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Rank-n antisymmetric tensor gauge fields in 2n-dimensional space-time and mass generation mechanism
Volume 126B, number 3,4
PItYSICS LETTERS
30 June 1983
RANK.n ANTISYMMETRIC TENSOR GAUGE FIELDS IN 2n-DIMENSIONAL SPACE-TIME AND MASS GENERATION MECHANISM H. ARATYN
Department of Physics, Technion-lsrael Institute of Technology, Haifa, lsrael Received 4 November 1982
By introducing mutually dual gauge fixing conditions we achieve a realization of a selfdual rank-n antisymmetric tensor gauge field in 2n-dimensional space-time as an object completely described by lower rank auxiliary tensor fields. Our construction can be extended to yield a mechanism of mass generation corresponding to the bosonized version of the generalized Schwinger model.
Antisymmetric tensor gauge fields (atgfs) have growing application in many branches o f quantum field theory. They have been discussed in connection with string models [ 1], the confinement problem [2], supergravity [3] and recently they also had received attention in lattice field theory [4]. In this paper we consider the class o f completely antisymmetric rank-n tensor gauge fields Aul...un(X) in 2n-dimensional s p a c e - t i m e . The field dual to Aul ...un(x) is defined by
7%~..~ = (1/n !) e,l ...~ u~ ...un A u~ ...Un
( l)
and has the same rank (number o f indices)as the original field. Our class thus contains: the Maxwell field Au in two dimensions, the K a l b - R a m o n d field Auv in four, atgfAuz,~" o f third rank in six and so on. The corresponding antisymmetric field tensors
Ful...Un+, (x) = 3lulAuz...Un÷l ](x) ,
(2)
are invariant under the generalized gauge transformation
Au,...un(X)-'> Aul...un(X) + 3[ulfu2...unl(X) ,
(3)
where ful ...Un-1 (x)are arbitrary antisymmetric tensor functions o f rank-(n - 1) and where we have introduced the generalized rotation 3[ulAu2 ...un] defined by
~[u Au2...unl =cycli~cperm!-l)"~u~qAu,r2...Unn .
(4)
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © i 983 North-Holland
As already pointed out in our previous work [5], which we from now on will refer to as I, in the cases of two- and four-dimensions the invariance under the extended gauge transformation
"4ut...un -" Au, ...Un + fu~...Un '
(5)
can be established provided we introduce the set of two auxiliary antisymmetric tensor fields Aul.." Un-i (x), But.-.Un -t (x) with rank one lower than that o f Au~...un(x). If we assume that the tensor fields Aul...Un -1 (x) and Bul...un_l (x) transform as
Aul...Un_ 1 --" Bul...un_ 1 + OPfpal...Un_ 1 ,
(6)
Bux...un_l -* Bu,...Un_, + O°?oul...un_l
(7)
[ w i t h f , l... ~ related toful...u n through (1)] then it can be shown tha the field equation
eUl ..~n" I...~n O~l Bv~.,.~
(8)
= ( - 1 ) n 2 - 1 ( n _ l)!(E3Aul...u n -b[ulAu2...unl ) , is invariant under the transformation (5) accompanied by (6) and (7). The p r o o f of this is based on the trivial identity obeyed by the generalized rotation [6]
3U(Oluful...unl ) = [--]ful...Un - 3[Ul Ouflulu2...unl ' (9) where the index isolated by the vertical bars I 1does not participate in the antisymmetrization process. One can easily see that for two and four dimensions 189
V o l u m e 1 2 6 B , n u m b e r 3,4
30 J u n e 1 9 8 3
PHYSICS LETTERS
(8) reduces to eqs. (19) and (39e) in I. Thus in four dimensions formula (8) yields EIAu ~ expressed in terms of the "electric-" and "magnetic-like" vector fields Au and Bu. Let us not introduce two spin-one projection operators in the space of the antisymmetric rank two tensors, defined in momentum space by
= - ( 1/2n!) i~xAul ...UnaXAUl ...V.n
+PuPog.,o),
X ~l'"VnXt'"XnAvl ...Un~XiBx2...hn 1
~ [a/(n - 1)!]Aul...Un_IA ul"''un-t
+ ½ [B/(n - 1)!]Bvt...Vn_xB ~q'''vn-I
(10) -
(~uv,o a)H" = Iuv,,oo - (~uv,.oa)F'r,
(11)
1)!]Aul...Uni~U2A "2"''un
+ [(-1)n2/(n - 1)!n!]
-
( ~ uv,oo)FT = (1/2p2) (P uP ogvo - P uP OgUa - PuP oguo
[l/(n-
-
-
[ 1/(n -
2)!1Aut .-.Un-I ~1AU2...~n-i
[1/(n - 2)!] Bht...xn_ l ~hlBh2""Xn~l .
(16)
where Iuv,o a = !2 (~" ",,upovo ~ - gla~gvo,'~ And we notice, with respect to the four-dimensional duality relation
The equations of motion derived from (16) are
I-'lAuv = i~[uAul - eu~aoaOBa ,
OUlAulu2...Un-I = 0 ,
(17)
that the following formulas hold
~XlB~qx2...Xn_ l = 0 ,
(18)
~ uv,o~[OA°] = ~[uAu] , ~uu,ooeo°Xr~xBr = 0 ,
~Ut Aulu2...V.n - otAu2...u n - i~[u2Au3...Un] = 0 , (19)
~u~,ao~[° A °l
~)alA~.la2...x n -/3Bx2...x n
=
(12)
0 , ff uu~ooea°Xr~xBr = e~hr~XB r.
By introducing the tensor -,,
o
.
8x~..-xn
¢~l.tl...#nvl...u n - 6 / a r k I . . . 6 # n ^ n Vl...~n ,
(13)
where 8~x~.iTu~nis the generalized Kronecker delta in 2n-dimensional space, we can define the projection operators in the space ofantisymmetric rank-n tensors by ( ~l~U~...lan , v l . . . ~ ) F T =
(14)
(1/n!p2)p[u~pU8 lulu~...un l,uv..u n ,
( ~ u ~ ...Un,Vt ...vn)FT = (1/n! p2) pU(p [v.S ui ...Un ] ,V~...Vn) . (15) These operators when acting on the right hand side of eq. (8) will project the Au~...Un -1 -term or Bux...Un_~ term depending on the use of ~ o r ~ operator, exactly as it happened in the four dimensional case above. We will now present the 2n-dimensional lagrangian which leads to the field equation (8).
+ a[h2Bh3...kn]
= 0 ,
and eq. (8) obtained from variation of the lagrangian density (16)with respect to Aul...u n. Furthermore the lagrangian density (16) is invariant up to a total derivative under the combined gauge transformations (5), (6) and (7) as can be shown using again the identity (9). As seen from the equations of motions (8) and (17)--(20) there are only two consistent values for the gauge parameters a and/3 given by a = 13= 1 or a =/3 = 0. The commutation relations for Aul...Un, Aul .--Un-~ and Bu~...Un_x consistent with the equations of motion have the following form
[Au I ...#n(X), Au I ...vn(Y)] = iabux ...,n,Vl...vnD( x - Y ) + iaeul ...un ux..-UnD ( x - y ) '
(21)
[Aui...un-i (x), A~I ...~0')1 = i3"6,,~...,n_1 ,~, ...~, X D ( x - y ) + i s 8Ueuu ~...un_l ~l...Vn D ( x - y ) ,
[Bul ...Un-,
(22)
(x), A ~1 ..-.n (Y) 1 = i ~Ueu., ...Un-i ~, ...~n ( 2 3 )
X D(x-y)-
• --
n2
1( 1)
a~6uul...Un_t,vl...unD(x-y),
where
D(x) = - i ( 2 ~ ) -2n+l
190
(20)
f dZ~p e(P0)8(p 2) exp(ipx).
Volume 126B, number 3,4
PHYSICS LETTERS
30 June 1983
the atgfAta~...tan acquires the mass m2 = g2; however, the model is still gauge invariant under
As it was the case with two and four dimensions in I, by imposing canonical quantization we must put and/3 to be equal to zero. For a = 13= 0 the lagrangian density (16) is invariant under the transformations:
Atal ...tin ~ Atil ...un + 3[tal Ati2...tan] ,
Aul ...tin "-~'Atil ...tan + 3[tilfti:2 ...Unl '
Btil ...Un -l ~ Btia...~n-a + ~ [tal Xti2...tan-i ] '
(24)
provided ftix ...tin-l satisfies 3ta(3ltifti, ""ta/"/-I ] ) = 0 .
(25)
The relation (25) ensures the invariance of the eq. (8), because o f I--13[ufu I ...tan-l} = 3[ta(3[tal 3°fwlu2...tan-1 ]) = 0. With c~= j3= 0 our model can be analyzed entirely in terms of the Lagrange multiplier fields Ati~...tan -I and Bu~ ...tan-~. In particular, in two dimensions we have shown in I that it is possible to express A u solely in terms of two scalar fields. Clearly, such a situation is reminiscent o f the Schwinger model [7]. It is straightforward to establish the more precise analogy by introducing a modified version of the lagrangian density (16):
(28)
with Ati I ...tan -1 which satisfies (25). Note that if we introduced the free lagrangian density forAul ...tan-x instead of that for Bu~...tan_z into the expression (26), then we would obtain the massive equation (27) for 3tiAtita2...ti n . The two-dimensional version of (26) is well-known in the literature as the bosonized Schwinger model [8]. We will now concentrate on a four-dimensional model which includes fermions and we start, by analogy with the two-dimensional Schwinger--Thirring model [9], with .~ = ~ 3x A tiv3X AtiV + ½ ~x A ov OOA XV - AUV~taA v - A t i ~taA - ½ m, l tiA ti - ~1.14'7ti ~3ta 4' + ½Ati,,e,,oti"~t/ + g
2(~Otiv~)(~oti,~) .
(29)
The gamma matrices here are real, (75)2 = --1, ½euu,~t3o'~t3 = otiz,')'5 and K is a parameter with the dimension of mass which sets a mass scale in the model. The equations of motion for A, Ati and Ati v are
= - ( 1 / 2 n ! ) 3xAti I ...un 3xA ul ...tin
+ [1/2(n - I)!] 3xAtal...tinaUlAXti2 ...tin - [l/(n - 1)!]Aul ...tan3UlA u2"''tan + [g(-l)n2/n!(n-1)!]
3taAti = 0 ,
(30a)
X e vl''blxl'xnAvl...u n 3hBx~...an
~3OAata = aAti + ~3u A ,
(30b)
- ~ [ed(n - 1)!1Ati~ ...Un -1
HAta v - 3[u3OA,aI~I - 3[tiAvl +~tar = 0,
(30c)
[3A = []Ati = 0 ,
(30d)
-
[ 1/(n - 2)!]Aut ...Un-~ 3ti~Ata:'"Un -~
+ [I/2(n - 1)!] 3xBta~...tan_~ 3hB t a t t i n - ~ -
[l/2(n - 2 ) ! ] 3xBu~...tan_~3U~B xta~'''tin-~ . (26)
Note, that g is a constant with the dimension of mass and that we have added the conventional lagrangian density for the Bu~..4~n_~ field• We can now derive from (26) the Euler-Lagrange equations of motion from which it follows that (VI + g2)OUAuta~...u n = 0 ,
(27)
or
(V1 + g2) * Fu2...tin = 0 . with *Fta~...tan = [1/n~• ]G t a 2 . . . l a , , ,vu, .
where/u~, = ~o,~ff is the conserved fermionic current. It is easy to verify that the dual currentTu v = ~otav75 ~k is not conserved, by using an argument analogous to that of Jackiw [10]. Because of the conservation of the current the relevant correlation function must be written as (0 ~ u u ( x ) / a o ( Y ) l O ) = ~ uv,aa[l(x - y ) ,
[~tavoo is defined by (11)]. From (31) we obtain (OI3UTu~,(x)]oo(y)lO) 4= 0 and this indicates the existence of the anomaly. If we postulate the following form for the anomaly equation 3ta Tti u = c K 3ti.4 ti z, ,
...1;n
F vv~'''vn Hence
(31)
(32)
with an unknown proportionality constant c, then 191
Volume 126B, number 3,4
PHYSICS LETTERS
from (32) and eqs. of motion (30a)--(30d) we will get (l-] + ctc2) 0u.4~v = 0 .
(33)
Observe now that the current conservation implies the existence of a vector potential Bu such that (34)
/ . u = ~ c . . o o aa B a ,
apart from some proportionality factor, which we ignore. With the equivalence relation (34) we are led to the bosonized version of (29)
.~ ~ - } ~ x A u , a x A u" + } OxA,oz,aaA x" - A,v~uA~, - AuaUA
_
1 ~oeAUA u+ ½K e~UP°Am, Oag a
+ ¼0 [uBv113[uBvl .
(35)
Because o f B o ~ Auv mixing the field a#.4uu will acquire mass as seen from (I--1+ k2)a~.4~u = 0. A model similar to (35) exhibiting the same mass generation mechanism, has already been found in the literature [11]. Here we want to draw attention to the features equivalent to those encountered in the Schwinger model. The four-dimensional version o f the anomaly equation has now the form
Oa(i3 [aBol
+ K eao,~Aa~
) = 0.
(36)
The "magnetic" field tensor F u r = aiuBul is gauge invariant under B u -* B u + a u A but is not conserved, hence the "magnetic" symmetry B . -* B~ + 0uA is broken through the anomaly. The equations of motion ensure that the conserved quantity is aiuBv I - tCeuvaoAa° , which leads to a definition o f the conserved magnetic current as
gu = ~ eua~w OaAOW •
(37)
By analogy with the Schwinger model we expect then the dyon field ~b to carry a "magnetic" charge. To summarize: We have presented here a consistent theory o f the atgf which contains both the massless and
192
30 June 1983
the massive case. In the massless limit, by requiring extended gauge invariance we were led to the set of auxiliary tensor fields which absorbed all degrees of freedom of the original atgf. When the free lagrangian density o f one of the auxiliary fields is added to the lagrangian system the generalized Schwinger-Higgs mechanism takes place and this time the degrees o f freedom o f the auxiliary field are absorbed providing the original atgf with a mass in a gauge invariant way. It is a pleasure to thank my colleagues at the Department o f Physics at Technion for their kind hospitality. Thanks are due to T. Jacobson for correcting my English. This work was supported by the Lady Davis Fellowship Foundation.
References [ 1] M. Kalb and P. Mamond, Phys. Rev. D9 (1974) 2273; Y. Nambu, Phys. Rep. 23C (1976) 250. [2] M. Lusher, Phys. Lett. 78B (1978) 465; M. Aurilia, Phys. Lett. 81B (1979) 203. [3] P. van Nieuwenhuizen, Five Lectures on Supergravity (March 1982), University of Toronto; M.J. Duff, in: Supergravity 81, eds. S. Ferrara and J.G. Taylor (Cambridge U.P.) and references cited therein, to be published. [4] N. Lehto, H.B. Nielsen and M. Ninomiya, Phys. Lett. 115B (1982) 129; P. Orland, Nucl. Phys. B205 (FS5] (1982) 107. [51 H. Aratyn, Phys. Lett. I13B (1982) 248. [6] Z. Tokuoka, Phys. Lett. 87A (1982) 215. [7] J.H. Lowenstein and J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172; N. Nakanishi, Prog. Theor. Phys. 57 (1977) 580. [8] J. Kogut and L. Susskind, Phys. Rev. DI 1 (1975) 3594; P.A. Marchetti, Nuovo Cimento 60A (1980) 13. [9] J. Kogut and P.K. Sinclair, Phys. Rev. D10 (1974) 4181 ; H. Aratyn, Z. Phys. C9 (1981) 243. [ 10] R. Jackiw, Gauge invariance and mass, III. MIT CTP 965 (1982); R. Jackiw, in: Laws of hadronic matter, ed. A. Zichichi, (Academic Pre~, New York, 1975). [ 1 i ] A. Aurilia and Y. Takahashi, Prog. Theor. Phys. 66 (1981) 693.
PItYSICS LETTERS
30 June 1983
RANK.n ANTISYMMETRIC TENSOR GAUGE FIELDS IN 2n-DIMENSIONAL SPACE-TIME AND MASS GENERATION MECHANISM H. ARATYN
Department of Physics, Technion-lsrael Institute of Technology, Haifa, lsrael Received 4 November 1982
By introducing mutually dual gauge fixing conditions we achieve a realization of a selfdual rank-n antisymmetric tensor gauge field in 2n-dimensional space-time as an object completely described by lower rank auxiliary tensor fields. Our construction can be extended to yield a mechanism of mass generation corresponding to the bosonized version of the generalized Schwinger model.
Antisymmetric tensor gauge fields (atgfs) have growing application in many branches o f quantum field theory. They have been discussed in connection with string models [ 1], the confinement problem [2], supergravity [3] and recently they also had received attention in lattice field theory [4]. In this paper we consider the class o f completely antisymmetric rank-n tensor gauge fields Aul...un(X) in 2n-dimensional s p a c e - t i m e . The field dual to Aul ...un(x) is defined by
7%~..~ = (1/n !) e,l ...~ u~ ...un A u~ ...Un
( l)
and has the same rank (number o f indices)as the original field. Our class thus contains: the Maxwell field Au in two dimensions, the K a l b - R a m o n d field Auv in four, atgfAuz,~" o f third rank in six and so on. The corresponding antisymmetric field tensors
Ful...Un+, (x) = 3lulAuz...Un÷l ](x) ,
(2)
are invariant under the generalized gauge transformation
Au,...un(X)-'> Aul...un(X) + 3[ulfu2...unl(X) ,
(3)
where ful ...Un-1 (x)are arbitrary antisymmetric tensor functions o f rank-(n - 1) and where we have introduced the generalized rotation 3[ulAu2 ...un] defined by
~[u Au2...unl =cycli~cperm!-l)"~u~qAu,r2...Unn .
(4)
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © i 983 North-Holland
As already pointed out in our previous work [5], which we from now on will refer to as I, in the cases of two- and four-dimensions the invariance under the extended gauge transformation
"4ut...un -" Au, ...Un + fu~...Un '
(5)
can be established provided we introduce the set of two auxiliary antisymmetric tensor fields Aul.." Un-i (x), But.-.Un -t (x) with rank one lower than that o f Au~...un(x). If we assume that the tensor fields Aul...Un -1 (x) and Bul...un_l (x) transform as
Aul...Un_ 1 --" Bul...un_ 1 + OPfpal...Un_ 1 ,
(6)
Bux...un_l -* Bu,...Un_, + O°?oul...un_l
(7)
[ w i t h f , l... ~ related toful...u n through (1)] then it can be shown tha the field equation
eUl ..~n" I...~n O~l Bv~.,.~
(8)
= ( - 1 ) n 2 - 1 ( n _ l)!(E3Aul...u n -b[ulAu2...unl ) , is invariant under the transformation (5) accompanied by (6) and (7). The p r o o f of this is based on the trivial identity obeyed by the generalized rotation [6]
3U(Oluful...unl ) = [--]ful...Un - 3[Ul Ouflulu2...unl ' (9) where the index isolated by the vertical bars I 1does not participate in the antisymmetrization process. One can easily see that for two and four dimensions 189
V o l u m e 1 2 6 B , n u m b e r 3,4
30 J u n e 1 9 8 3
PHYSICS LETTERS
(8) reduces to eqs. (19) and (39e) in I. Thus in four dimensions formula (8) yields EIAu ~ expressed in terms of the "electric-" and "magnetic-like" vector fields Au and Bu. Let us not introduce two spin-one projection operators in the space of the antisymmetric rank two tensors, defined in momentum space by
= - ( 1/2n!) i~xAul ...UnaXAUl ...V.n
+PuPog.,o),
X ~l'"VnXt'"XnAvl ...Un~XiBx2...hn 1
~ [a/(n - 1)!]Aul...Un_IA ul"''un-t
+ ½ [B/(n - 1)!]Bvt...Vn_xB ~q'''vn-I
(10) -
(~uv,o a)H" = Iuv,,oo - (~uv,.oa)F'r,
(11)
1)!]Aul...Uni~U2A "2"''un
+ [(-1)n2/(n - 1)!n!]
-
( ~ uv,oo)FT = (1/2p2) (P uP ogvo - P uP OgUa - PuP oguo
[l/(n-
-
-
[ 1/(n -
2)!1Aut .-.Un-I ~1AU2...~n-i
[1/(n - 2)!] Bht...xn_ l ~hlBh2""Xn~l .
(16)
where Iuv,o a = !2 (~" ",,upovo ~ - gla~gvo,'~ And we notice, with respect to the four-dimensional duality relation
The equations of motion derived from (16) are
I-'lAuv = i~[uAul - eu~aoaOBa ,
OUlAulu2...Un-I = 0 ,
(17)
that the following formulas hold
~XlB~qx2...Xn_ l = 0 ,
(18)
~ uv,o~[OA°] = ~[uAu] , ~uu,ooeo°Xr~xBr = 0 ,
~Ut Aulu2...V.n - otAu2...u n - i~[u2Au3...Un] = 0 , (19)
~u~,ao~[° A °l
~)alA~.la2...x n -/3Bx2...x n
=
(12)
0 , ff uu~ooea°Xr~xBr = e~hr~XB r.
By introducing the tensor -,,
o
.
8x~..-xn
¢~l.tl...#nvl...u n - 6 / a r k I . . . 6 # n ^ n Vl...~n ,
(13)
where 8~x~.iTu~nis the generalized Kronecker delta in 2n-dimensional space, we can define the projection operators in the space ofantisymmetric rank-n tensors by ( ~l~U~...lan , v l . . . ~ ) F T =
(14)
(1/n!p2)p[u~pU8 lulu~...un l,uv..u n ,
( ~ u ~ ...Un,Vt ...vn)FT = (1/n! p2) pU(p [v.S ui ...Un ] ,V~...Vn) . (15) These operators when acting on the right hand side of eq. (8) will project the Au~...Un -1 -term or Bux...Un_~ term depending on the use of ~ o r ~ operator, exactly as it happened in the four dimensional case above. We will now present the 2n-dimensional lagrangian which leads to the field equation (8).
+ a[h2Bh3...kn]
= 0 ,
and eq. (8) obtained from variation of the lagrangian density (16)with respect to Aul...u n. Furthermore the lagrangian density (16) is invariant up to a total derivative under the combined gauge transformations (5), (6) and (7) as can be shown using again the identity (9). As seen from the equations of motions (8) and (17)--(20) there are only two consistent values for the gauge parameters a and/3 given by a = 13= 1 or a =/3 = 0. The commutation relations for Aul...Un, Aul .--Un-~ and Bu~...Un_x consistent with the equations of motion have the following form
[Au I ...#n(X), Au I ...vn(Y)] = iabux ...,n,Vl...vnD( x - Y ) + iaeul ...un ux..-UnD ( x - y ) '
(21)
[Aui...un-i (x), A~I ...~0')1 = i3"6,,~...,n_1 ,~, ...~, X D ( x - y ) + i s 8Ueuu ~...un_l ~l...Vn D ( x - y ) ,
[Bul ...Un-,
(22)
(x), A ~1 ..-.n (Y) 1 = i ~Ueu., ...Un-i ~, ...~n ( 2 3 )
X D(x-y)-
• --
n2
1( 1)
a~6uul...Un_t,vl...unD(x-y),
where
D(x) = - i ( 2 ~ ) -2n+l
190
(20)
f dZ~p e(P0)8(p 2) exp(ipx).
Volume 126B, number 3,4
PHYSICS LETTERS
30 June 1983
the atgfAta~...tan acquires the mass m2 = g2; however, the model is still gauge invariant under
As it was the case with two and four dimensions in I, by imposing canonical quantization we must put and/3 to be equal to zero. For a = 13= 0 the lagrangian density (16) is invariant under the transformations:
Atal ...tin ~ Atil ...un + 3[tal Ati2...tan] ,
Aul ...tin "-~'Atil ...tan + 3[tilfti:2 ...Unl '
Btil ...Un -l ~ Btia...~n-a + ~ [tal Xti2...tan-i ] '
(24)
provided ftix ...tin-l satisfies 3ta(3ltifti, ""ta/"/-I ] ) = 0 .
(25)
The relation (25) ensures the invariance of the eq. (8), because o f I--13[ufu I ...tan-l} = 3[ta(3[tal 3°fwlu2...tan-1 ]) = 0. With c~= j3= 0 our model can be analyzed entirely in terms of the Lagrange multiplier fields Ati~...tan -I and Bu~ ...tan-~. In particular, in two dimensions we have shown in I that it is possible to express A u solely in terms of two scalar fields. Clearly, such a situation is reminiscent o f the Schwinger model [7]. It is straightforward to establish the more precise analogy by introducing a modified version of the lagrangian density (16):
(28)
with Ati I ...tan -1 which satisfies (25). Note that if we introduced the free lagrangian density forAul ...tan-x instead of that for Bu~...tan_z into the expression (26), then we would obtain the massive equation (27) for 3tiAtita2...ti n . The two-dimensional version of (26) is well-known in the literature as the bosonized Schwinger model [8]. We will now concentrate on a four-dimensional model which includes fermions and we start, by analogy with the two-dimensional Schwinger--Thirring model [9], with .~ = ~ 3x A tiv3X AtiV + ½ ~x A ov OOA XV - AUV~taA v - A t i ~taA - ½ m, l tiA ti - ~1.14'7ti ~3ta 4' + ½Ati,,e,,oti"~t/ + g
2(~Otiv~)(~oti,~) .
(29)
The gamma matrices here are real, (75)2 = --1, ½euu,~t3o'~t3 = otiz,')'5 and K is a parameter with the dimension of mass which sets a mass scale in the model. The equations of motion for A, Ati and Ati v are
= - ( 1 / 2 n ! ) 3xAti I ...un 3xA ul ...tin
+ [1/2(n - I)!] 3xAtal...tinaUlAXti2 ...tin - [l/(n - 1)!]Aul ...tan3UlA u2"''tan + [g(-l)n2/n!(n-1)!]
3taAti = 0 ,
(30a)
X e vl''blxl'xnAvl...u n 3hBx~...an
~3OAata = aAti + ~3u A ,
(30b)
- ~ [ed(n - 1)!1Ati~ ...Un -1
HAta v - 3[u3OA,aI~I - 3[tiAvl +~tar = 0,
(30c)
[3A = []Ati = 0 ,
(30d)
-
[ 1/(n - 2)!]Aut ...Un-~ 3ti~Ata:'"Un -~
+ [I/2(n - 1)!] 3xBta~...tan_~ 3hB t a t t i n - ~ -
[l/2(n - 2 ) ! ] 3xBu~...tan_~3U~B xta~'''tin-~ . (26)
Note, that g is a constant with the dimension of mass and that we have added the conventional lagrangian density for the Bu~..4~n_~ field• We can now derive from (26) the Euler-Lagrange equations of motion from which it follows that (VI + g2)OUAuta~...u n = 0 ,
(27)
or
(V1 + g2) * Fu2...tin = 0 . with *Fta~...tan = [1/n~• ]G t a 2 . . . l a , , ,vu, .
where/u~, = ~o,~ff is the conserved fermionic current. It is easy to verify that the dual currentTu v = ~otav75 ~k is not conserved, by using an argument analogous to that of Jackiw [10]. Because of the conservation of the current the relevant correlation function must be written as (0 ~ u u ( x ) / a o ( Y ) l O ) = ~ uv,aa[l(x - y ) ,
[~tavoo is defined by (11)]. From (31) we obtain (OI3UTu~,(x)]oo(y)lO) 4= 0 and this indicates the existence of the anomaly. If we postulate the following form for the anomaly equation 3ta Tti u = c K 3ti.4 ti z, ,
...1;n
F vv~'''vn Hence
(31)
(32)
with an unknown proportionality constant c, then 191
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PHYSICS LETTERS
from (32) and eqs. of motion (30a)--(30d) we will get (l-] + ctc2) 0u.4~v = 0 .
(33)
Observe now that the current conservation implies the existence of a vector potential Bu such that (34)
/ . u = ~ c . . o o aa B a ,
apart from some proportionality factor, which we ignore. With the equivalence relation (34) we are led to the bosonized version of (29)
.~ ~ - } ~ x A u , a x A u" + } OxA,oz,aaA x" - A,v~uA~, - AuaUA
_
1 ~oeAUA u+ ½K e~UP°Am, Oag a
+ ¼0 [uBv113[uBvl .
(35)
Because o f B o ~ Auv mixing the field a#.4uu will acquire mass as seen from (I--1+ k2)a~.4~u = 0. A model similar to (35) exhibiting the same mass generation mechanism, has already been found in the literature [11]. Here we want to draw attention to the features equivalent to those encountered in the Schwinger model. The four-dimensional version o f the anomaly equation has now the form
Oa(i3 [aBol
+ K eao,~Aa~
) = 0.
(36)
The "magnetic" field tensor F u r = aiuBul is gauge invariant under B u -* B u + a u A but is not conserved, hence the "magnetic" symmetry B . -* B~ + 0uA is broken through the anomaly. The equations of motion ensure that the conserved quantity is aiuBv I - tCeuvaoAa° , which leads to a definition o f the conserved magnetic current as
gu = ~ eua~w OaAOW •
(37)
By analogy with the Schwinger model we expect then the dyon field ~b to carry a "magnetic" charge. To summarize: We have presented here a consistent theory o f the atgf which contains both the massless and
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30 June 1983
the massive case. In the massless limit, by requiring extended gauge invariance we were led to the set of auxiliary tensor fields which absorbed all degrees of freedom of the original atgf. When the free lagrangian density o f one of the auxiliary fields is added to the lagrangian system the generalized Schwinger-Higgs mechanism takes place and this time the degrees o f freedom o f the auxiliary field are absorbed providing the original atgf with a mass in a gauge invariant way. It is a pleasure to thank my colleagues at the Department o f Physics at Technion for their kind hospitality. Thanks are due to T. Jacobson for correcting my English. This work was supported by the Lady Davis Fellowship Foundation.
References [ 1] M. Kalb and P. Mamond, Phys. Rev. D9 (1974) 2273; Y. Nambu, Phys. Rep. 23C (1976) 250. [2] M. Lusher, Phys. Lett. 78B (1978) 465; M. Aurilia, Phys. Lett. 81B (1979) 203. [3] P. van Nieuwenhuizen, Five Lectures on Supergravity (March 1982), University of Toronto; M.J. Duff, in: Supergravity 81, eds. S. Ferrara and J.G. Taylor (Cambridge U.P.) and references cited therein, to be published. [4] N. Lehto, H.B. Nielsen and M. Ninomiya, Phys. Lett. 115B (1982) 129; P. Orland, Nucl. Phys. B205 (FS5] (1982) 107. [51 H. Aratyn, Phys. Lett. I13B (1982) 248. [6] Z. Tokuoka, Phys. Lett. 87A (1982) 215. [7] J.H. Lowenstein and J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172; N. Nakanishi, Prog. Theor. Phys. 57 (1977) 580. [8] J. Kogut and L. Susskind, Phys. Rev. DI 1 (1975) 3594; P.A. Marchetti, Nuovo Cimento 60A (1980) 13. [9] J. Kogut and P.K. Sinclair, Phys. Rev. D10 (1974) 4181 ; H. Aratyn, Z. Phys. C9 (1981) 243. [ 10] R. Jackiw, Gauge invariance and mass, III. MIT CTP 965 (1982); R. Jackiw, in: Laws of hadronic matter, ed. A. Zichichi, (Academic Pre~, New York, 1975). [ 1 i ] A. Aurilia and Y. Takahashi, Prog. Theor. Phys. 66 (1981) 693.