On 2D massless field wave equations
The 2D wave equations of massless fields with arbitrary spin are obtained by application of the Extended Little Lorentz Group. The possibility of the existence of an interaction of charged 2D massless fields of spin one or higher with the external electromagnetic field is investigated. С помощью рас...
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| Datum: | 2012 |
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Institute of Problems of Mechanical Engineering of the RAS
2012
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| Zitieren: | On 2D massless field wave equations / I.E. Ovcharenko, Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 53-54. — Бібліогр.: 7 назв. — англ. |
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| author | Ovcharenko, I.E. Stepanovsky, Yu.P. |
| author_facet | Ovcharenko, I.E. Stepanovsky, Yu.P. |
| citation_txt | On 2D massless field wave equations / I.E. Ovcharenko, Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 53-54. — Бібліогр.: 7 назв. — англ. |
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| description | The 2D wave equations of massless fields with arbitrary spin are obtained by application of the Extended Little Lorentz Group. The possibility of the existence of an interaction of charged 2D massless fields of spin one or higher with the external electromagnetic field is investigated.
С помощью расширенной малой группы Лоренца получены 2D волновые уравнения для безмассовых полей произвольного спина. Выяснено, возможно ли существование взаимодействия безмассовых частиц со спинами, равными или большими единицы, с электромагнитным полем.
За допомогою розширеної малої групи Лоренця одержані 2D хвильові рівняння для безмасових полів довільного спіну. З'ясовано, чи можливе існування взаємодії безмасових частинок зі спінами, рівними або більшими одиниці, з електромагнітним полем.
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| first_indexed | 2025-12-07T16:22:45Z |
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ON 2D MASSLESS FIELD WAVE EQUATIONS
I.E. Ovcharenko 1 and Yu.P. Stepanovsky 2∗
1Institute of Problems of Mechanical Engineering of the RAS, 199178, St.-Petersburg, Russia
2National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 31, 2011)
The 2D wave equations of massless fields with arbitrary spin are obtained by application of the Extended Little
Lorentz Group. The possibility of the existence of an interaction of charged 2D massless fields of spin one or higher
with the external electromagnetic field is investigated.
PACS: 03.65.Pm, 13.40.-f
1. INTRODUCTION
The great success in the study of fundamental elec-
tronic properties of two-dimensional graphene [1] re-
cently attracted attention to relativistic 2D massless
field wave equations. In application to the graphene,
usually only 2D Dirac-Weyl wave equations are con-
sidered but this restriction is not well founded and
we want to clarify the question whether or not this
restriction can be removed if possible. So our sub-
ject now is the investigation of the relativistic wave
equations of 2D massless fields with arbitrary spin.
2. LITTLE LORENTZ GROUP,
EXTENDED LITTLE LORENTZ GROUP
AND CORRESPONDING WAVE
EQUATIONS
In [2] we considered some properties of 2D Dirac-
Weyl equations for charged massless spin 1/2 par-
ticles. Wave equations in 3D case for massless ar-
bitrary spin particles can be obtained by applica-
tion of the Little Lorentz Group [3, 4] or by appli-
cation of the Extended Little Lorentz Group [5, 6].
The Little Lorentz Group [3] consists of the Lorentz
transformations leaving invariant particle momentum
(i = 1, 2, 3). The Extended Little Lorentz Group [5,6]
is the Little Lorentz Group enlarged by dilatations.
The Extended Little Lorentz Group includes Lorentz
boosts of the plane-wave solutions of massless arbi-
trary spin equations. The wave equations of massless
fields with arbitrary spin result from the very simple
assumption: the state of an elementary particle may
be completely described by its momentum [4]. In 3D
case (4-dimensional space-time) in both cases the ob-
tained wave equations are identical [5]. But this is
not the case for relativistic 2D wave equations, when
the Little Lorentz Group leads to the equations:
Si
∂
∂xi
ψ = 0 (1)
and the Extended Little Lorentz Group leads to the
equations: (
S
∂
∂xi
+ iεiklSl
∂
∂xk
)
ψ = 0 (2)
(Si are relativistic 2D spin operators, SiSi = S(S+1),
S is the value of spin, SiSk − SkSi = iεiklSl). These
equations are equivalent only in the Dirac-Weyl case
of S = 1/2. In this case the multiplication of (2)
by Si at once gives (1). The action on (2) by ∂/∂xi
immediately gives the Klein-Gordon equation
∂2
∂x2
i
ψ = 0,
i.e. equation (2) describes the massless particle
whereas equation (1) has only not massless solu-
tions.
3. CAN 2D MASSLESS PARTICLES BE
CHARGED?
It is well known that 3D massless particles of spin one
or higher cannot be charged owing to inconsistency
of corresponding wave equations [7]. It is of interest
to consider this problem of equations inconsistency
in 2D case. Let us consider charged massless particle
which is described by the equation:
(SDi + iεiklSlDk)ψ = 0, (3)
where
Di =
∂
∂xi
− ieAi,
and Ai is the relativistic 3-vector potential of the elec-
tromagnetic field. Note that
DiDk −DkDi = ieεiklFl, (4)
where relativistic 3-vector of electromagnetic field is
Fi =
1
2
εimn
(
∂Am
∂xn
− ∂An
∂xm
)
. (5)
∗Corresponding author E-mail address: yustep@kipt.kharkov.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 53-54.
53
Instead of Klein-Gordon equation we have now the
following equation:
(SD2
i − eSiFi)ψ = 0. (6)
The multiplication of (3) by Si gives
SiDiψ = 0. (7)
To verify the compatibility of the equations (7)
and (3) we have to verify that
[SjDj , (SDi + iεiklSlDk)]ψ = 0. (8)
After calculation we obtain the relation:
[SiD
2
j − SSie(SjFj) + e(SjFj)Si(1 + S)
−S(S + 1)eFi]ψ = 0. (9)
Excluding D2
j from (9) with the help of (6) we obtain
[(
1
S
−S)Si(SjFj)Si(1+S)−S(S+1)Fi]ψ = 0. (10)
The latter equation is physically meaningless be-
cause according to (10) the arbitrarily small elec-
tromagnetic field puts the perceptible influence on
the particle under consideration. Only in the case
of spin 1/2 the equation (10) turns to identity. So
we can conclude that 2D massless particles with spin
S > 1/2 cannot be charged.
4. CONCLUSIONS
We investigated the wave equations of 2D massless
fields with arbitrary spin obtained by application of
the Extended Little Lorentz Group. We showed that
2D massless particles of spin one or higher cannot be
charged due to inconsistency of corresponding mass-
less particles wave equations.
References
1. S.D. Sarma, S. Adam, E.H. Hwang. Electronic
transport in two-dimensional graphene // Rev.
Mod. Phys. 2011, v. 83, p. 407-475.
2. I.E. Ovcharenko, Yu.P. Stepanovsky. On some
properties of 2D Weyl equation for charged mass-
less spin-1/2 particle // Problems of Atomic Sci-
ence and Technology. 2007, v. 3 (1), p. 56-60.
3. E.P. Wigner. On unitary representations of the
inhomogeneous Lorentz group // Annals of
Math. 1939, v. 40, p. 149-204.
4. Yu.P. Stepanovsky. Little Lorentz group and
wave equations of free massless fields with arbi-
trary spin // Ukrainskij fizicheskij zhurnal. 1964,
v. 9, p. 1165-1168 (in Ukrainian).
5. Yu.P. Stepanovsky. On wave equations of mass-
less fields // Teoreticheskaja i matematicheskaja
fizika. 1981, v. 47, p. 343-351 (in Russian).
6. Y.S. Kim, E.P. Wigner. Covariant phase-space
representation for localized light waves // Phys.
Rev. 1987, v. A36, p. 1293-1297.
7. K.M. Case, S.G. Gasiorowicz. Can massless par-
ticles be charged? // Phys. Rev. 1962, v. 125,
p. 1055-1058.
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| id | nasplib_isofts_kiev_ua-123456789-106981 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:22:45Z |
| publishDate | 2012 |
| publisher | Institute of Problems of Mechanical Engineering of the RAS |
| record_format | dspace |
| spelling | Ovcharenko, I.E. Stepanovsky, Yu.P. 2016-10-10T15:50:04Z 2016-10-10T15:50:04Z 2012 On 2D massless field wave equations / I.E. Ovcharenko, Yu.P. Stepanovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 53-54. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 03.65.Pm, 13.40.-f https://nasplib.isofts.kiev.ua/handle/123456789/106981 The 2D wave equations of massless fields with arbitrary spin are obtained by application of the Extended Little Lorentz Group. The possibility of the existence of an interaction of charged 2D massless fields of spin one or higher with the external electromagnetic field is investigated. С помощью расширенной малой группы Лоренца получены 2D волновые уравнения для безмассовых полей произвольного спина. Выяснено, возможно ли существование взаимодействия безмассовых частиц со спинами, равными или большими единицы, с электромагнитным полем. За допомогою розширеної малої групи Лоренця одержані 2D хвильові рівняння для безмасових полів довільного спіну. З'ясовано, чи можливе існування взаємодії безмасових частинок зі спінами, рівними або більшими одиниці, з електромагнітним полем. en Institute of Problems of Mechanical Engineering of the RAS Вопросы атомной науки и техники Section A. Quantum Field Theory On 2D massless field wave equations О волновых 2D-уравнениях безмассовых полей Про хвильові 2D-рівняння безмасових полів Article published earlier |
| spellingShingle | On 2D massless field wave equations Ovcharenko, I.E. Stepanovsky, Yu.P. Section A. Quantum Field Theory |
| title | On 2D massless field wave equations |
| title_alt | О волновых 2D-уравнениях безмассовых полей Про хвильові 2D-рівняння безмасових полів |
| title_full | On 2D massless field wave equations |
| title_fullStr | On 2D massless field wave equations |
| title_full_unstemmed | On 2D massless field wave equations |
| title_short | On 2D massless field wave equations |
| title_sort | on 2d massless field wave equations |
| topic | Section A. Quantum Field Theory |
| topic_facet | Section A. Quantum Field Theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106981 |
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