CC BY
27
UDC 530.1; 539.1
Review of BRST approach to higher spin field theory
V. Krykhtin
Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia.
E-mail: krvkhtin@tspu.edu.ru
We review the method which allows to construct Lagrangians for different type of fields: massive and massless, bosonic and fermionic with different index symmetry.
Keywords: higher spin Gelds, gauge theory.
1 Introduction
The method reviewed here allows to construct Lagrangians for different kind of field: in Minkowski and AdS spacetime both for bosonic and fermionic fields (massive and massless) with any index symmetry, see e.g. [1-10], in Einstein space for massive and massless spin 2 and 3/2 fields [11,12] and for antisymmetric bosonic fields in arbitraty curved spacetime [13].
The plan of the review is as follows. First we consider basic steps of the Lagrangian construction method using the example of bosonic massive spin-s field. Next we comment on some peculiar features of Lagrangian construction for other kind of fields: the fields corresponding to arbitrary k-row Young tableau, the fields in AdS space and the fermionic fields. Finally we consider generalization of the method to the case of spin-2 and spin-3/2 in Einstein manifold.
2 Basic steps of the Lagrangian construction
Let us review the basic steps of the Lagrangian construction using example of the massive bosonic spin-s field in Minkowski spacetime.
As is known the totally symmetric tensor field <PMi...Ms(x), describing irreducible spin-s massive representation of the Pioncare group must satisfy the following constraints
(d2 - rn2)^Mi...Ms =0 (!)
dMi <p»i...»s =0 nMiM2 ^Mi...Ms =0. (2)
Our purpose is to construct Lagrangian which reproduce these equations of motion.
The first step of the Lagrangian construction method is to rewrite these equations in the form of operator constraints
For this purpose we introduce Fock space generated by creation and annihilation operators a+, aM satisfying the commutation relations
[aM,a+] = Vmv = (-, +,..., +)
Then we define operators
l0 = d2 — m2, l\ = iaMdM, l2 = 2 aMaM
These operators act on states in the Fock space
TO
№ = ^ \tfs) \tfs) = ^Mi-Ms (x)aMi+ ••• aMs+|°)
s=0
Each component <^Mi...Ms(x) will satisfy the equations of motion (1), (2) if the following constraints on the states are fulfilled
lo\v) =°, lib) =°, I2\^) =°.
Thus the equations defining the irreducible representation are treated as the operators of first class constraints in some auxiliary Fock space.
lo l1 l2
other.
To construct real Lagrangian we must construct
l1 l2
are not Hermitian
(li)+ = ll (l2)+ = l2
we can’t construct such operator on the base of opera-
lo l1 l2
the underlying set of operators must be invariant under Hermitian conjugation.
The second step is to add more operators in order to provide such invariance.
Thus we add two more operators
l+ = iaM+dM l+ = 1 aM+ a+
As a result, the set of operators l0, l^ l2, l+ l+ is invariant under Hermitian conjugation.
Algebra of the operators l0, l^ l2, l+ l+ is open in terms of commutators of these operators. To use the BRST construction in the simplest (minimal) form the underlying algebra must be closed.
Third step is to get such an algebra. For this purpose we add to the above set of operators, all operators
generated by the commutators of Zo, /i, Z2, Z+, Z+. Doing such a way we obtain two new operators
gm = m2 and go = a+aM + d/2.
Thus we have set of operators
Zo Zi /2 Z+ Z+ go gm
which is invariant under Hermitian conjugation and form an algebra.
Zo Z1 Z2
in the ket-vector space
Zo|^} = Zi|^} = Z21^) = 0,
operators Zo, Z+, Z+ are constraints in the bra-vector space
(^|Zo = (^|Z+ = (^|Z+ = 0
gm go
[Z+ ,Zi]= Zo + gm [Z2,Z+] = go
and they are not constraints neither in the bra-vector space nor in the ket-vector space
Now if we construct BRST operator we find that one of the equation on the physical state |^s) will have the form
gm |^s} = 0 ^ |^s} = 0
Thus we will not reproduce the proper equations of motion. This happens due to the presence of the operators
gm go
Forth step. We enlarge the representation space of the operator algebra by introducing additional (new) creation and annihilation operators and enlarge expressions for the operators
oi —> Oi = oi + oi,
oi ^ Z^ Z^ ^ Z+ g0, gm}
The enlarged operators must satisfy two conditions:
1. They must form an algebra [Oi, Oj ] ~ Ok;
2. The operators which can’t be regarded as constraints must be zero or contain linearly an arbitrary parameter.
For our case the enlarged expressions for the opareators can be taken in the form
Lo = d2 — m2
L1 = + mbi
L+ = + m6+
L2 = 5®^®^ + 5 bi + (6+ 62 + h)b2
L+ = 2^ib+2 + 6+
Go = + 6+ 6i + 26+ 62 + h
Gm =0 [6i, 6+] = [62, 6+] = 1
It is easy to see that the operators L2 and L+ are not Hermitian conjugate to each other if we use the usual rules for Hermitian conjugation of the additional creation and annihilation operators.
Fifth step. We change the definition of scalar product of vectors in the new representation as follows
($l|$2>new = ($i|K |$2>,
with some operator K. This operator K can be found in the form
K = £ |n>(n|, |n> = (b+)n|0},
n=o *
C(n, h) = h(h + 1)(h + 2) ... (h + n — 1),
C(0, h) = 1.
Now we are ready to construct BRST operator. Sixth step is construction of the BRST operator
Q = noLo + Li + niL+ + L2 + n2L+ + Vg Go + n+niP o — n+^P G
+ (nGn+ — n+ ni)P i + (ninG — n+ n2 )P+
+ 2nGn+P 2 + 2n2nGP+
The ghost operators satisfy the usual commutation relations
{no, P o} = {nG, P g} = {ni, P + } =
= {n+, Pi} = {n2, P + } = {n+, P2} = 1
and act on the vacuum state as follows P o|0} = P g|0} = ni|0} = P110} = n210} = P 210} = 0.
The introduced operators act in the enlarged space of state vectors depending on a+M, 6+, 6+ and on the ghost operators no> n+ P + n+ P + but one assumes that the state vectors must be independent of the ghost nG corresponding to the operator Go. The general structure of such state is
|x} = £(b+)k1 (b+)k2(no)k3(n+)k4(P+)k5 x
ki
x (n2+)k6(P+)k7a+M1 ■ ■ ■ (x)|0}.
The sum is taken over ko, ki5 k2, running from 0 to infinity and over k3, k5, k^ k7 running from 0 to 1. Besides for the ’physical’ states we must leave in the sum only those terms which have ghost number is zero. The last step is Lagrangian construction.
We extract the dependence of the BRST operator nG PG
Q = Q + nG (a + h) — n+n2PG,
q2 = n+n2(a + h) [Q,a]=0,
a = a+Ma+ + 6+6i +26+62 + n+P i + P +ni
+ 2n2+P 2 + 2P+n2 + ^
After this, the equation on the ’physical’ states in the BRST approach Q|x} =0 yields two equations
Q|x} = 0,
(a + h)|x} = 0.
The last equation is the eigenvalue equation for the operator a with the corresponding eigenvalues — h
— h = s + ^, s = 0,1, 2,... .
a
corresponding to the eigenvalues s + as |xs}
TO
a|Xs} = (s + ) |Xs} |X} = £ |Xs}
s=o
and
Qs = QU^s + ^ Ks = K |-h^s+ ^i-5
s
Qs|Xs} =0 Q2|Xs}= 0
and this equation of motion can be obtained from La-gran gi an
Ся
dno (Xs|KsQs|xs)
The integral is taken over Grassmann odd variable n0-The equations of motion and Lagrangian are invariant ander the gauge transformations
¿|Xs) = Qs |Л8). 5|ЛЯ) = Qs|Os)
s ^1s) , = 0, ^^(|Ля)) = —1
q2|^) = 0, gh(|fis)) = -2.
3 Fields corresponding to an arbitrary Young tableau
Let us consider the Lagrangian construction for the fields with index symmetry corresponding to Young tableau with 2 rows (s5 > s2)
Ф
Mr"Ms1, V1 •
,(x)
Mi M2 M s 1
V1 V2 vs 2
The tensor field is symmetric with respect to permutation of each type of the indices
$M1-MS1 ,*1-V.2(x) = $(M1 ),(vi-v.2)(x) and in ad-
dition must satisfy the following equations
(d2 - m2)$M1-•.„ ,V1 • • Vs2 (x) = 0,
dM1 Фм
^Mr • • Msi , Vi-• • Vs 2 ПМ1М2 Ф
(x) = дV1Ф
Mr • • Ms1 , Vi • • • Vs
= nV1 V2 ^ =
M1M2-• • Ms1 ,V1-• • Vs 2 = '/ ^M1-• • Ms1 ,V1V2-• • Vs 2 =
= n^1V2 * = 0
n ^M1-• • Ms1 ,V1-• • Vs 2 0,
^(M1-• • Ms1, V1) • • • Vs2 (x) = °.
Then we define Fock space generated by creation and annihilation operators
K,a+V ] = nMV ^ij,
nMV = diag( —, +, +, ••• , +) i, j = 1, 2.
The number of pairs of creation and annihilation operators one should introduce is determined by the number of rows in the Young tableau corresponding to the symmetry of the tensor field. Thus we introduce two pairs of such operators. An arbitrary state vector in this Fock space has the form
|Ф) ^E$Mr • Ms1,
S1=0 S2 =0
M1-• • Ms1 , Vr • • Vs2 (x) X
x a
+M1
+Msi +Vi
11
+ Vs.
Thus the Lagrangian is constructed.
Let us repeat the basic steps of the Lagrangian construction procedure. At the starting point Lagrangian is unknown, but we know the equations of motion. Schematically the procedure looks like as follows
Equations of motion are known ^
^ we rewrite the EoM in the form of operator constraints ^ we add Hermitian conjugated constraints ^ we add operators to form an algebra ^ we construct enlarged expressions for the operators ^ we change scalar product ^ we construct BRST charge ^ we find Lagrangian
‘1 • • • a1 a2 • • • a2 lu/.
To get the equations on the coefficient functions we introduce the following operators
2
"0 = d mPm - m
I m
"j = о ai ajM
iaM дм,
+M
a1 а2м
¿0
lii — ~a g12
One can check that the EoM are equivalent to
/о|Ф) — 0 , /i|Ф) — 0 , lij|ф) — 0, gi2|Ф) — 0
Now we can generalize this construction to the fields corresponding to k-row Young tableau. For this purpose one should introduce Fock space generated by k pairs of creation and annihilation operators, where i,j — 1,2,..., k, and then introduce operators /0, /i, /ij gij but tow with i, j — 1, 2,..., k. Operator g12 is generalized to operator gij- — where i > j.
After this the Lagrangian construction can be carried out as usual.
V
2
4 Fields in AdS
The difference of the Lagrangian construction in AdS space is that the algebra generated by the constraints is nonlinear, but it has a special structure
[ li,lj]= fij+ Z/TWrn, /ij“ ^ R
where /ij, /ijm are constants. The constants /ijm are proportional to the scalar curvature and disappear in the flat limit. This has two consequences.
1) The algebra of the enlaged operators is changed in comparison with the algebra of the initial operators
r t T 1 -pk t < j*km i j*mk\7/ t i .ckmr t
[ Li,Lj] = fijLk — (Jij + fij )lmLk + Jij LkLm,
with the additional parts satisfying the algebra additional parts
r 7/ 7/] _ /k 7/ _ /kmi/ 7/
L li,lj] fijlk fij lmlk.
2) Due to the algebra is nonlinear the BRST-BFV operator is defined unambiguously.
There exist different possibilities to order operators in the right hand sides of the commutators. All possible ways to order the operators can be described by some arbitrary parameters & [6,7]. As a result the BRST operator is defined unambiguously. The arbitrariness in the BRST operator stipulated by the parameters &i is resulted in arbitrariness of introducing the auxiliary fields in the Lagrangians and hence does not affect the dynamics of the basic field.
5 Fermionic fields
The Lagrangian construction for the fermionic higher spin theories have specific difference compared to the bosonic ones and demands some comments.
As before the equations of motion are reproduced from
Qs|^s} =0, Q2 |^s} = 0,
¿|*s} = QS|AS}
Specific features is that in the fermionic theory we must obtain Lagrangian which is linear in derivatives. But if we try to construct Lagrangian similar to the bosonic case
£~ <*|Q|*}
we obtain Lagrangian which has second order in derivatives. To overcome this problem one first partially fixes the gauge and partially solves some field equations. Then the obtained equations are Lagrangian ones and thus we can derive the correct Lagrangian.
6 Fields in Einstein space
Usually the Lagrangian construction in the BRST approach is carried out for fields of all spins simultaneously. The equations of motion and gauge transformations
Q|^s} =0 ¿|*s} = Q|AS}
Q
Nilpotency of the BRST operator provides us the gauge transformations and fields |^s} mid |^s} + Q|As} are both physical. Since we consider all spins simultaneously
Q2|AS} =0 ^ Q2 =0
. But if we want to construct Lagrangian for the field
s
require a weaker condition.
The BRST operator for given spin Qs may be not nilpotent in operator sense but will be nilpotent only-on the specific Fock vector parameter |As} corresponds
q2|as} = 0 but q- = 0
on states of general form. Just this point allows us to construct Lagrangians for spin-2 and spin-3/2 fields in Einstein background [11,12].
7 Summary
• We have considered the basic principles of gauge invariant Lagrangian construction for higher spin fields.
•
— any bosonic and fermionic fields in Minkowski and (A)dS spaces with any index symmetry
— antisymmetric bosonic fields in arbitrary-curved space-time
— spin-2 and spin-3/2 fields in Einstein space parameters are imposed.
gauge invariance.
Acknowledgement
This research has been supported in parts by the grants Kadry, project No 14.B37.21.1301, RFBR-Ukraine grant, project No 11-02-90445, RFBR grant project No. 12-02-00121-a and the grant for LRSS, project No 224.2012.2 .
References
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Received 01.10.2012
В. A. Крыхтин
БРСТ ПОДХОД К ТЕОРИИ ПОЛЕЙ ВЫСШИХ СПИНОВ. ОБЗОР
Даётся обзор БРСТ подхода к построению лагранжианов для полей высших спинов. На примере массивных бозонных полей высших спинов в пространстве Минковского показаны все этапы построения лагранжинов с помощью данного подхода. Рассказано также о специфике построения лагранжианов для фермионных полей высших спинов, полей высших спинов в пространстве постоянной кривизны и полей спина 2 и 3/2 в пространстве Эйнштейна.
Ключевые слова: поля высших спинов, калибровочная симметрия.
Крыхтин В. А., кандидат физико-математических наук.
Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: krykhtin@tspu.edu.ru
