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Massless fermionic free fields
Volume 186, number 3,4
PHYSICS LETTERSB
12 March 1987
MASSLESS FERMIONIC FREE FIELDS J.M.F. LABASTIDA 1 The institutefor Advanced Study, Princeton, NJ 08540, USA
Received 17 November 1986
The covariant description of massless fermionic free particles in any dimemsion which carry arbitrary representationsof the Lorcntzgroup is presented.
Recent work [ 1 ] has shown that gauge invariance plays a crucial role in constructing free field theories for massless bosonic particles. Using the principle of gauge invarianee the description of free massless bosonic particles in any space-time dimension carrying arbitrary representations of the Lorentz group has been obtained. The importance of gauge invariante was first pointed out in this context by Curtright [2], who constructed free field theories for totally symmetrized representations. He obtained in an elegant manner the same results as in previous treatments [ 3 ] of the problem, in which the massless case was considered as a limit of the massive one [4]. Later studies [ 5,6 ] have culminated in the work presented in ref. [ 1 ] in which the gauge principle combined with methods inspired by string-field theories have provided the description of all mixed-symmetry representations, which were only slightly analyzed in previous works [ 5,6 ]. The analysis of fermionic representations of the Lorentz group started long ago with the classical works of Dirac [ 7 ] and Rarita and Schwinger [ 8 ]. Totally symmetric fermionic representations have been also studied by Curtright [ 2] using the principle of gauge invariance. His elegant formulation agrees with the one obtained previously [ 9 ] considering the massless limit of the massive formulation [ 10]. Mixed-symmetry representations have been considered only very slightly [ 5 ] for fermionic particles. In this note we present the description of 'Research supported by US DOE contract DE-AC0276ER02220.
massiess fermionic particles carrying arbitrary representations of the Lorentz group. This formulation has the same structure as the one presented in ref. [ 1 ] for massless bosonic particles. The little group of a massless fermionic particles in d dimensions is S I ( d - 2 ) . However, we will be concerned with irreducible representations of O ( d - 2 ) . For d odd this is enough because erach irreducible representation of O ( d - 2 ) remains irreducible on restriction to the unimodular subgroup SO ( d - 2 ) . Some of these irreducible representations become equivalent (the so-called associated in O ( d - 2 ) ) . They account for the ordinary and dual descriptions. For d even the situation is different. Besides the equivalence between associated representations (which simply accounts for the ordinary and dual descriptions), each spinor irreducible representation of O ( d - 2) splits into two spinor irreducible representations of S O ( d - 2 ) under the restriction to the unimodular subgroup. We will be dealing in this note with the covariant description of the spinor irreducible representations of O ( d - 2 ) . To reduce further these representation for the case of even d one needs to impose certain conditions on the covariant fields. How this will be carried out will be discussed elsewhere [ 11 ]. We will denote the irreducible spinot representations of O ( d - 2 ) by using the standard [ 12] Young tableaux (YT), notation (al, a2..., aM), where a~indicates the number of boxes in the ith row. These representations contains a~ + a2 +..., ipaM vector indices and one spinor index. We always suppress the spinor 365
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index. Representations of G L ( d ) are labeled by the standard notation [ al, a2 ..... aM]. Before presenting our analysis we will review the principle of gauge invariance as stated in ref. [6]. Given an arbitrary spinor representation of the little group of a massless fermionic particle in d dimensions, (al, a2 .... , aM), we will postulate that the covariant description can be formulated with a field g/whose vector space-time indices have the structure of the YT corresponding to the irreducible representation o f G L ( d ) [ al, a2, ..., aM]. Additionally, we postulate that the action of this field is invariant under gauge transformations whose gauge parameters have vector index structures cotrresponding to all the YT that one can make by removing one box from the YT of the original field. The gauge ntransformations postulated above can be expressed in a very simple way [ 1 ] using the context of string-field theory [13]. Consider a vector-field I ~ ( x ) ) ~ in the Fock space spanned by a set of N covariant oscillators a ~ which satisfy [ a ~ , a~] =~m,r/~,
(1) d--I
where rff'= (1, - 1 , - 1.... , - I i. The vector field I ~ ( x ) ) repre"gents a collection of local Majorana spinor fields which are just the coefficients of its expansion,
I~(x)>=~/(x)l >+~/2(x)a~l > + ~t)umnl~at(l~nt v) 2v'lzv x ~ , . . . .
-~- L/~mnl v h ~ t [ ~ a tv]
2v-~
~,~,
....
[ )
I ) +...,
(2)
where [ ) represents the vacuum. For N>~ d all the representations appear in the expansion (2). Gauge transformations are defined by
81 ~'(x) ) =a~ OuIA"( x) ),
(3)
where lAin(x)) are spinor gauge parameters with expansion )
~
,
.m
IA,(x) ) - 2 ~ ( x ) I ) + ; t T ( x ) a ~ I ) + ....
(4)
As in the bosonic case [ 1 ], one can verify expanding both sides of (3) that these gauge transformations are of the form postulated above. We now construct the vector-field equation that I ~ ( x ) ) must satisfy. ~ I ~V(x)) carries one spinor index. 366
12 March 1987
This involves the construction of an operator (9 such that the equation (91 ~ ( x ) ) =0,
(5)
possesses the following properties: (a) (9 must be a first-order differential operator; (b) if we normal order the terms entering into (9, each term must have the same number of creation and annihilation operators; (c) the terms entering into (9 can not be operators that involve the Dirac operator and one or more pairs of creation and annihilation operators; and, (d) the vector-field equation (5) must be invariant under the gauge transformations (3). The reasons for requiring these propereties are entirely equivalent to the ones discussed in ref. [ 1 ] and we will not discuss them here. These four properties determine the operator (9 uniquely up to overall normalization and provides us with the constraints that the gauge parameter must satisfy. We describe now the construction of (9. The possible normal-ordered terms entering into (9 are classified according to their number of creation operators (degree) and then by the inequi .¢alent ways that their indices can be contracted. In this classification there are one operator of degree zero, the Dirac operator, 7uOu, two of degree two,TuamtUa~O~, ~ 7 a~a~O (notice that the ones with three 7-matrices can be expressed in terms of these ones and operators that are forbidden by property (c)), etc. In the analysis of the gauge invariance of (5) the following observation is crucial. After performing the gaa'ge transformation (3) in (5), each term in (9 gets an extra creation operator. This operator must be commuted to the left to obtain normal-ordered operators. In doing this, two kinds of operators are generated. Suppose we started with an operator in (9 of degree n. This operator generates normal-ordered operators with n creation operators and n - 1 annihilation operators and a remainder which consists of n + 1 creation operators and n annihilation operators. For the case of degree zero there is a reminder only. Performing a gauge transforming in (5), cancellations can occur if the coefficients of the terms entering (9 are arranghed in such a way that the remainders of the operators of degree n are cancelled by the non-remainders originated by the operators of degree n + 1. To analyze the arrangement of the coefficients we start with the Dirac operator (which has to be nee-
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essarily in ¢ in order to describe spin-l/2 massless particles properly) and the operators of degree one. Let us assume that we fix the overall normalization in such a way that the Dirac operator has coefficient one. After performing a gauge transformation, this operator generates a reminder which has to be canceled by the non-reminder operators generated from the action of the gauge transformation on the operators of degree one. It is very simple to observe that to achieve this only of the operators have a non-vanishing coefficient. In fact, the only operator of degree one entering (9 is ?ua~aUO~ with coefficient 1. This operator generates a reminder, tp tp amOvOplAn(x)), u 7ua,,,an whose only chance for cancellation may come from the non-reminder operators of degree two. There exist operators of degree two in agreement with properties ( a ) - ( c ) such that they posses more than one ?-matrix. However, there operators do not play any role when trying to cancel the reminder from the operator of degree one because they are such that the number ofT-matrices that they contain do not decrease after performing a gauge transformation. There are 13 inequivalent of degree rwo with one ?-matrix. The non-reminder terms generated from these operators after performing the gauge transformation (3) are 16 inequivalent ones (which are the complete set which could appear). One can verify that the resulting linear system does not possess any solution in such a way that the reminder of degree one is canceled. Gauge invariante forces us to stop here and to impose the following constraint on the gauge parameters: ?ua~n IA,,)(x) ) =0.
(8)
where IAn,,(x)) constitute a set of second-generation gauge parameters. The gauge parameters IA,,m(X) ) are such that I ~U(x) )remains invariant under a double gauge transformation. This implies that they must be constrained according to
atm
(9)
This constraint indicates that not all the components of the gage parameters IAn,,(x)) are independent. This fact is very well known and, in fact, the classification of generation of gauge parameters presented in ref. [6] also applies here and is in agreement with (9). This formalism provides more information about the constraints satisfied by the gauge parameters. It is well known that the constraint imposed on the first generation implies new constraints on higher generations. To keep track of these constraint in the approach presented in ref. [6] is rather difficult. However, in this context, those constraints appear simply as consistency between (6) and (8) and therefore they can be stated in a simple way:
?ua~naT~l IA,,)p(x) ) =0,
(I0)
where Iq [ means that the index q must be excluded from the symmetrization. The conditions satisfied by higher generations can be analyzed in a similar way. For example, for the third one:
8 IA,m(X) ) =atpUOuIAn,.p(x) ), at~uatv) IA,m~,(x) ) =0, rrt ~ p
(6)
The other conditions that could be imposed to get rid of the reminder are too restrictive or differential. Constraint (6) is consistent with the results known from totally symmetric representations [ 2 ]. We conelude that gauge invariance singles out the operator (9 to be "1"~' .u (9=?/z Ou+Tua,,,a,,O~,
8 IAn(x)) =a~O u I A , , , ( x ) ) ,
12 March 1987
(7)
and forces the gauge parameters to be constrained according to (6). As in the bosonic case, this formalism provides a very simple way to deal with higher-generation gauge invariance. The gauge transformation of the firstgeneration gauge parameters _+An( x ) ) is given by
t(u ant~) apta IAn.,p(x))=0, am v ~ct 7ua /x
(11 )
The number of conditions increases with the number of generations and the series of gauge invariances eventually stops. As we observe from (11 ), the gauge parameter [Amnp(x) ) is antisymmetric in np and in ran. This implies that for N= 2 the parameters vanish. In general, the gauge parameter of the ith generation, [A. . . . . . . . . . . . (X)), is antisymmetric in mlm2, m2m3, ..., rni_ ~mi. This means that for i > N t h e constraints are so strong that the generation vanishes identically. A consequence of this is that the number of generations for a given local field is smaller or equal 367
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than d. IfN~< dthis is obvious. In fact one has a lower bound which is the expected [ 6 ] one, since with N different oscillators the maximum number of rows of the YT corresponding to local fields is N. If N > d the gauge parameters IA. . . . 2....... ( x ) ) with i> d do not correspond to any of the local fields of I ~v( x ) ) and so, although they do not vanish as a consequence of the constraints above, they are irrelevant. The theory presented in this note involves fermionic local fields carrying vector indices which correspond to all the irreducible representations of GL(d) and a series of generations of gauge parameters for each local field. To verify that this theory do indeed correspond to a covariant description of fermionic particles carrying arbitrary representations of the Lorentz group one needs to analyze the physical degrees of freedom that the theory possesses. One way to do this analysis is to study the field equation and gauge invariances associated to each of the local fields. The counting of physical degrees of freedom turns out to be right for representations whose YT have at most two columns. However, for representations with three or more columns the counting is not correct. One has too many physical degrees of freedom. This does not surprise us since we know from the analysis of totally symmetric representations that one is forced to constraint the local field itself [2,9]. In this context, the constraint that seems to work and that reduces to the known one for totally symmetric representations is
Lmnp I ~V(x) ) =0,
(12)
where L , ~ p -.- ?utl~a(manap). ~ a u
(13)
In definition (12) the symmetrization comprises the three indices m, n and p. This constraint is not a consequence of gauge invariance. It must be imposed on the field I ~V(x)). The first question that has to be answered is whether or not this constraint is compatible with the field equation (5) with d7 as expressed in (7). In other words, we need to know if imposing the constraint on I ~ ( x ) ) at an initial time, I ~V(x) ) verifies it at later times. The answer to this question is affirmative since one obtains, after some algebra, that
[L,,,,,p, (9] = - 37UOuL,.,p - 27,aqtu aqOuL,nn r ~ + ~7~aqtu a~qOuLmno). 368
(14)
12 March 1987
This tells us that the modes corresponding to the traces" involved in (12) are effectively decoupled in the field equation. In fact, one can verify that constraint (12) is the only one that reduces properly to the known cases (totally symmetric representations) and is compatible with the field equation (5). Another question related to constraint (12) that has to be answered is whether or not it implies certainm constraints on the gauge parameters. Based on the knowledge from totally symmetric representations one expects a negative answer to this question. To prove this we show now that constraint (12) is gauge invariant under the gauge transformation (3) with gauge parameters constrained according to (6). Performing a gauge transformatioon in (12) one finds, after commuting the resulting creation operator to the left,
7url~a~,~a~Ou[Ap) ) +27ua~%aUO~ lAp) ) tP ~ P /~ -- ~,urh~Baq a (ma,,ap) Op IAq ) =0.
(15)
We now prove that because of the constraint (6), eq. (15) is satisfied. The first two terms vanish trivially in virtue of (6). To prove that the last term does also vanish we operate on (6) with ~,,,),,,a~,a~: ~ u IAm))=O. ~a~Fv~uapaqa(n
(16)
Symmetrizing in p, q and n one finds O"
V
/g
rh,~9,ua(paqa~) IA,,, ) "t-
.u o V rlo~,uama(paq IA,) ) =0.
(17)
On the other hand, taking again (6), acting on it with 2)~a~,, and symmetrizing in all roman indices one finds /.t v rl~a(pa, lA,,) )=O,
(18)
which implies that the last term in (17 ) vanishes and so, (17) implies that the last term in (15) also vanishes, completing the proof. The fact that constraint (12) is gauge invariant is in agreement with known results for totally symmetric representations. What is remarkable is that this occurs in general, i.e. constraint (101) reduces the number of independent components of the field I ~ ( x ) ) but is does not imply any condition on the ghost content of the theory.
Volume 186, number 3,4
PHYSICS LETTERS B
We have analyzed m a n y o d the representations and the conjecture that ( 5 ), ( 6 ), ( 7 ) a n d ( 11 ) constitute the complete description seems to be correct (details will be shown elsewhere [11 ]). The counting o f degrees o f f r e e d o m becomes very c o m p l i c a t e d for representations corresponding to Y T with m a n y boxes. Certainly, a general p r o o f in this respect should be preferred. The f o r m u l a t i o n presented in this note for the covariant description o f fermioonic fields is simple a n m d unique. This makes us suspect that this f o r m u l a t i o n m a y c o r r e s p o n d to an o p e r a t o r representation o f some new k i n d o f theories which involve fields carrying arbitrary representations o f the Lorentz group. In this note a n d in a previous one [ 1 ] we have developed the free a n d bosonic parts o f these theories. These formulations can generate the a d e q u a t e framework to describe interactions a m o n g massless particles in a r b i t r a r y representations. However, we believe that the right way to proceed towards interactions m u s t begin with the u n d e r s t a n d i n g o f the (geometric?) structure behind these free formulations.
12 March 1987
References [ 1] J.M.F. Labastidas, IAS preprint (August 1986). [2] T. Curtight, Phys. Lett B 85 (1979); B. de Wit and D.Z. Freedman, Phys. Rev. D 21 (1980) 358. [3] C. Fronsdal, Phys. Rex'. 161 (1978) 3624. [4] L.P.S.Singh and C.R. Hagen, Phys. Rev. D 9 (1974) 898. [5] T. Curtright and P.G.O. Freund, Nucl. Phys. B 172 (1980) 413; T. Curtright, Phys. Lett. 165 B 304 (1985); University of Florida preprint, UFTP-82-22; G.R.E. Black and B.G. Wybourne, J. Phys. A 16 2405 (1983); R. Delbourgo and P. Jarvis, J. Phys. A 16 (1983) L275; C.S. Aulakh, I.G. Koh and S. Ouvry, Phys. Lett. B 173 (1986) 284. [6] J.M.F. Labastida and T.R. Morris, Phys. Lett. B 180 (1985) 101. [7] P.A.M. Dirac, Proc. R. Soc. A 155 (1936 447. [8] W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61. [9] J. Fang and C. Fronsdal, Phys. Rev. D 18 (1978) 3630. [ 10] L.P.S. Singh and C.R. Hagen, Phys. Rev. D 9 (1974) 910. [ 11 ] J.M.F. Labastida, IAS preporint, in preparation. [ 12 ] M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley,Reading, MA, 1962). [13] W. Siegel, Phys. Lett. B 149 (1984) 157, 162;B 151 (1985) 391,396; W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105.
369
PHYSICS LETTERSB
12 March 1987
MASSLESS FERMIONIC FREE FIELDS J.M.F. LABASTIDA 1 The institutefor Advanced Study, Princeton, NJ 08540, USA
Received 17 November 1986
The covariant description of massless fermionic free particles in any dimemsion which carry arbitrary representationsof the Lorcntzgroup is presented.
Recent work [ 1 ] has shown that gauge invariance plays a crucial role in constructing free field theories for massless bosonic particles. Using the principle of gauge invarianee the description of free massless bosonic particles in any space-time dimension carrying arbitrary representations of the Lorentz group has been obtained. The importance of gauge invariante was first pointed out in this context by Curtright [2], who constructed free field theories for totally symmetrized representations. He obtained in an elegant manner the same results as in previous treatments [ 3 ] of the problem, in which the massless case was considered as a limit of the massive one [4]. Later studies [ 5,6 ] have culminated in the work presented in ref. [ 1 ] in which the gauge principle combined with methods inspired by string-field theories have provided the description of all mixed-symmetry representations, which were only slightly analyzed in previous works [ 5,6 ]. The analysis of fermionic representations of the Lorentz group started long ago with the classical works of Dirac [ 7 ] and Rarita and Schwinger [ 8 ]. Totally symmetric fermionic representations have been also studied by Curtright [ 2] using the principle of gauge invariance. His elegant formulation agrees with the one obtained previously [ 9 ] considering the massless limit of the massive formulation [ 10]. Mixed-symmetry representations have been considered only very slightly [ 5 ] for fermionic particles. In this note we present the description of 'Research supported by US DOE contract DE-AC0276ER02220.
massiess fermionic particles carrying arbitrary representations of the Lorentz group. This formulation has the same structure as the one presented in ref. [ 1 ] for massless bosonic particles. The little group of a massless fermionic particles in d dimensions is S I ( d - 2 ) . However, we will be concerned with irreducible representations of O ( d - 2 ) . For d odd this is enough because erach irreducible representation of O ( d - 2 ) remains irreducible on restriction to the unimodular subgroup SO ( d - 2 ) . Some of these irreducible representations become equivalent (the so-called associated in O ( d - 2 ) ) . They account for the ordinary and dual descriptions. For d even the situation is different. Besides the equivalence between associated representations (which simply accounts for the ordinary and dual descriptions), each spinor irreducible representation of O ( d - 2) splits into two spinor irreducible representations of S O ( d - 2 ) under the restriction to the unimodular subgroup. We will be dealing in this note with the covariant description of the spinor irreducible representations of O ( d - 2 ) . To reduce further these representation for the case of even d one needs to impose certain conditions on the covariant fields. How this will be carried out will be discussed elsewhere [ 11 ]. We will denote the irreducible spinot representations of O ( d - 2 ) by using the standard [ 12] Young tableaux (YT), notation (al, a2..., aM), where a~indicates the number of boxes in the ith row. These representations contains a~ + a2 +..., ipaM vector indices and one spinor index. We always suppress the spinor 365
Volume 186, number 3,4
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index. Representations of G L ( d ) are labeled by the standard notation [ al, a2 ..... aM]. Before presenting our analysis we will review the principle of gauge invariance as stated in ref. [6]. Given an arbitrary spinor representation of the little group of a massless fermionic particle in d dimensions, (al, a2 .... , aM), we will postulate that the covariant description can be formulated with a field g/whose vector space-time indices have the structure of the YT corresponding to the irreducible representation o f G L ( d ) [ al, a2, ..., aM]. Additionally, we postulate that the action of this field is invariant under gauge transformations whose gauge parameters have vector index structures cotrresponding to all the YT that one can make by removing one box from the YT of the original field. The gauge ntransformations postulated above can be expressed in a very simple way [ 1 ] using the context of string-field theory [13]. Consider a vector-field I ~ ( x ) ) ~ in the Fock space spanned by a set of N covariant oscillators a ~ which satisfy [ a ~ , a~] =~m,r/~,
(1) d--I
where rff'= (1, - 1 , - 1.... , - I i. The vector field I ~ ( x ) ) repre"gents a collection of local Majorana spinor fields which are just the coefficients of its expansion,
I~(x)>=~/(x)l >+~/2(x)a~l > + ~t)umnl~at(l~nt v) 2v'lzv x ~ , . . . .
-~- L/~mnl v h ~ t [ ~ a tv]
2v-~
~,~,
....
[ )
I ) +...,
(2)
where [ ) represents the vacuum. For N>~ d all the representations appear in the expansion (2). Gauge transformations are defined by
81 ~'(x) ) =a~ OuIA"( x) ),
(3)
where lAin(x)) are spinor gauge parameters with expansion )
~
,
.m
IA,(x) ) - 2 ~ ( x ) I ) + ; t T ( x ) a ~ I ) + ....
(4)
As in the bosonic case [ 1 ], one can verify expanding both sides of (3) that these gauge transformations are of the form postulated above. We now construct the vector-field equation that I ~ ( x ) ) must satisfy. ~ I ~V(x)) carries one spinor index. 366
12 March 1987
This involves the construction of an operator (9 such that the equation (91 ~ ( x ) ) =0,
(5)
possesses the following properties: (a) (9 must be a first-order differential operator; (b) if we normal order the terms entering into (9, each term must have the same number of creation and annihilation operators; (c) the terms entering into (9 can not be operators that involve the Dirac operator and one or more pairs of creation and annihilation operators; and, (d) the vector-field equation (5) must be invariant under the gauge transformations (3). The reasons for requiring these propereties are entirely equivalent to the ones discussed in ref. [ 1 ] and we will not discuss them here. These four properties determine the operator (9 uniquely up to overall normalization and provides us with the constraints that the gauge parameter must satisfy. We describe now the construction of (9. The possible normal-ordered terms entering into (9 are classified according to their number of creation operators (degree) and then by the inequi .¢alent ways that their indices can be contracted. In this classification there are one operator of degree zero, the Dirac operator, 7uOu, two of degree two,TuamtUa~O~, ~ 7 a~a~O (notice that the ones with three 7-matrices can be expressed in terms of these ones and operators that are forbidden by property (c)), etc. In the analysis of the gauge invariance of (5) the following observation is crucial. After performing the gaa'ge transformation (3) in (5), each term in (9 gets an extra creation operator. This operator must be commuted to the left to obtain normal-ordered operators. In doing this, two kinds of operators are generated. Suppose we started with an operator in (9 of degree n. This operator generates normal-ordered operators with n creation operators and n - 1 annihilation operators and a remainder which consists of n + 1 creation operators and n annihilation operators. For the case of degree zero there is a reminder only. Performing a gauge transforming in (5), cancellations can occur if the coefficients of the terms entering (9 are arranghed in such a way that the remainders of the operators of degree n are cancelled by the non-remainders originated by the operators of degree n + 1. To analyze the arrangement of the coefficients we start with the Dirac operator (which has to be nee-
Volume 186,number 3,4
PHYSICSLETTERSB
essarily in ¢ in order to describe spin-l/2 massless particles properly) and the operators of degree one. Let us assume that we fix the overall normalization in such a way that the Dirac operator has coefficient one. After performing a gauge transformation, this operator generates a reminder which has to be canceled by the non-reminder operators generated from the action of the gauge transformation on the operators of degree one. It is very simple to observe that to achieve this only of the operators have a non-vanishing coefficient. In fact, the only operator of degree one entering (9 is ?ua~aUO~ with coefficient 1. This operator generates a reminder, tp tp amOvOplAn(x)), u 7ua,,,an whose only chance for cancellation may come from the non-reminder operators of degree two. There exist operators of degree two in agreement with properties ( a ) - ( c ) such that they posses more than one ?-matrix. However, there operators do not play any role when trying to cancel the reminder from the operator of degree one because they are such that the number ofT-matrices that they contain do not decrease after performing a gauge transformation. There are 13 inequivalent of degree rwo with one ?-matrix. The non-reminder terms generated from these operators after performing the gauge transformation (3) are 16 inequivalent ones (which are the complete set which could appear). One can verify that the resulting linear system does not possess any solution in such a way that the reminder of degree one is canceled. Gauge invariante forces us to stop here and to impose the following constraint on the gauge parameters: ?ua~n IA,,)(x) ) =0.
(8)
where IAn,,(x)) constitute a set of second-generation gauge parameters. The gauge parameters IA,,m(X) ) are such that I ~U(x) )remains invariant under a double gauge transformation. This implies that they must be constrained according to
atm
(9)
This constraint indicates that not all the components of the gage parameters IAn,,(x)) are independent. This fact is very well known and, in fact, the classification of generation of gauge parameters presented in ref. [6] also applies here and is in agreement with (9). This formalism provides more information about the constraints satisfied by the gauge parameters. It is well known that the constraint imposed on the first generation implies new constraints on higher generations. To keep track of these constraint in the approach presented in ref. [6] is rather difficult. However, in this context, those constraints appear simply as consistency between (6) and (8) and therefore they can be stated in a simple way:
?ua~naT~l IA,,)p(x) ) =0,
(I0)
where Iq [ means that the index q must be excluded from the symmetrization. The conditions satisfied by higher generations can be analyzed in a similar way. For example, for the third one:
8 IA,m(X) ) =atpUOuIAn,.p(x) ), at~uatv) IA,m~,(x) ) =0, rrt ~ p
(6)
The other conditions that could be imposed to get rid of the reminder are too restrictive or differential. Constraint (6) is consistent with the results known from totally symmetric representations [ 2 ]. We conelude that gauge invariance singles out the operator (9 to be "1"~' .u (9=?/z Ou+Tua,,,a,,O~,
8 IAn(x)) =a~O u I A , , , ( x ) ) ,
12 March 1987
(7)
and forces the gauge parameters to be constrained according to (6). As in the bosonic case, this formalism provides a very simple way to deal with higher-generation gauge invariance. The gauge transformation of the firstgeneration gauge parameters _+An( x ) ) is given by
t(u ant~) apta IAn.,p(x))=0, am v ~ct 7ua /x
(11 )
The number of conditions increases with the number of generations and the series of gauge invariances eventually stops. As we observe from (11 ), the gauge parameter [Amnp(x) ) is antisymmetric in np and in ran. This implies that for N= 2 the parameters vanish. In general, the gauge parameter of the ith generation, [A. . . . . . . . . . . . (X)), is antisymmetric in mlm2, m2m3, ..., rni_ ~mi. This means that for i > N t h e constraints are so strong that the generation vanishes identically. A consequence of this is that the number of generations for a given local field is smaller or equal 367
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than d. IfN~< dthis is obvious. In fact one has a lower bound which is the expected [ 6 ] one, since with N different oscillators the maximum number of rows of the YT corresponding to local fields is N. If N > d the gauge parameters IA. . . . 2....... ( x ) ) with i> d do not correspond to any of the local fields of I ~v( x ) ) and so, although they do not vanish as a consequence of the constraints above, they are irrelevant. The theory presented in this note involves fermionic local fields carrying vector indices which correspond to all the irreducible representations of GL(d) and a series of generations of gauge parameters for each local field. To verify that this theory do indeed correspond to a covariant description of fermionic particles carrying arbitrary representations of the Lorentz group one needs to analyze the physical degrees of freedom that the theory possesses. One way to do this analysis is to study the field equation and gauge invariances associated to each of the local fields. The counting of physical degrees of freedom turns out to be right for representations whose YT have at most two columns. However, for representations with three or more columns the counting is not correct. One has too many physical degrees of freedom. This does not surprise us since we know from the analysis of totally symmetric representations that one is forced to constraint the local field itself [2,9]. In this context, the constraint that seems to work and that reduces to the known one for totally symmetric representations is
Lmnp I ~V(x) ) =0,
(12)
where L , ~ p -.- ?utl~a(manap). ~ a u
(13)
In definition (12) the symmetrization comprises the three indices m, n and p. This constraint is not a consequence of gauge invariance. It must be imposed on the field I ~V(x)). The first question that has to be answered is whether or not this constraint is compatible with the field equation (5) with d7 as expressed in (7). In other words, we need to know if imposing the constraint on I ~ ( x ) ) at an initial time, I ~V(x) ) verifies it at later times. The answer to this question is affirmative since one obtains, after some algebra, that
[L,,,,,p, (9] = - 37UOuL,.,p - 27,aqtu aqOuL,nn r ~ + ~7~aqtu a~qOuLmno). 368
(14)
12 March 1987
This tells us that the modes corresponding to the traces" involved in (12) are effectively decoupled in the field equation. In fact, one can verify that constraint (12) is the only one that reduces properly to the known cases (totally symmetric representations) and is compatible with the field equation (5). Another question related to constraint (12) that has to be answered is whether or not it implies certainm constraints on the gauge parameters. Based on the knowledge from totally symmetric representations one expects a negative answer to this question. To prove this we show now that constraint (12) is gauge invariant under the gauge transformation (3) with gauge parameters constrained according to (6). Performing a gauge transformatioon in (12) one finds, after commuting the resulting creation operator to the left,
7url~a~,~a~Ou[Ap) ) +27ua~%aUO~ lAp) ) tP ~ P /~ -- ~,urh~Baq a (ma,,ap) Op IAq ) =0.
(15)
We now prove that because of the constraint (6), eq. (15) is satisfied. The first two terms vanish trivially in virtue of (6). To prove that the last term does also vanish we operate on (6) with ~,,,),,,a~,a~: ~ u IAm))=O. ~a~Fv~uapaqa(n
(16)
Symmetrizing in p, q and n one finds O"
V
/g
rh,~9,ua(paqa~) IA,,, ) "t-
.u o V rlo~,uama(paq IA,) ) =0.
(17)
On the other hand, taking again (6), acting on it with 2)~a~,, and symmetrizing in all roman indices one finds /.t v rl~a(pa, lA,,) )=O,
(18)
which implies that the last term in (17 ) vanishes and so, (17) implies that the last term in (15) also vanishes, completing the proof. The fact that constraint (12) is gauge invariant is in agreement with known results for totally symmetric representations. What is remarkable is that this occurs in general, i.e. constraint (101) reduces the number of independent components of the field I ~ ( x ) ) but is does not imply any condition on the ghost content of the theory.
Volume 186, number 3,4
PHYSICS LETTERS B
We have analyzed m a n y o d the representations and the conjecture that ( 5 ), ( 6 ), ( 7 ) a n d ( 11 ) constitute the complete description seems to be correct (details will be shown elsewhere [11 ]). The counting o f degrees o f f r e e d o m becomes very c o m p l i c a t e d for representations corresponding to Y T with m a n y boxes. Certainly, a general p r o o f in this respect should be preferred. The f o r m u l a t i o n presented in this note for the covariant description o f fermioonic fields is simple a n m d unique. This makes us suspect that this f o r m u l a t i o n m a y c o r r e s p o n d to an o p e r a t o r representation o f some new k i n d o f theories which involve fields carrying arbitrary representations o f the Lorentz group. In this note a n d in a previous one [ 1 ] we have developed the free a n d bosonic parts o f these theories. These formulations can generate the a d e q u a t e framework to describe interactions a m o n g massless particles in a r b i t r a r y representations. However, we believe that the right way to proceed towards interactions m u s t begin with the u n d e r s t a n d i n g o f the (geometric?) structure behind these free formulations.
12 March 1987
References [ 1] J.M.F. Labastidas, IAS preprint (August 1986). [2] T. Curtight, Phys. Lett B 85 (1979); B. de Wit and D.Z. Freedman, Phys. Rev. D 21 (1980) 358. [3] C. Fronsdal, Phys. Rex'. 161 (1978) 3624. [4] L.P.S.Singh and C.R. Hagen, Phys. Rev. D 9 (1974) 898. [5] T. Curtright and P.G.O. Freund, Nucl. Phys. B 172 (1980) 413; T. Curtright, Phys. Lett. 165 B 304 (1985); University of Florida preprint, UFTP-82-22; G.R.E. Black and B.G. Wybourne, J. Phys. A 16 2405 (1983); R. Delbourgo and P. Jarvis, J. Phys. A 16 (1983) L275; C.S. Aulakh, I.G. Koh and S. Ouvry, Phys. Lett. B 173 (1986) 284. [6] J.M.F. Labastida and T.R. Morris, Phys. Lett. B 180 (1985) 101. [7] P.A.M. Dirac, Proc. R. Soc. A 155 (1936 447. [8] W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61. [9] J. Fang and C. Fronsdal, Phys. Rev. D 18 (1978) 3630. [ 10] L.P.S. Singh and C.R. Hagen, Phys. Rev. D 9 (1974) 910. [ 11 ] J.M.F. Labastida, IAS preporint, in preparation. [ 12 ] M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley,Reading, MA, 1962). [13] W. Siegel, Phys. Lett. B 149 (1984) 157, 162;B 151 (1985) 391,396; W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105.
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