From BRST to light-cone description of higher spin gauge fields

0.19 MB·English

by  G. Barnich

ULB-TH/05-04, FIAN-TD/05-05
Sissa 17/2005/EP, hep-th/0502232
From BRST to light-cone description of higher
spin gauge fields
6
0
0
2 Proceedings of the Workshop “Quantum Field Theory and Hamiltonian Systems”,

n Caciulata, Romania, 16 - 21 Oct, 2004.
a
J

7 G. Barnicha, G. Bonellib and M. Grigorievc

4
v
Physique Th´eorique et Math´ematique
2
3 Universit´e Libre de Bruxelles
2
and
2
0 International Solvay Institutes
5
Campus Plaine C.P. 231, B-1050 Bruxelles
0
/ Belgium
h
t
-
p
e
h
: Abstract
v
i
X
r
a In this short note we show, at the level of action principles, how the light-
coneactionofhigherspingaugefieldscaneasilybeobtainedfromtheBRST
formulation through the elimination of quartets. We analyze how the al-
gebra of cohomology classes is affected by such a reduction. By applying
the reduction to the Poincar´e generators, we give an alternative way of an-
alyzing the physical spectrum of the Fronsdal type actions, with or without
trace condition.
∗ SeniorResearchAssociateoftheNationalFundforScientificResearch(Belgium).
b nowat: SISSA,HighEnergySectorandINFNTrieste,viaBeirut4,34014, Trieste,Italy.
c nowat: TammTheoryDepartment, LebedevPhysicsInstitute, Leninskyprospect53,119991Moscow,Russia.
From BRST to light-cone description of higher spin gauge fields 2
1 Introduction
The title of the lectures given by G. Barnich during the workshop was “Higher spin
gauge fields: Basics”. The discussion was restricted to free theories in 4 dimensions
with integer spin. The following material was covered1:
1. Representations of the Poincar´e group [3], [4]
2. Mass-shell field representations [5]
3. Variational principles [6], [7], [8]
4. BRST formulation [9], [10], [11], [12]
5. Connection to string theory [9], [10], [11], [12], [14], [13], [15], [16]
6. Trivial pairs and auxiliary fields [17], [18]
7. Connection to the light-cone formulation [19],[20],[21]
8. Connection to Vassiliev’s formulation [22], [18]
InordertobridgebetweentheselecturesandthelecturesbyL.Brink“Light-cone
frame formulation of field theories and string theories, a non-BRST formulation”,
we will elaborate on item 7 above and show, as a pedagogical exercice, how the
light-cone action can be explicitly reached from the BRST formulation.
More precisely, after reviewing the BRST formulation of the Fronsdal action, we
recall that in BRST language, the reduction to the light-cone gauge corresponds to
the elimination of quartets [23]. The free light-cone action for higher spin gauge
fields [19] is then recovered directly from the reduced BRST operator. We develop
in general terms the reduction of the Lie algebra of BRST cohomology classes under
the elimination of trivial pairs. As an application, we recover the expression of the
Poincar´e generators [19] in the light cone gauge. Finally, we use these expressions in
4 dimensions to give an alternative proof of the fact that the Fronsdal type action
at level s describes
(i) massless particles of helicities s, s+2,...,s 2,s if no trace constraint is
− − −
required,
(ii) massless particles of helicity s if the trace constraint is imposed [8, 24, 25].
±
1See also the reviews [1, 2] and the references cited therein.
From BRST to light-cone description of higher spin gauge fields 3
2 BRST formulation of the Fronsdal Lagrangian
We will mostly follow the notations and conventions of [18], to which we refer for
further details.
2.1 Degrees of freedom, constraints and BRST charge
The variables are xµ,p ,aµ,a µ, where µ = 0,...,d 1. Classically, xµ and p
µ † µ
−
correspond to coordinates on T Rd 1 and aµ,a µ correspond to internal degrees of
∗ − †
freedom. After quantization, they satisfy the canonical commutation relations
(2.1) [p ,xµ] = ıδµ, [aµ,aν ] = ηµν,
ν − ν †
where η = diag( 1,1,...,1). We assume that xµ,p are Hermitian, (xµ) = xµ,
µν µ †
−
p = p , while aµ and a µ are interchanged by Hermitian conjugation.
†µ µ †
The constraints of the system are
(2.2) ηµνp p = 0, p aµ = 0, p a µ = 0.
µ ν µ † µ †
L ≡ S ≡ S ≡
For the ghost pairs (θ, ), (c ,b), and (c,b ) corresponding to each of these con-
† †
P
straints, we take the canonical commutation relations in the form2
(2.3) [ ,θ] = ı, [c,b ] = 1, [b,c ] = 1.
† †
P − −
The ghost-number assignments are
(2.4) gh(θ) = gh(c) = gh(c ) = 1, gh( ) = gh(b) = gh(b ) = 1.
† †
P −
The hermitian BRST operator is then given by
(2.5) Ω = θ +c + c ı c c,
† † †
L S S − P
and satisfies Ω2 = 0.
2.2 Representation space
Contrary to reference [18], we now choose the momentum instead of the coordinate
representation for the operators (xµ,p ) and also3 the holomorphic instead of the
ν
Fock representation for the oscillators (a ,a ),(c,b ),(b,c ), while the pair (θ, )
µ †µ † † P
continues to be quantized in the coordinate representation.
2We use the “super” convention that (ab)† =( 1)|a||b|b†a†.
3 −
This step will allows us to write rather compact formulas below, but is not really necessary
for the analysis.
From BRST to light-cone description of higher spin gauge fields 4
The “Hilbert space” consists of the “wave functions” ψ(p,θ,α ,β γ ) =
∗ ∗ ∗
H
pθα β γ ,ψ . On these functions, the operators p , θ, aµ , b , c act as multi-
∗ ∗ ∗ µ † † †
h i
plication by p , θ, αµ , β , γ , while the operators xµ, , aµ, c, b act as ı ∂ , ı ∂ ,
µ ∗ ∗ ∗ P ∂pµ − ∂θ
∂ , ∂ , ∂ . The inner product is defined by
∂α∗µ ∂β∗ −∂γ∗
(2.6) ψ,χ = ddpdθ(d−1 dαµ∗dαµ)dβ dγdγ dβ
∗ ∗
h i 2πı
Z µ=0
Y
exp( αµ α β γ +γ β)[ψ(p,θ,α ,β γ )] χ(p,θ,α ,β ,γ ).
∗ µ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
− −
The string field becomes
d 1 dα dα
(2.7) Ψ = ddpdθ( − µ∗ µ)dβ dγdγ dβ
∗ ∗
2πı
Z µ=0
Y
exp( αµ α β γ +γ β) pθαβγ ϕ(p,θ,α ,β ,γ ).
∗ µ ∗ ∗ ∗ ∗ ∗
− − | i
The ghost numbers of the component fields of ϕ(p,θ,α ,β ,γ ) are defined to be
∗ ∗ ∗
1/2 minus the ghost number (2.4) of the corresponding state in the first quantized
theory. The ghost-number-zero component of the string field can be parameterized
as
(2.8) Ψ(0) = Φ ıθb C +c b D,
† † †
−
where
pθα β γ ,Φ = Φ(p,α ), pθα β γ ,C = C(p,α ),
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
(2.9) h i h i
pθα β γ ,D = D(p,α ).
∗ ∗ ∗ ∗
h i
2.3 Physical and master actions
The action
(2.10) S[Ψ] = 1 ddp Ψ,ΩΨ
−2 h i
Z
is the solution of the Batalin-Vilkovisky master action associated with the physical
action
(2.11) Sph[Ψ(0)] = 1 ddp Ψ(0),ΩΨ(0) .
−2 h i
Z
The trace and level operators
(2.12) = η aµaν +2cb, N = a aµ c b+b c s
T µν s †µ − † † −
From BRST to light-cone description of higher spin gauge fields 5
commute with the BRST charge Ω and satisfy [N , ] = 2 so that they can be
s
T − T
consistently imposed as restrictions on the Hilbert space and on the string field:
˜ = Ker KerN , N Ψ˜ = 0 = Ψ˜. Assuming appropriate reality conditions, the
s s s
H T ∩ T
restricted action
(2.13) Sph[Ψ˜(0)] = 1 ddp Ψ˜(0),ΩΨ˜(0)
−2 h i
Z
is the gauge invariant Fronsdal action [8] (written with additional auxiliary fields
contained in C(p,α )), while
∗
(2.14) S[Ψ˜] = 1 ddp Ψ˜,ΩΨ˜
−2 h i
Z
is the associated proper solution of the Batalin-Vilkovisky master equation.
3 Reduction to the light-cone formulation
As an alternative to the analysis in [8], [24] or [25], we will analyze the physical
spectrum of actions (2.11) and (2.13) by going to the light-cone formulation. In
order to do so, we use the grading introduced in [23] to relate the BRST to the
light-cone formulation in the context of string theory. This grading allows one to
identify quartets for the full BRST charge on the level of the lowest part in the
expansion.
3.1 Unconstrained reduction
Hereandinthefollowing,indicesareloweredandraisedwithη = diag( 1,1...,1).
µν
−
Thelight-coneoperatorsaredefinedthroughp = 1 (p0 pd 1),a = 1 (a0 ad 1),
± √2 ± − ± √2 ± −
with [a−,a+†] = 1 = [a+,a−†] with transverse momenta and oscillators pi,ai,ai†,
−
i = 1,...,d 2, unchanged. In the following, we denote the transverse momenta
−
and oscillators collectively by a superscript T. The BRST operator becomes
(3.15) Ω = θ(p pi 2p p+)+c ( p a+ p+a +p ai)+
i − † − − i
− − −
+( p−a+† p+a−† +piai†)c ı c†c.
− − − P
The grading is defined through the eigenvalues of the operator
(3.16) G = 2a−†a+ 2a+†a− b†c c†b,
− − −
so that
a+ = 2, a+† = 2, a−† = 2, a− = 2,
(3.17) | | | | | | − | | −
c = 1, c = 1, b = 1, b = 1,
† †
| | | | | | − | | −
From BRST to light-cone description of higher spin gauge fields 6
with all other operators having grading zero. The operator G acts on the wave
functions φ(p,θ,α ,β ,γ ) and the associated string fields as
∗ ∗ ∗
∂ ∂ ∂ ∂
(3.18) G = 2α +2α+ β +γ .
−∗ ∗ ∗ ∗
− ∂α ∂α+ − ∂β ∂γ
−∗ ∗ ∗ ∗
The Hilbert space can be decomposed according to this grading and, as the wave
H
functions are polynomial in the holomorphic variables, each wave function has a
component with lowest degree. The BRST operator decomposes as Ω = 3 Ω ,
i= 1 i
−
with
P
Ω 1 = p+(c†a− +a−†c), Ω0 = θ( 2p−p+ +pipi),
(3.19) − − −
Ω1 = c†piai +piai†c, Ω2 = ı c†c, Ω3 = p−(c†a+ +a+†c).
− P −
If we assume p+ = 0, which is a standard assumption in the light-cone approach,
6
states depending non trivially on α+ ,α ,β ,γ form quartets for the differential
∗ −∗ ∗ ∗
Ω
. Indeed, we have
1
−
1
(3.20) K1 = −p+(a+†b−b†a+), [K1,Ω−1] = Nq,
Nq = a+†a− a−†a+ c†b+b†c,
− − −
where the operator N acts on states and the associated string fields as the operator
q
that counts the unphysical variables,
∂ ∂ ∂ ∂
(3.21) N = α +α+ +β +γ .
q −∗ ∗ ∗ ∗
∂α ∂α+ ∂β ∂γ
−∗ ∗ ∗ ∗
The Hilbert space can accordingly be decomposed as = with
H H E ⊕ F ⊕ G
= ImΩ , = ImK and KerΩ corresponding to the states ψ(p,θ,α T)
1 1 1 ∗
G − F E ⊂ −
that are eigenvectors of N with eigenvalue zero.
q
Proposition 3.6 of [18] then implies that the full BRST charge is invertible
between and : if FΘG (GΩF) 1 denotes the inverse of the BRST charge GΩF
−
F G ≡
between and and ρ = (GΩF ) 1, we have
1 −
F G −
(3.22) FΘG = ( 1)nρ[( GΩF )ρ]n.
i
−
n>0 i>0
X X
Notethattheinverseexistsbecausethedegreeofeachwavefunctionisboundedfrom
Ω
below. Furthermore, because the cohomologyof is concentrated in degree 0, the
1
−
system (Ω, ) canbe reduced to the system (Ω , ). According to proposition3.4 of
0
H E
[18], the associated field theories are related through the elimination of generalized
auxiliary fields4. In addition, the elimination of the quartets is compatible with the
4Becausewearenotconcernedwithissuesconcerninglocalityoftheassociatedfieldtheoriesin
our discussion, the coordinates xµ and the momenta p are considered on the same foot with the
µ
other degrees of freedom. The discussion of [18] is then simplified because only the Hilbert space
is involved in the discussion.
H
From BRST to light-cone description of higher spin gauge fields 7
inner product: after elimination of the generalized auxiliary fields the inner product
on reduces to an inner product on that makes ΩEE formally self-adjoint. This
0
H E
means that the elimination can be done on the level of the corresponding master
equations, and corresponds to the elimination of generalized auxiliary fields in the
sense of [26]. Explicitly, the reduced master action obtained from (2.10) reads
(3.23) S[Ψ ] = 1 ddp Ψ ,Ω Ψ =
E E 0 E
−2 h i
Z
d 2
= 1 ddp − dαi∗dαi exp( α iα )[ϕ(p,α T)] ( 2p p+ +p pi)ϕ(p,α T).
∗ i ∗ ∗ − i ∗
− 2 2πı − −
!
Z i=1
Y
As it depends only on ghost number 0 fields, it coincides with the physical action
and there is no gauge invariance left.
3.2 Imposing the level and trace constraints
BecausebothΩ andK commutewithN theeliminationcanbedoneconsistently
1 1 s
−
also in KerN .
s
The reduction can also be done in Ker . To see this we use an expansion with
T
respect to the eigenvalues of N . The trace operator decomposes as = 2+ 0,
q −
T T T
with 2 = 2a a+ +2cb, 0 = aia T, while each wave function decomposes
− − i
T − T ≡ T
as ψ = ψ0 + ...ψM. Suppose ψ = 0. If Ω ψ = 0, we have, for M = 0, ψM =
1
T − 6
Ω ( 1 K ψM). Because [ ,K ] = 4 a+b, the term K ψ is double traceless. In
−1 M 1 T 1 −p+ 1 T
fact one can adjust K ψ to make it traceless using the expansion in homogeneity
1
T
in all oscilators. Namely, for each monomial ψ of definite homogeneity d in all
(d)
the oscillators, φ K ψ k( 2α +α +2γ β +α iα ) K ψ is traceless
(d) ≡ 1 (d)− − ∗ ∗− ∗ ∗ ∗ i∗ T 1 (d) T
for an appropriate choice of k. Since Ω commutes with both and 2α +α +
1 ∗ ∗−
− T −
2γ β +α iα , it follows, for M = 0, that Ω ( 1 φ ) = ψM +..., where the dots
∗ ∗ ∗ i∗ 6 −1 M (d) (d)
denotetermsoforderstrictlylower thanM. Hence, bysumming over allmonomials,
ψ Ω ( 1 φ)is traceless, Ω closedanditsexpansion stopsatthelatestatM 1.
− −1 M T −1 −
Byinductionweconcludethat ψ = 0 = Ω ψ impliesψ = ψ0+Ω χ, with χ = 0
1 1
T − − T
and Tψ0 = ψ0 = 0.
T T
The same reasoning as in the previous subsection then shows that the Fronsdal
master action (2.14) can be reduced to the master and physical action
(3.24) S[Ψ˜ ] = 1 ddp Ψ˜ ,Ω Ψ˜ =
E −2 h E 0 Ei
Z
d 2
= 1 ddp − dαl∗dαl exp( α iα )[ϕ˜(p,α T)] ( 2p p+ +p pi)ϕ˜(p,α T)
∗ i ∗ ∗ − i ∗
− 2 2πı − −
!
Z l=1
Y
with TΨ˜ = 0 = Tϕ(p,α T) and also N Ψ˜ = 0 = N ϕ(p,α T).
E ∗ s E s ∗
T T
From BRST to light-cone description of higher spin gauge fields 8
3.3 Reduction of observables
By observables, we mean in this context BRST cohomology classes of operators, i.e.,
equivalence classes of operators A which commute with the BRST charge Ω, up to
Ω
such operators which are in the image of the adjoint action of ,
(3.25) [Ω,A] = 0, A A+[Ω,B],
∼
As discussed for instance in [14, 17, 18], if one restricts to antihermitian operators in
ghost number zero, these cohomology classes describe equivalence classes of global
symmetries of the action (2.11), with
(3.26) δ Ψ(0) = AΨ(0).
A
In order for such observables to be well defined for the constrained action (2.13),
they have to commute with the trace and the level constraint,
(3.27) [N ,A] = 0 = [ ,A].
s
T
In the following discussion we have in mind in particular the observables that
describe the Poincar´e transformations. There are several ways to reduce such ob-
servables to the light-cone formulation. One possibility, discussed in [19], is to work
out the compensating gauge transformation needed to stay in the light-cone gauge
when performing the covariant Poincar´e transformations. Another possibility, fol-
lowed in [12], is to use the quantum Dirac bracket. Alternatively, one could use a
Dirac-type antibracket for representatives Ψ,AΨ of antifield BRST cohomology
h i
classes in ghost number 1, which describe the global symmetry on the level of the
−
master action (2.10). This approach has the advantage to extend to the non linear
and non Lagrangian setting and will be discussed elsewhere. Here we will follow a
first quantized BRST approach.
Let (h ) (e ,f ,g ) denote a basis for = . Assuming only
A α a a
≡ H E ⊕ F ⊕ G
existence of (GΩF) 1 FΘG, we can consider the following invertible change of basis:
−
≡
(3.28) e˜ = e f (FRE)b, FRE = FΘGGΩE
α α − b α
(3.29) f˜ = f (FΘG)b,
a b a
(3.30) g˜ = e (ELG)β +f (FΩF(FΘG)b +g , ELG = EΩFFΘG.
a β a b a a
In other words, h˜ = hC where
EδE 0 ELG EδE 0 ELG
−
(3.31) C =  ERE FΘG FΩFFΘG , C−1 =  GΩE GΩF GΩG .
−
   
 0 0 GδG   0 0 GδG 
   
   
From BRST to light-cone description of higher spin gauge fields 9
with δ denoting the identity matrix. The new basis is chosen in such a way that the
Ω
expression for simplifies to
Ω˜
0 0
(3.32) Ωˆ = C−1ΩC =  0 0 0 ,
 0 GδF 0 
 
 
with
(3.33) Ω˜ = EΩE ELGGΩE EΩFFRE +ELGGΩFFRE = EΩE EΩFFΘGGΩE.
− − −
If Aˆ = C 1AC, and A is of parity A , [Ω,A] = 0 implies
−
| |
A˜ 0 0
(3.34) Aˆ = 0 0 0 +[Ωˆ,A ],
  E
0 0 0
 
 
where
(3.35) A˜ = EAE ELGGAE EAFFRE +ELGGAFFRE,
− −
0 0 ( 1)A(EAF ELGGAF)FΘG
| |
− −
 
(3.36) AE = (GAE GAFFRE) GAFFΘG ( 1)AGΩHHAFFΘG .
| |
 − − 
 
 0 0 0 
 
 
In particular,
0 0 0
(3.37) ΩE =  0 FδF 0 .
 0 0 0 
 
 
Equation(3.34)impliesthatthemapg fromlinearoperatorson tolinearoperators
H
on defined through g(A) = A˜ induces a well defined map g# from the cohomology
E
of[Ω, ]tothecohomologyof[Ω˜, ]. Italsofollowsdirectlyfrom(3.34)thatg#([A]) =
· ·
[A˜]is an isomorphism which preserves the Lie algebra of observables: g#([[A],[B]] =
[[A˜],[B˜]].
3.4 Lorentz generators in the light-cone
Let ξ = δ + ω xν, with ω antisymmetric. The Poincar´e generators are then
µ µ µν µν
obtained from the antihermitian generating operators
(3.38) L(ω,a) = ı ξµp +a [pµ,ξν]a ,
− µ †µ ν
(cid:0) (cid:1)
From BRST to light-cone description of higher spin gauge fields 10
satisfying
(3.39) [Ω,L] = 0, [L(ω ,δ ),L(ω ,δ )] = L([ω ,ω ],ω δ ω δ )
1 1 2 2 1 2 1 2 2 1
−
through
∂L
(3.40) Jµν 2ı = xµpν xνpµ ı(aµ aν aν aµ) ,
† †
≡ − ∂ω − − −
µν
∂L (cid:0) (cid:1)
(3.41) Pµ ı = pµ.
≡ ∂δ
µ
Now we apply the reduction formula (3.35) to the Poincar´e generators in the
light-cone basis, P+,P ,Pi,Jij,J+i,J i,J+ . First we note that the translations
− − −
and the orbital parts of the Lorentz transformations carry vanishing degree and
map to . Moreover all the terms in (3.35) besides the first one vanish for these
E E
operators because R and L carry strictly positive degree.
The same reasoning applies to the spin part Sij = ı(ai†aj aj†ai) of Jij, which
− −
are unchanged in the reduction. The spin operator S+− = ı(a+†a− a−†a+) also
− −
carries vanishing degree, but since its restriction to vanishes, S˜+ = 0. The spin
−
E
operator S+i = ı(a+†ai ai†a+) carries degree +2 and therefore S˜+i = 0. Finally,
− −
the spin operator S−i = ı(a−†ai ai†a−) has degree 2: therefore the first term
− − −
in (3.35) vanishes. Furthermore, the fourth term also vanishes. Indeed, it contains
both R and L operators whose expansion contains terms with degree greater or
equal to 2. This is so because Θ is of strictly positive degree while the only possible
contributions to GΩE and EΩF are coming from Ω given in (3.19). Explicitly, one thus
1
has
(3.42) S˜ i = SEFiρGΩE ΩEF ρSGEi.
− − 1 1 −
− −
Choosing = Ω , = 1 K Ω M, where the expansion is with respect
G −1H F M>0 M 1 −1H
to the eigenvalues of N , we have ρ = K and, for an arbitrary state ψ ,
q P 1 E ∈ E
1
(3.43) ρGΩE1ψE = ρc†pjajψE = K1c†pjajψE = p+a+†pjajψE
ı
(3.44) ρSG−EiψE = ıρa−†aiψE = ıK1a−†aiψE = b†aiψE .
− − p+
Summing up, one finds
ı
(3.45) S˜−i = p+(ai†pjaj −pja†jai).