The causal approach to scalar QED via Duffin-Kemmer-Petiau equation
In this work we consider the scalar QED via Duffin-Kemmer-Petiau equation in the framework of Bogoliubov-Epstein-Glaser causal perturbation theory. We calculate the lowest order distributions for Compton scattering, vacuum polarization, the self energy and, by using a Ward identity, the vertex corre...
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irk-123456789-794152015-04-02T03:02:10Z The causal approach to scalar QED via Duffin-Kemmer-Petiau equation Lunardi, J.T. Manzoni, L.A. Pimentel, B.M. Valverde, J.S. Quantum electrodynamics In this work we consider the scalar QED via Duffin-Kemmer-Petiau equation in the framework of Bogoliubov-Epstein-Glaser causal perturbation theory. We calculate the lowest order distributions for Compton scattering, vacuum polarization, the self energy and, by using a Ward identity, the vertex correction. The causal method provides a mathematically well defined and noneffective theory which determines, in a natural way, the propagator and the vertex of the usual effective theory. 2001 Article The causal approach to scalar QED via Duffin-Kemmer-Petiau equation / J.T. Lunardi, L.A. Manzoni, B.M. Pimentel, J.S. Valverde // Вопросы атомной науки и техники. — 2001. — № 6. — С. 25-29. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 29.27.Hj 29.90. +r http://dspace.nbuv.gov.ua/handle/123456789/79415 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum electrodynamics Quantum electrodynamics Lunardi, J.T. Manzoni, L.A. Pimentel, B.M. Valverde, J.S. The causal approach to scalar QED via Duffin-Kemmer-Petiau equation Вопросы атомной науки и техники |
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In this work we consider the scalar QED via Duffin-Kemmer-Petiau equation in the framework of Bogoliubov-Epstein-Glaser causal perturbation theory. We calculate the lowest order distributions for Compton scattering, vacuum polarization, the self energy and, by using a Ward identity, the vertex correction. The causal method provides a mathematically well defined and noneffective theory which determines, in a natural way, the propagator and the vertex of the usual effective theory. |
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Article |
| author |
Lunardi, J.T. Manzoni, L.A. Pimentel, B.M. Valverde, J.S. |
| author_facet |
Lunardi, J.T. Manzoni, L.A. Pimentel, B.M. Valverde, J.S. |
| author_sort |
Lunardi, J.T. |
| title |
The causal approach to scalar QED via Duffin-Kemmer-Petiau equation |
| title_short |
The causal approach to scalar QED via Duffin-Kemmer-Petiau equation |
| title_full |
The causal approach to scalar QED via Duffin-Kemmer-Petiau equation |
| title_fullStr |
The causal approach to scalar QED via Duffin-Kemmer-Petiau equation |
| title_full_unstemmed |
The causal approach to scalar QED via Duffin-Kemmer-Petiau equation |
| title_sort |
causal approach to scalar qed via duffin-kemmer-petiau equation |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
| topic_facet |
Quantum electrodynamics |
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http://dspace.nbuv.gov.ua/handle/123456789/79415 |
| citation_txt |
The causal approach to scalar QED via Duffin-Kemmer-Petiau equation / J.T. Lunardi, L.A. Manzoni, B.M. Pimentel, J.S. Valverde // Вопросы атомной науки и техники. — 2001. — № 6. — С. 25-29. — Бібліогр.: 13 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-06T03:28:13Z |
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| fulltext |
THE CAUSAL APPROACH TO SCALAR QED
VIA DUFFIN-KEMMER-PETIAU EQUATION
J.T. Lunardi*1, L.A. Manzoni2, B.M. Pimentel1 , J.S. Valverde1
1Instituto de Física Teórica, UNESP, Rua Pamplona 145, São Paulo, SP, Brazil
2Instituto de Física, USP, Caixa Postal 66318, São Paulo, SP, Brazil
In this work we consider the scalar QED via Duffin-Kemmer-Petiau equation in the framework of Bogoliubov-
Epstein-Glaser causal perturbation theory. We calculate the lowest order distributions for Compton scattering, vacu-
um polarization, the self energy and, by using a Ward identity, the vertex correction. The causal method provides a
mathematically well defined and noneffective theory which determines, in a natural way, the propagator and the ver-
tex of the usual effective theory.
PACS: 29.27.Hj 29.90. +r
1. INTRODUCTION∗
The usual way to approach scalar quantum electro-
dynamics (SQED) is by performing the electromagnetic
minimal coupling in the free Lagrangian of Klein-Gor-
don (KG) scalar field theory [1]. An alternative way is
to start from the free Duffin-Kemmer-Petiau (DKP) La-
grangian instead of the KG one [2].
It is known that in the free field case the DKP and
KG theories are equivalent, both in classical and quan-
tum pictures [2,3,4]. However, there are still no general
proofs of equivalence between these theories when inter-
actions and decays of unstable particles are taken into
account [5,6] (in this context see also references [7,8],
which relate different results by using both DKP and KG
formalisms with strong interactions). Some progress in
this direction has been made recently. For instance, it
was shown that this equivalence holds at the classical
level when minimal interaction with electromagnetic
[4,9] and gravitational [10] fields are present. Also,
strict proofs of this equivalence were given for the quan-
tized scalar field interacting with classical and quantized
electromagnetic, Yang-Mills and external gravitational
fields [6,11].
Perhaps the most evident advantages in working with
DKP theory are the formal similarity with spinor QED,
the fact that do not appear derivative couplings between
DKP and the gauge field and that this theory allows an
unified treatment of the scalar and vector fields. Despite
this, the theory shares some difficulties with SQED
based on KG one (SQED-KG), which are usually sur-
passed by dealing with an effective theory [2,6].
In this work we shall consider SQED-DKP theory in
the framework of Bogoliubov-Epstein-Glaser causal per-
turbative method [12], which gives a mathematical rig-
orous treatment of ultraviolet divergences in quantum
On leave from Departamento de Matematica e Estatistica,
Universidad Estadual de Ponta Grossa, Ponta Grossa, PR,
Brazil.
field theory. Our goals are to obtain a non-effective and
mathematically well defined theory for SQED-DKP, at
the same time recovering the results of the effective the-
ory already obtained in the usual perturbative approach.
In addition, this work must be viewed as an initial step
in the attempts to rigorously establish the renormaliz-
ability of the theory.
2. THE DUFFIN-KEMMER-PETIAU THEO-
RY
The free DKP theory is given by the Lagrangian
[2,4]
ψψψβψ µ
µ miL −∂=
2
, (1)
where ψ is a multicomponent wave function, 0ηψψ += ,
and ( ) 12
200 −= βη . µβ are a set of matrices (
3,2,1,0=µ ) satisfying the algebraic relations
µ νρν ρµµνρρνµ ββββββββ gg +=+ .
It is known that the above algebra has only three ir-
reducible representations, whose degrees are 1, 5 and 10
[13]. The first one is trivial, having no physical content.
The second and the third correspond, respectively, to
scalar and vectorial representations. In this work we
shall restrict us to the scalar case.
The standard procedure of canonical quantization for
the free lagrangian (1) gets [2]
)(1)(),( yxS
i
yx abba −=
++− ψψ
and
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 25-29. 25
)(1)(),( xyS
i
yx baba −−=
−+− ψψ ,
where −ψ and +ψ contains only annihilation and cre-
ation operators, respectively, and
( )[ ] )(1 xmii
m
S ab
def
ab
±± ∆+∂/∂/= , (2)
where )(x±∆ are the positive (negative) frequency parts
of the Pauli-Jordan distribution +∆=∆ + )()( xx )(x−∆ .
This later distribution has causal support [12] and can be
written as )()()( xxx advret ∆−∆=∆ , where )(xret∆ and
)(xadv∆ has, respectively, retarded and advanced sup-
ports with respect to the point x . Analogously we de-
fine
)()()()()( xSxSxSxSxS advret
def
−=+= −+ (3)
where again )()( xS advret have retarded (advanced)
support with respect to x and it is defined by equation
(2) just replacing ±∆ by )(advret∆ . The above splitting
of )(xS into retarded and advanced parts is not the
unique possible, as we will see in the next sections.
The interaction with the electromagnetic field is in-
troduced by the minimal substitution µµµ ieA−∂→∂
in the Lagrangian (1), which becomes
IF LLL += ,
where =FL is the free lagrangian (1) and
µ
µ ψβψ AeLI ::= . (4)
is the interaction lagrangian. In this work we use
0>e .
3. THE BOGOLIUBOV-EPSTEIN-GLASER
APPROACH
In the Bogoliubov-Epstein-Glaser causal method
[12] the S -matrix is introduced as an operator-valued
functional given by the perturbative series
)()(
),,(
!
11)(
1
1
1
4
1
4
n
n
nnn
xgxg
xxTxdxd
n
gS
∑ ∫
∞
=
×+=
, (5)
where )(xg is a c-number test function supposed to be-
long to the Schwartz space, ∈)(xg S4( 4R ). The sym-
metric n -point functions }),,{()( 1 n
def
n xxXXT = are
distributions written in terms of free fields and are the
basic building blocks to be inductively constructed from
the knowledge of )(1 xT by means of the requirements
of causality
)()()( 2121 gSgSggS =+ , if supp >1g supp 2g
and translational invariance. The above causality condi-
tion implies that
),,(),,(),,( 111 nmmnmmnn xxTxxTxxT +−= ,
if
},,{},,{ 11 nmm xxxx +> ; (6)
Based on general arguments such as correspondence
[3], we have that )1(
1 IiLT = , where )1(
IL is the term of
first order in the coupling constant in the interaction La-
grangian.
The inductive procedure works as follows. From the
assumption that all )(XTm , with 1−≤ nm , are known
and satisfy (6), we can construct the distributions
∑ −=
2
11
),()(~),,( 1
'
P
nnnnnn xYTXTxxA , (7)
∑ −=
2
11
)(~),(),,( 1
'
P
nnnnnn XTxYTxxR , (8)
where 2P stands for all partitions ,{: 12 xP }, 1−nx
= 0, /≠/∪ XYX into disjoint subsets with 1nX =
2−= ≤ nY . In these expressions )(~ XTn refers to the n
-point distributions corresponding to a series for the
1−S -matrix analogous to (5), which can be obtained
from the formal inversion of )(gS . If in the above ex-
pressions the sums are extended in order to include the
empty set 0/=X we get
),,(),,(
),()(~),,(
11
'
1
0
2
11
nnnn
P
nnnnnn
xxTxxA
xYTXTxxA
+=
= ∑ −
(9)
),,(),,(
)(~),(),,(
11
'
1
0
2
11
nnnn
P
nnnnnn
xxTxxR
XTxYTxxR
+=
= ∑ −
(10)
where now 0
2P stands for all partitions ,,{: 1
0
2 xP
YXxn ∪=− }1 . A glance at Eqs. (9) and (10) shows that
nA and nR are not known because they contain the un-
known nT . However, the distribution defined by
''
1 ),,( nnnn
def
nn ARARxxD −=−= (11)
26
is known. It can be proved that the supports of nR
and nA are retarded and advanced with respect to nx ,
respectively. Then, nD has a causal support with re-
spect to this point, i.e.,
supp )()(),( nnnnnn xxxXD −+ Γ∪Γ⊆ ,
After splitting nD into its advanced and retarded
parts we can obtain nT from the relations (9) or (10).
The above splitting is the nontrivial step of the in-
ductive procedure. In dealing with this problem we need
only to consider the numerical distribution d associated
with nD (i.e., we neglect the operator field distributions
that appear as factors in the causal distribution nD ).
Then, we write ard −= , where r and a are respec-
tively the associated retarded and advanced numerical
distributions. To solve the splitting problem we must de-
termine the singular order ω of a distribution. We have
to consider two distinct cases: i) 0<ω - in this case the
solution of the splitting problem is unique and the r and
a distributions can be found by multiplication of d by
step functions; ii) 0≥ω - now the solution can be no
longer obtained by multiplying d by step functions and,
after a careful mathematical treatment, it may be shown
that the retarded distribution (which suffices to deter-
mine nT ) is given, in momentum space, by the central
splitting solution
∫
+ ∞
∞−
+ +−−
=
)01()0(
)(ˆ
2
)(ˆ
1 itit
tpddtipr
ωπ , (12)
where the symbol ^ denotes the Fourier transform. How-
ever, in contrast with the case 0<ω , this expression
does not give a unique solution for the splitting problem.
By assuming that the splitting procedure cannot raise the
singular order of d , it can be shown that the general so-
lution in momentum space is given by
∑
=
+=
ω
0
)(ˆ)(~
a
a
a pCprpr , (13)
where the aC are constant coefficients which are not
fixed by the causal structure - we need additional physi-
cal conditions in order to determine them.
We now apply the inductive steps above to construct
the two point distributions for SQED-DKP theory.
Looking for (4) we see that the one-point distribution for
the DKP field interacting with electromagnetic field is
given by )(:)()(:)(1 xAxxiexT µ
µ ψβψ= , where e is
the physical charge. To go from 1=n to 2=n we con-
struct the distribution ),( 212 xxD , which in this case is
given by
)](),([),( 2111212 xTxTxxD =
:)(:)(::)()({: 1122
2 xxxxe ψβψψβψ µν= (14)
)()(:})()(::)()(: 212211 xAxAxxxx νµ
νµ ψβψψβψ−
We can verify explicitly that this distribution is
causal from the fact that 0)](),([ 2111 =xTxT if
0)( 2
21 <− xx .
4. APPLICATIONS
We now apply the formalism sketched above to de-
termine some lowest order processes in SQED, namely
the Compton scattering, vacuum polarization, self-ener-
gy of the scalar particle and the vertex correction at the
limit of zero transferred momentum.
4.1. COMPTON SCATTERING
By using Wick's theorem [12] we normally expand
(14) and keep only those terms contributing to Compton
scattering, given by
)(:)()(:
:)()(:),(
2121
21
2
212
xxSxAxA
xxiexxD
bc
dacdab
I
−
×ψψββ=
νµ
νµ
(15)
)}({:)()(:
:)()(:),(
1221
21
2
21
xxSxAxA
xxiexxD
da
cbcdab
II
−−
×ψψββ=
νµ
νµ
(16)
In both these terms, the numerical distribution we
have to split is )(xS , which is causal. Thus, a splitting
solution is trivially obtained from (3). But it can be
shown that )(xS has singular order 0=ω , which im-
plies that this splitting is not unique. The general solu-
tion for the numerical retarded distribution in configura-
tion space is )()()(~ xCxSxr ret δ+= , where C is an ar-
bitrary constant. The general result for the numerical
distribution ),( 21 xxt I is
)()(),( 212121 xxCxxSxxt FI −+−−= δ , (17)
where we have defined )()()( 1 xmiixS F
m
def
F ∆+∂/∂/−=−
( )(xF∆ is the usual scalar Feynman propagator). In a
similar way we find +−−= )(),( 1221 xxSxxt FII
)(' 12 xxC −δ . The condition of charge conjugation in-
variance of the theory requires that 'CC = . The gauge
invariance requirement yields m
IC = , where I is the 5
× 5 identity matrix. Turning this result into (17) we ob-
tain
),()(),( 122121 xxtxxTxxt IICI =−−= , (18)
where we have defined )()()( 1 xxSxT m
FC δ−≡ . It is
straightforward to see that this distribution is the Green
function for DKP equation, i. e., )()()( xxTmi C δ=−∂/ .
So, in this sense the causal approach gives, in a natural
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
27
way, the correct effective propagator for the DKP scalar
particle, which agrees with the results of [2,6].
4.2. VACUUM POLARIZATION
Considering now those terms contributing to vacuum
polarization we have
:)()(:)}()({),( 21212 xAxAyPyPxxDVac
νµ
ν µµ ν −−= (19)
where
21 xxy
def
−= and ×−= µµ ν β{)( 2TreyP
def
)}()( ySyS −−+ νβ . It turns out that the expression into
curl brackets in the above expression is the numerical
distribution we have to split, which have singular order
2=ω . Using the central splitting formula (12) to deter-
mine its retarded part )(ˆ kr we can determine the corre-
sponding numerical distribution )(ˆ kt . The final result
for the complete two-point distribution for the vacuum
polarization in configuration space is given by
:)()()(:),( 2211212 xAxxxAixxT Vac
ν
µ ν
µ −Π−= ,
where
)(ˆ
2
1)(ˆ
24
kg
k
kkk Π
−=Π µ ν
νµ
µ ν
π
;
2/32
4
2
4
2 41
)0(
1
12
)(ˆ
2
−
+−
=Π ∫
∞
s
m
isks
dskek
m
.
However, from the fact that the singular order of the
distribution we had to split was 2=ω , the above result
is not unique. The most general solution is given by
2
20)(ˆ)(~ kCkCCkk +++Π=Π µ
µ , where the normaliza-
tion constants 0C , µC and 2C are not determined by
causality, but from the requirements of parity invariance,
zero mass for the gauge field, and the identification of e
with the physical charge. These conditions imply that
020 === CCC µ .
4.3. SELF-ENERGY AND VERTEX CORRECTION
Now, the terms in (14) contributing to the scalar self-
energy are
)()([)(:),( 120211
2
21 xxDxxSxexxD self
I −−−= +−µβψ
:)()]()( 221021 xxxDxxS ψβ−−+ µ
++ ; (20)
)()([)(:)( 120122
2
2,1 xxDxxSxexxD self
II −−= ++µβψ
:)()]()( 121012 xxxDxxS ψβ−−+ µ
+− . (21)
The numerical distributions we have to split are the
terms into brackets, which have singular order 1=ω .
The final general solution, involving two arbitrary con-
stants, is given by
:)()()(:),( 2211
2
21 xxxxiexxT self
I ψψ −Σ=
:)()()(: 1122
2 xxxxie ψ−Σψ+ (22)
where
( )[ ]{ 222
4
2
1log
)2(4
)(ˆ mpibiep −−−=Σ π θ
π
×
/++/−
−/+
− pCC
b
p
b
p
b
m 10242 2
11
2
11 . (23)
By considering the complete propagator for the
scalar particle and by requiring its normalized mass to
be m , we find that these constants satisfy the condition
−= 10 2
1 CmC ,
which can be used to eliminate one of them, say 0C .
The remaining constant 1C is related to another one,
which appears in the splitting of the vertex causal distri-
bution, by the following Ward identity
)(ˆ
)2(
1),(ˆ
2
p
p
pp Σ
∂
∂=Λ
µ
µ
π
, (24)
where µΛ is the vertex numerical distribution. Substi-
tuting the explicit form of the self-energy (23) into this
identity we obtain
( )[ ]{ 222
6
2
1log
)2(4
),(ˆ mpibiepp −−−=Λ π θ
π
µ ×
−+
/+
424
11
2
12
bmb
p
mb
p µµ β
+
−
−−
−
+ 2
2
22
11)1(
1
1..2
b
mbi
b
VP
m
p π δ
µ
+−/+
−/ µ
µµ
β
β
12424 2
11
2
C
bbm
pp
b
p
(25)
This result is well defined at 0=p , but it is singular
on mass shell 22 mp = due to the logarithmic term. The
above form of the vertex function suffices to study the
physical meaning of the constant 1C , which is done in
28
connection to charge normalization. The physical charge
is defined in the scattering of a scalar particle by an ex-
ternal electromagnetic field at low energies. Thus we
must consider the contributions to S matrix (in the limit
of zero transferred momentum) from the terms contain-
ing 1C in both the self-energy and vertex distributions.
Because of the above mentioned mass shell singularity,
we must be care in taking the adiabatic limit. Making so,
we can prove that all these contributions cancel them-
selves and conclude that this constant has no physical
meaning. Nevertheless it can be specified by requiring
the vertex function to satisfy the central splitting condi-
tion, i.e., )0,0(ˆ µΛ . Using this condition into (25) we ob-
tain 4
1
1 =C .
5. CONCLUSIONS AND FINAL REMARKS
In this work we considered SQED-DKP theory in the
framework of Bogoliubov-Epstein-Glaser causal
method. The starting point was the identification of the
one point distribution )(1 xT , which was determined by
the interaction Lagrangian (4) written in terms of free
fields. The causal method thus dictated completely the
form of the interaction, giving us a non-effective and
mathematically well-defined theory. In calculating the
two-point distribution for Compton scattering, and by
using the requirements of charge conjugation and gauge
invariance, we recovered in a natural way the scalar
propagator of the DKP usual effective theory. At one
loop level we calculate the vacuum polarization, the
scalar self-energy and the vertex correction in the limit
of zero transferred momentum. We determined the phys-
ically meaningful normalization constants by using
physical requirements, as charge and mass normaliza-
tion, parity and gauge invariance. All our results agree
with those obtained in the context of the effective
SQED-DKP theory as well as in the context of SQED-
KG, what corroborates the belief about the equivalence
of KG and DKP theories. A complete analysis at one
loop level, as well as the study of the renormalizability
of the theory, is in course. As future perspectives we can
quote the use of the causal approach to study DKP field
interacting with external fields.
ACKNOWLEDGMENT
J.T.L. and B.M.P. thank respectively to CAPES-
PICDT and CNPq for partial support. L.A.M. and J.S.V.
thank FAPESP for full support.
REFERENCES
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Singapore: Mc-Graw Hill, 1980.
2. A. I. Akhiezer and V. B. Berestetskii. Quantum
Electrodynamics. New York: Interscience, 1965.
3. N. N. Bogoliubov and D. V. Shirkov. Introduction
to the Theory of Quantized Fields. New York: Wi-
ley, 1980.
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
29
J.T. Lunardi*1, L.A. Manzoni2, B.M. Pimentel1 , J.S. Valverde1
1Instituto de Física Teórica, UNESP, Rua Pamplona 145, São Paulo, SP, Brazil
2Instituto de Física, USP, Caixa Postal 66318, São Paulo, SP, Brazil
PACS: 29.27.Hj 29.90. +r
1. INTRODUCTION
2. THE DUFFIN-KEMMER-PETIAU THEORY
3. THE BOGOLIUBOV-EPSTEIN-GLASER APPROACH
acknowledgment
REFERENCES
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