On Equivalene of DuÆn-Kemmer-Petiau
and Klein-Gordon Equations
V. Ya. Fainberg
and B. M. Pimentel
Instituto deFsiaTeoria,
UniversidadeEstadual Paulista,
Rua Pamplona145,S~aoPaulo,S.P,01405-900, Brazil.
E-mails: fainbergift.unesp.br,pimentelift.unesp.br
Reeived7January,2000
AstritproofofequivalenebetweenDuÆn-Kemmer-Petiau(DKP)andKlein-Gordon(KG)
the-oriesis presentedforphysialS-matrixelementsinthease ofhargedsalar partilesinterating
inminimal waywithanexternalor quantizedeletromagneti eld. First,Hamiltoniananonial
approahto DKPtheoryisdevelopedinmatrixform. Thetheoryisthenquantizedthroughthe
onstrutionofthegeneratingfuntionalforGreenfuntions(GF)andthephysialmatrixelements
ofS-matrixareprovedtoberelativisti invariants. Theequivalenebetweenboththeoriesisthen
provedusing the onnetion between GF and the elements of S-matrixin redution formulas of
Lehmann,Symanzik,Zimmermann.
I Introdution
Morethan60yearsagoG.Petiau[1℄, RDuÆn[2℄and
N.Kemmer[3℄proposedtherstorderequation(DKP
equation)fordesriptionofspin0and1partiles. This
period of time is onventionally divided in three
peri-ods: therstonefrom1939until,approximately1970;
theseondfrom1970to1980andthelastonefrom1980
on. Duringtherstperiodthe majorityof thepapers
aboutDKP equation was devoted to the development
of DKP formalism and to the investigation of DKP
hargedpartilesinterationwitheletromagnetield
(EMeld). For manylassesofproesses(suhas QE
of spin0 mesons, meso-atom and others) alulations
basedonDKPandKGequationsyieldidentialresults
[5℄,inludingone-looporretions[4,5,6℄ 1
.
Theseond periodanbeharaterizedasbysome
disappointments and hesitations. By this time two
great disoveries had been made: parity violationand
reation of unied theory of eletro-weak interation
(Weinberg-Salam theory or Standard Model). The
question about theequivalene of bothDKP and KG
theories arises again at the attempts to desribe new
proesses. Manyworks(seereferenesin[8℄)havebeen
made applying DKP formalism to deays of K and
otherunstable mesonsand to strong interation. The
onlusionpresentedinreferene[8℄wasnotoptimisti:
DKP formalism in some ases yield dierent results
fromaseondorderformalism 2
. Thethirdperiodgoes
under the sign of unertainness: are both DKP and
KGequivalentornot? Not somanypapershavebeen
published on this theme. In our opinion one of main
reasonsfor thederease of interestin DKP formalism
in the last period is the onlusion about
nonequiva-lenebetweenDKPandKGtheories 3
. Webelievethat
the equivalene between these theories in the ase of
nonstablepartilesanbeprovedaswellasforall
pro-esseswhih are desribed by renormalizabletheories.
Thisquestion, however,goesbeyondthe sopeof this
paper. As we know there are no strit proof of the
equivalene between DKP and KG theories in
Quan-tum Eletrodynamis of spin 0partiles, too.
Coini-dent resultshavebeen obtainedfor manyproessesin
rstorderpertubationtheory,onelooporretionsand
theinfraredapproximation[5,6,7℄. Themain goalof
Permanentaddress:P.N.LebedevInstituteofPhysis,Mosow,Russia.
1
Arihlistofrefereneswithhistorialommentsanbefoundinreferene[8℄. Unfortunatelyinthisworktherearenoreferenes
totheworksbyI.GelfandandA.Yaglom,whoobtainedtherstorderequationforpartileswithxedarbitraryspin. Forreferenes
totheseandothersworkssee[9℄.
2
Moreover, inwork[10℄itisaÆrmedthatDKPtheory givesforKmesondeayqualitativelydierentresultswhenomparedto
KG-formalism
this paper is to give astrit proof of the equivalene
of DKP and KG theories for harged salar partiles
interating with an externalor quantizedEM eld in
minimal way for physial matrix elements in any
or-der of pertubation theory. In setion II Hamiltonian
anonialapproahtoDKPtheoryisdevelopedin
ma-trixform. Theonstrutionofgeneratingfuntionalof
Greenfuntion(GF)ofDKPtheoryisusedfor
quanti-zation ofthe theoryand the physial matrixelements
ofS-matrix areprovedto berelativistiinvariants. In
setionIII theequivalene ofDKPand KGtheoriesis
proved for physial matrix elements utilizing the
on-netion betweenGF and theelementsof S-matrix,
in-ludingtheaseofmanyphotonsstates. Fortheproof
weusethe redutionformulasof Lehmann,Symanzik,
Zimmermann[11℄.
InsetionIVweshortlydisussthebasiresultsand
questions about onstrution of renormalizable DKP
theoryforspin0partiles.
II Canonial Quantization
II.1 Hamiltonian approah in matrix
form
Ouraim is to onstrutthe Hamiltonian for DKP
theory whih is onewith onstraintsdue to
degenera-tionof matries. TheLagrangiandensityis
L= ( i
D
m) ; (1)
where D = ieA ; = x ; g =
diagf1; 1; 1; 1g. PrimarilyoneonsidersA
asan
external EM eld. We hoose the
matries in the
followingform: 0 =
0 i 0 0 0
i 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 ; 1 =
0 0 i 0 0
0 0 0 0 0
i 0 0 0 0
0 0 0 0 0
0 0 0 0 0
2 =
0 0 0 i 0
0 0 0 0 0
0 0 0 0 0
i 0 0 0 0
0 0 0 0 0
; 3 =
0 0 0 0 i
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
i 0 0 0 0 (2) = ( ) = ' ;' 0 ; ' 1 ; ' 2 ; ' 3 ;
= 2(
0 ) 2 1; = ';' 0 ;' 1 ;' 2 ;' 3 : (3)
Startingfrom equation(1)weandenethemomenta
p = L : =i(
0 ) ; p = L :
=0; (4)
p = L : =0; p
= L :
=0; (5)
for=1;2;3i. TheinitialHamiltonianH isequal
to: H = Z d 3 xfp : + p : Lg = Z d 3 x i k D k
+ m e
0 A 0 = Z d 3 x i k k
+m e
A
; (6)
where k=1;2;3. Herewewrite downallthe1 st
stage
andthe2 st
stageonstraintsandLagrangian
multipli-ers in matrix form, omitting alulations: - 1 st stage onstraints 4 = p i 0 (7) = p (8) -2 nd stageonstraints 2 = h 1 ( 0 ) 2 i k D k m i 2 = i k D k +m 1 ( 0 ) 2 9 > = > ; (9)
-Lagrangianmultipliers
2 0 = i 0 i k D k m ; 2 0 = i k D k +m 0 ; (10) h 1 ( 0 ) 2 m i k D k i = 0; i k D k +m 1 ( 0 ) 2 = 0:(11) 4
Wefollowtheterminologyofthebook[12℄. Quantizationoftheorieswiththe2 nd
Forgenerating funtional in external EM eld we get
thefollowingexpression[13℄
Z(I; I)=Z 1
0 Z
D D
Æ
1 (
0 )
2
i
k D
k
m
Æ
i
k D
k
+m
1 (
0 )
2
exp
i Z
d 4
x ( i
D
m)
+ I + I : (12)
IntroduingtheauxiliaryeldsCand Cinstead
fun-tional Æ funtionin equation(12)weget
Z( I; I) = Z 1
0 Z
D D DCD C
exp
i Z
d 4
x (i
D
m)
+ C
1 (
0 )
2
(i
D
m)
i
D
+m
1 (
0 )
2
C+
I + I ; (13)
where one used that
0
1 (
0 )
2
= 0. Integrating
overallelds , ,C and C wenallyobtain:
Z(I; I) = exp
i Z
d 4
xd 4
yI(x)( S(x;y;A)
+ 1
m
1 (
0 )
2
Æ 4
(x y)
I(y)
; (14)
where wehaveintroduedthetotalGFofDK partile
in externalEMeld A
:
S(x;y;A)=( i
D
m) 1
Æ 4
(x y) (15)
One an make some important omments about
ex-pression (15). 1) When we integrate over , , C
and C divergenesappearandinniteexpression for
det(i
D
m) 1
. All this multipliers also arise in
Z
0
anddisappearfromthenalequation(14). 2)The
nonrelativistiinvariantterm
1 (
0 )
2
in
equa-tion(14)arisesatexludingnonphysialomponent ,
due to the seond stage onstraints( 2
i ,
2
i
in
equa-tion (9)). One an show that this term (whih does
notdependonharge)doesnotontributetophysial
matrix elements of S-matrix[13℄. 3)If one generalizes
equations(12)to(15)totheaseofinterationofDK
partileswithquantizedEMeldswegetthefollowing
theory(in-gauge):
Z(I; I;J
) = Z 1
0 Z
DA
exp
i Z
d 4
x
Trln
S(x;x;A)
S(x;x;0) 1
4 F
F
J
A
1
2 (
A
) 2
Z
d 4
y I( x)(S(x;y;A)
+ 1
m
1 (
0 )
2
Æ 4
(x y)
I(y)
; (16)
Here weinsert in denominatorand in Z
0
innity
on-stant( detS( x;x;0)) 1
. As itis well known, theterm
TrlnS(x;x;A) in equation (16) is responsible for
appearane of all vauum polarizations diagrams. 4)
Startingfrom(16)itiseasytoprovethatmanyphotons
GFoinidein DKPandKGtheories. Thegenerating
funtionalofGFin KGtheoryhastheform:
Z(J
;J;J
) = Z 1
0 Z
DA
exp
i Z
d 4
x
Tr
ln
G(x;x;A)
G( x;x;0) 1
4 F
F
1
2 (
A
) 2
+J
A
Z
d 4
yJ
( x)G( x;y;A)J( y)
: (17)
Here
G(x;y)= D
D
+m 2
1
Æ 4
(x y): (18)
To get the generating funtional of GF only for
pho-tons we have to put J
= J = 0 in equation (17)
and I = I =0 in equation (16). Equalityof these
equationswill beestablishedifweprovethat
Z
A
det
S(x;y;A)
S( x;y;0)
=expTrln
S(x;x;A)
S( x;x;0)
= det
G( x;y;A)
G( x;y;0)
: (19)
Ontheotherhand
Z
A
= Z
1
0 Z
D D
exp
i Z
d 4
x (i^D m)
; (20)
where
Z
0 =
Z
D D exp
i Z
d 4
x ( i^ m)
Inomponentformexpression(20)equalsto:
Z
A
= Z
1
0 Z
D'D'
D'
D'
exp
i Z
d 4
x('
D
' '
D
'
m('
'+'
'
))g;
Afterintegrationover'
and'
weget
Z
A =
e
Z 1
0 Z
D'D'
exp
i
m Z
d 4
x'
D
D
+m
2
'
; (22)
wherenow
e
Z 1
0 =
Z
D'D'
exp
i
m Z
d 4
x'
+m
2
'
: (23)
Doingsubstitutionthe '
p
m
!'weseethatthe
deter-minant(22)is equaltotherighthandsideofequation
(19). Theequivalenewasproved.
III Equivalene between
phys-ial matrix elements of
S-matrix in DKP and KG
equations
We use LSZ redution formulas [11℄ for proof of the
equivalene between both theories. The main goal of
LSZ approah is to express the matrix elements of
S-matrixthroughtotalmanypartilesGF,making
min-imal assumptions aspossible. Toapply this approah
to DKPtheorywemustwritedowntheoperator's
so-lutionsofthefreeDKPequations. Takingintoaount
that in DKP theory there are only two linearly
inde-pendentsolutions[5℄ofthefreeequationoneanwrite
^
in;out =
1
(2) 3=2
Z
d 3
p
u (p) ^a
in;out (p)e
ipx
+u +
(p) ^ b +
in;out (p)e
ipx
; (24)
and
^
in;out =
1
(2) 3=2
Z
d 3
p
u +
( p) ^ a +
in;out (p)e
ipx
+u ( p) ^ b
in;out (p)e
ipx
: (25)
Here
(^ pm)u
( p)=0; u
(p)(^ pm)=0=0;
b p
p
; p
0 = p
2
+m 2
1=2
=!( p)=!; (26)
operatorsa
in;out andb
in;out
satisfytotheusual
ommu-tation relations; and the solutionsin omponent form
are
u
=
r
m
2!
1; i!
m ;
ip 1
m ;
ip 2
m ;
ip 3
m
: (27)
Itiseasyto hekthatthesalarprodutsare
u
(p)
0 u
(p) = u
(p)
0 u
(p)=1
u
(p)
0 u
(p) = 0: (28)
the operators a
in;out and b
in;out
are those of reation
andannihilationwithpositiveandnegativeharges
a-ordingly. Fromequations(24)and(25)onealsohave
^ a
in;out (p)=
Z
d 3
xe ipx
u ( p)
0 ^
in;out ( x);
(29)
^b +
in;out ( p)=
Z
d 3
xe ipx
u +
(p)
0 ^
in;out
(x); (30)
and
a +
in;out
(p)= a
in;out ( p)
;
b
in;out (p) b
+
in;out (p)
: (31)
VauumandS-matrixaredened asusualas
a
in;out (p)j0i
in;out = b
in;out (p)j0i
in;out =0;
Sj0i
in;out
= j0i
in;out
; (32)
a +
out =a
+
in
S: (33)
Foranyphysial matrixelementofS-matrixone has
hn 0
;m 0
;outjn;m;ini=hn 0
;m 0
;injSjn;m;ini; (34)
where: n+m=n 0
+m 0
is theonservation ofharge
and
n;m; in
=i=1nuj=1muainout +
(p
i
)binout +
(q
j
Nowwe an formulate themain assumptionof LSZapproah 5
. Foranymatrix elementsof Heisenberg operators
^ ( x)and ^ ( x)thefollowingasymptotirelationsareimplemented
x
0
!1lim
n 0
;m 0
; in
out
^ (x)
n;m; in
out
=
=
n 0
;m 0
; in
out
^ in (36)
out(x)
n;m; in
out
(37)
and the samerelationfor ^ (x) . In orderto do notompliatethe proof ofthe equivalenebetween DKP and
KGF theories we are restrited by onsideration of the matrix elements of S-matrix for partiles with the same
(positive)hargesandoneutilizesLSZredutionformula. Wehave
h 0ji=1ku a
out (p
i
)j=1kua +
in (q
j )j0i=
= h 0ji=1k 1u a
out (p
i )a
out (p
k
)j=1kua +
in ( q
j )j0i
= hn
1 ;:::;n
k 1 ;outj
Z
d 4
xe ipx
u (p)
i
!
x m
^ ( x)
jm
1 ;:::;m
k
;ini; (38)
d
wherep
0
=!(p)andisanotessentialonstant.
Uti-lizing equations (3) and (27) we an rewrite equation
(38)inomponentform
hn
1 ;:::;n
k 1 ;outj
1
m e
ipx
!
2
x +m
2
^'( x)+
+
x
e ipx
1
m
x
^'( x) ^ '
( x) :
jn
1 ;:::;n
k
;ini (39)
The mainideaofthe proofisto showthat theseond
term under total derivative in equation (39) is equal
zero. We havefor theseond term, omitting jiniand
houtj states,
Z
d 4
x
x
e ipx
1
m
x
^ '(x) ^'
(x)
=
= Z
d
e
ipx
1
m
x
^ '(x) ^'
(x)
: (40)
Oneanhoosethesurfae
sothat
:f T x
0
T; Lx
i
L;=1;2;3g: (41)
Therstterm,=0,equals
Z
d 3
x
e
i!T+ipx
1
m
T
^ '(x;T)
^ ' 0
( x;T)
e
i!T+ipx
1
m
T
^ '(x; T)
^ ' 0
( x; T)
: (42)
Sine
T !1lim
T
^'( x;T) =
T
^'out (43)
in(x;T)
= ^ 'out (44)
in 0
(x;T) (45)
thersttermin (40)disappearsinthelimitT !1.
Moredeliatesituationarisesattheproofof
disappear-ane of ontribution of terms d
i
(see equation (41))
whenL!1atxedT. Mathematiallystritlyitis
possibletodoprovesonlyintroduingthewavepakets
insteadof planewaveswith given momentum. Wedo
nothave a plae to go in details of the proof. If one
introduestotalsetoforthonormalizingpakets f
n (p)
theontributionwill beequalto
I
1
(L) Z
dx
0 dx
? Z
dpf
n (p)e
i!x0 ip?x?
(
e ip1x1
1
m
1
x
^'(x) ^ ' 1
(x)
x1=L1 (46)
e ip1x1
1
m
1
x
^ '(x) ^' 1
(x)
x1= L1 )
; (47)
wherep
? =(p
1 ;p
2 ),x
? =(x
1 ;x
2 )andf
n
(p)isawave
paket. ForGaussiantypeofpakets
f
n
(p)=exp
p 2
2m 2
i=13uH
n
p
i
m
; (48)
5
where H
n
(x) denotes a Hermite polynomial of
de-gree n, it is possible to prove that I
1
(L) dereases
exp
(Lm) 2
. Thus, all the I
i
(L), i = 1;2;3are
equalzeroinlimitL!1. IfwerepeattheLSZ
pro-edure for the seond operators a +
out (p
2
) in equation
(38)weget
h k;injk;outi=hk 2;inj Z
d 4
x
1 d
4
x
2
e ip
1 x
1 0
!
1 +m
2
m 1
A 8
<
: e
ip
2 x
2 0
!
2 +m
2
m 1
A
T(^'( x
1 ) ^ '(x
2 ))+
+ x
2
e ip
2 x
2
x
2
T(^'( x
1
) ^'( x
2 ))
T(^'(x
1 ) ^'
(x
2
))) ℄gjk;ini: (49)
Again one an prove that the last term under total
derivativeequalszero(if introduedpakets).
Contin-uingthisindutiveproedurewegotoonlusionthat
all physial matrix elementsof S-matrix in DKP
the-oryoinide withthose ofKGF theoryindependentof
harater interation (renormalizabletype) if in both
theoriestheLSZasymptotionditions(37)are
imple-mented.
IV Conlusions
1)StartingfromanonialapproahtoDKPtheory
in-teratingwithquantizedEMeldandonstrutingthe
generating funtional for GF of thetheory westritly
proved total equivalene between physial matrix
ele-ments of S -matrixin DKP and KG theories and
be-tween many photons GF in both theories. The proof
ofequivalenebetweenboththeorieshavebeenarried
oututilizingtheLehmann,SymanzikandZimmermann
redutionformalism[11℄.
Wealsoprovedtheequivaleneoftheboththeories,
startingfromLagrangianapproahtogenerating
fun-tional in DKP theory(see equation (16) withoutlast
term)andforgettingaboutonstraints. 2)Inpriniple,
theDKPaswellasKGtheoriesarenonrenormalizable
ones even for salarpartiles due to the logarithmial
divergeneof oneloopdiagrams ofsatteringtwo
par-tiles with exhange of two photons [5℄. As it is well
known that KG theory beomes renormalizable if we
introdueaselfinterationterm( '
') 2
. This
prob-lem anbesolvedin DKP theory in thesameway: it
is neessarytoadd toLinequation (1)terms
P
2
=( '
') 2
; (50)
whereP =u(
)
2
istheprojetoronthesalarpart
the same method (Setions II and III) formally it is
possibleto proveequivalenebetweenDKPandProa
equation for spin onepartiles, destruting from
non-renormalizabilityofthesetheories. 4)Wewouldliketo
stress that DKP theory until now did not nd wider
appliation althoughthistheory hassomeadvantages
just due to thedegeneration of
matries(one very
simple to alulate traethat of)and due to minimal
haraterof interationwithEMelds. Oneompares
expressionsforS-matrixinboththeories:
S
DKP
=Texp
i Z
e
A
d 4
x
; (51)
S
KG
= Texp
i Z
ie('
'
'
'
e'
A
')A
d 4
x : (52)
In the last ase the interation ontains proportional
termstoeande 2
. Duetothisinhigher(twoandmore
loops)approximationsombinatorialoeÆients given
toorder e 2
beforehavingaompliatedform.
5) About equivaleneof DKP andKGfor
desrip-tionof unstablepartileswewould liketo note that if
weanapplyoneption ofasymptoti statesto some
suhpartileandutilizeforphysialmatrixelementsof
S-matrixthesamemethodwhihhasbeenusedin
Se-tionsIIandIII,thentheproofofequivaleneisobvious:
forinstane,fordeayofK
l
mesonsweanalulate
the imaginarypart of the GF of K
l
meson and get
equivalenewithexatnessredeningthe'omponent
of DKP funtion: i'
KG = '
DKP
=m, see equation
(49).
Aknowledgments
WewouldliketothankI.V.Tyutin,whosuggested
to use redution formalism of LSZ (see Setion III)
to proof the equivalene of both theories, and D. M.
Gitman for stimulating ritiism . V.Ya.F. thanks to
FAPESP for support (grant number 98/06237-0)and
RFFI forpartial support (grantnumber99-01-00376).
B.M.P.thanksCNPqforpartialsupport.
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