≡ (10) d
are Klebsch-Gordan coefficients, that defined so as, for
example, in [16].
First six Chebyshev polynomials have next explicit
form:
( ) ;10 =mϕ
( ) ;21 mm =ϕ
( ) ( )[ ];11212 22
2 −+−= Smmϕ
( ) ( )[ ] ;71236120 23
3 mSmm −+−=ϕ
( ) ( )[ ]
( )[ ] ( )[ ];9121129
131231201680
22
224
4
−+−++
−+−=
SS
mSmmϕ
( ) ( )[ ] 3255 712840030240 mSmm −+−=ϕ
( ) ( )[ ] .40712230121530 24 mSS ++−++
Now we can write an expression for matrix elements
of electromagnetic current in the case when a particle is
described by Bargman-Wigner equations [13]:
( )
( )[ ]Spp
pjpEE
12
1
1221
3 42
+−
=µπ
( ) ( )
( ) ( ) ( ) ( )( )µ12222!!2!2
!2 {(2 ppqQpuqi lll
llS
lSl
l
+× −∑
( ) ( ) ( )νµνρσερρ
122
2
2
12 ppqMlm
pp
l
i
q
qS
l ++
+−ϕ×
( )( 11} puq q
qS
lq
× ∂
∂ ρρ
σ
ϕρ ) , (11)
where q = p2 – p1,
( )
( )
;
2
12
12
2
1
pp
pp
i SS
+−
+= σ
µνρµνσρ ε (12)
)2()2()1( ... SS µνµνµνµν σσσ +++= ; (13)
i4
µννµ γγγγ
µνσ
−
= (14)
overhead index shows on which index of spin-tensor
( u ) or conjugate spin-tensor ( 11 pu ) )( 1...1 221
p
Sααα
( 22 pu ) ( ( 2
...
2
221 pSααα )u ) acts matrix .
and
µνσ ( )11 pu
(p )22u are normalized by condition:
( ) ( )( ) ( ) ( ) ( ) .2 2
...
...
221
221 Smpupupupu
S
S =≡ ααα
ααα
Spin-tensor satisfies Bargman-Wigner equations: ( )pu
.0)( 221 =+ Si
i umip αααααµµγ …… (15)
)2q and in (11) are “physical” form factors,
that are electric and magnetic multipole moments of the
particle in the Breit system of reference [11,12].
)( 2ql
3. VAVILOV-CHERENKOV RADIATION
FOR PARTICLE WITH ARBITRARY SPIN
The calculated probability of the photon radiation by
particle with arbitrary spin per unit of length l and per
frequency unit ω is:
,)1(
)};()4(
sin)]()({[
2
22
2222
1
22
1
4
222
2
22
−=
++
+=
c
nq
qMqcm
EV
c
qMqE
c
e
ddl
w
ω
θ
ω
(16)
where V1 and E1 are the initial particle velocity and en-
ergy, θ is the angle of photon emission relative to direc-
tion of motion of the initial particle, n is the medium
refractive index,
);(1)(
);()(
222
2
1
22
222
2
0
22
qMqA
l
lqMe
qQqAqEe
l
l
S
l
l
l
l
S
l
l
∑
∑
=
=
+
=
=
(17)
)12)(12(
)!2()!12(
])!2()!2[(
2
2
2
++
−++
=
lS
lSlS
lS
A
l
l . (18)
4. DISCUSSION
The theory developed by Frank [6] is the theory for
the Vavilov-Cherenkov radiation of arbitrary classical
multipoles. We investigated the Vavilov-Cherenkov
radiation of arbitrary quantum multipoles. The quantum
multipoles are the particles with arbitrary spin S and
with electrical and magnetic multipole momenta that are
described by some form factors. This paper is intrinsi-
cally connected with our papers [17,18], where the
probability of the one-photon emission by relativistic
electron was calculated in the framework of quantum
electrodynamics and the behavior of the radiation inten-
sities for large energy–momentum transfer was ana-
lyzed. It is known that “physical” form factors for elec-
tron are [11]
2
1
2
2
2
1
2
0
4
1)()(
−
+==
m
qeqMqQ .
154
Putting these form factors in Eq. (16) gives the result
obtained and analyzed in [17].
REFERENCES
1. V.L. Ginzburg. Quantum theory of light radiation by
electron uniformly moving in medium //ZhETF.
1940, v. 10, p. 589-600 (in Russian).
2. I.M. Frank. Doppler effect in refracting medium //Izv.
AN SSSR, ser. Fiz. 1942, v. 6, p. 3-31 (in Russian).
3. V.L. Ginzburg. On Cherenkov radiation of magnetic
dipole //In memory of S.I. Vavilov. Moskow: Izdat.
AN SSSR, 1952, p. 193-199 (in Russian).
4. I.M. Frank. Cherenkov radiation of multipoles //In:
To the memory of S.I. Vavilov. Moscow: Izdat. AN
SSSR, 1952, p. 172-192 (in Russian).
5. V.L. Ginzburg. On the fields and radiation of “true”
and current magnetic dipoles being in medium //Izv.
Vuz., ser. Radiofizika. 1984, v. 27, p. 852-872 (in
Russian).
6. I.M. Frank. Cherenkov radiation of electric and
magnetic multipoles //UFN. 1984, v 144, p. 251-275
(In Russian).
7. V.L. Ginzburg. Theoretical Physics and Astro-
physics. Supplementary Chapters. M.: “Nauka”,
1987, 488 p. (in Russian).
8. I.M. Frank. Vavilov-Cerenkov Radiation. Theore-
tical Aspects. M.: “Hauka”, 1988, 288 p. (in Rus-
sian).
9. G.N. Afanasiev, Yu.P. Stepanovsky. Electromag-
netic fields of electric, magnetic and toroidal dipoles
moving in medium //Physica Scripta. 2000, v. 61,
p. 704-716.
10. G.N. Afanasiev, Yu.P. Stepanovsky. Radiation of
electric, magnetic and toroidal dipoles uniformly
moving in an unbounded medium //Problems of
Atomic Science and Technology. Series "Nuclear
Physics Investigations" (40). 2002, N 2, p. 25-39.
11. M. Gourdin. Electromagnetic form factors //Nuovo
Cim. 1965, v. 36, p. 129-149.
12. M. Gourdin, J. Micheli. Electromagnetic form fac-
tors II //Nuovo Cim. 1965, v. 40A, p. 225-235.
13. I.K. Kirichenko, Yu.P. Stepanovsky. “Physical”
form factors and covariant parameterization of elec-
tromagnetic current for particle with arbitrary spin
//Yadernaya Fyzika. 1974, v. 20, p. 554-561.
14. P.L. Chebyshev. About one new series //Bull. de la
Classe phys.-mathém. de l’Acad. Imp. de sciences de
St.-Pétersbourg. 1859, v. XVII, p. 257-261.
15. H. Bateman. Higher transcendental functions. New
York, Toronto, London: “McGraw-Hill Book Com-
pany, Inc.”, 1953, v. II, 356 p.
16. A.R. Edmonds. Angular Momenta in Quantum Me-
chanics //Atomic Nucleus Deformation. M.:
‘’Inostrannaya Literatura’’, 1958, p. 305-352.
17. G.N. Afanasiev, M.V. Lyubchenko, Yu.P. Ste-
panovsky. Fine structure of the Vavilov–Cherenkov
radiation //Proc. Roy. Soc. A. 2006, v. 462, p. 689-
699.
18. G.N. Afanasiev, M.V. Lyubchenko, Yu.P. Step-
anovsky. Polarization properties of the Vavilov-
Cherenkov radiation //Problems of Atomic Science
and Technology. 2007, N. 3(1), p. 149-152.
КВАНТОВАЯ ТЕОРИЯ ИЗЛУЧЕНИЯ ВАВИЛОВА-ЧЕРЕНКОВА
ЧАСТИЦЕЙ С ПРОИЗВОЛЬНЫМ СПИНОМ
Г.Н. Афанасьев, М.В. Любченко, Ю.П. Степановский
Рассмотрено излучение Вавилова-Черенкова частицей с произвольным спином. Вероятность излучения
одного фотона релятивистской частицей рассчитана в рамках квантовой электродинамики с использованием
ковариантной параметризации матричных элементов электромагнитного тока частицы с произвольным спи-
ном.
КВАНТОВА ТЕОРІЯ ВИПРОМІНЮВАННЯ ВАВІЛОВА-ЧЕРЕНКОВА
ЧАСТИНКОЮ З ДОВІЛЬНИМ СПІНОМ
Г.Н. Афанасьєв, М.В. Любченко, Ю.П. Степановський
Розглянуте випромінювання Вавілова-Черенкова частинкою з довільним спіном. Iмовірність випроміню-
вання одного фотона релятивістською частинкою обчислена в рамках квантової електродинаміки з викорис-
танням коваріантної параметризації матричних елементів електромагнітного струму частинки з довільним
спіном.
155
|
| id | nasplib_isofts_kiev_ua-123456789-110933 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:31:56Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Afanasiev, G.N. Lyubchenko, M.V. Stepanovsky, Yu.P. 2017-01-07T08:57:49Z 2017-01-07T08:57:49Z 2007 Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin / G.N. Afanasiev, M.V. Lyubchenko, and Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 153-155. — Бібліогр.: 18 назв. — англ. 1562-6016 PACS: 13.40.Gp, 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/110933 We consider the Vavilov-Cherenkov radiation by a particle with arbitrary spin using the relativistic quantum treatment of the one-photon Cherenkov emission. The probability of the one-photon emission by relativistic particle is calculated in the framework of quantum electrodynamics using the covariant parameterization of electromagnetic current for particle with arbitrary spin. Розглянуте випромінювання Вавілова-Черенкова частинкою з довільним спіном. Iмовірність випромінювання одного фотона релятивістською частинкою обчислена в рамках квантової електродинаміки з використанням коваріантної параметризації матричних елементів електромагнітного струму частинки з довільним спіном. Рассмотрено излучение Вавилова-Черенкова частицей с произвольным спином. Вероятность излучения одного фотона релятивистской частицей рассчитана в рамках квантовой электродинамики с использованием ковариантной параметризации матричных элементов электромагнитного тока частицы с произвольным спином. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники High-energy electrodynamics in matter Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin Квантова теорія випромінювання вавілова-черенкова частинкою з довільним спіном Квантовая теория излучения вавилова-черенкова частицей с произвольным спином Article published earlier |
| spellingShingle | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin Afanasiev, G.N. Lyubchenko, M.V. Stepanovsky, Yu.P. High-energy electrodynamics in matter |
| title | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin |
| title_alt | Квантова теорія випромінювання вавілова-черенкова частинкою з довільним спіном Квантовая теория излучения вавилова-черенкова частицей с произвольным спином |
| title_full | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin |
| title_fullStr | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin |
| title_full_unstemmed | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin |
| title_short | Quantum theory of the Vavilov-Cherenkov radiation for a particle with arbitrary spin |
| title_sort | quantum theory of the vavilov-cherenkov radiation for a particle with arbitrary spin |
| topic | High-energy electrodynamics in matter |
| topic_facet | High-energy electrodynamics in matter |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110933 |
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