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UDC 530.1; 539.1
GAUGE INVARIANT LAGRANGIAN FORMULATIONS FOR MIXED-ANTISYMMETRIC FERMIONIC
HIGHER SPIN FIELDS
H. Takata1
Department of Theoretical Physics, Tomsk State Pedagogical University, Kievskaya str., 60, 634061 Tomsk, Russia.
E-mail: takata@tspu.edu.ru
A Lagrangian formulation of irreducible half-integer higher-spin representations of the Poincare algebra with a Young tableaux having two columns is presented based on the BRST approach. Starting from Casimir constraints written by oscillator representation of Poincare algebra, which is the necessary condition of the irreducibility, we find closed higher spin superalgebra. In order to convert all constraints to the first class we introduce four auxiliary oscillators with Y-matrix and use Verma module method. To get nilpotent BRST operators we further introduce ghosts. After using restrictions on spin number and ghost number we construct Lagrangian having gauge symmetry with finite stage of reducibility.
Keywords: higher-spin Gelds, gauge theories, BRST method, Lagrangian formulation.
1 Introduction
Higher spin theory has been studied in the hopes as an unified description of elementary particles [2]. For the progress in higher-spin field theory, see the reviews [3]. For our study, we will construct un-constraint Lagrangian by using BRST approach [4]. In this construction, we make a connection between Lagrangian and space-time symmetry, like Poincare symmetry. Considering constraints from suitable representation of Casimir operators in Poincare algebra, one may find constraint algebra, that is, higher spin algebra which leads nil-potent BRST operators [5]. Once BRST operator is found, gauge invariant Lagrangian can be constructed straightforwardly.
BRST approach has been applied for various cases of constructing free Lagrangian. In the metric-like approach for bosonic tensors (x) or fermionic tensors (x) with general type of indexes yU4, ••• ,yU,s, there are two different approaches; symmetric base approach and anti-symmetric base approach. In the former approach one start from totally symmetric tensor [4,6] and then generalize it to mixed symmetric case [7]. One introduce bosonic oscillators
lаiß, a'+] = -S'jVpv, i,j = 1, 2
Totally sym. Mixed sym.
Bosonic dßdß aßdß , ß 1 aßa'ß dßdß a'ißdß , 1 a' a'ß 2 a'ßa 3 n'+ n'ß a 1ßa 2
Fermionic Yßdß, iß Yßa Yßd ß, ß 1ßai n'+ n'ß a 1ßa 2
In other word, these approaches correspond row based classification of associated Young-tableau. While totally symmetric case corresponds one row Young-tableau, mixed symmetric case corresponds Young-tableau with more than one row.
On the other hand, in the later approach one start from totally anti-symmetric tensor [8] and then generalize it to mixed anti-symmetric case [1]. One introduce fermionic oscillators
jaiM,a+v} = -Sij,i,j = 1, 2,
(2)
to construct Fock space and consider suitable vanishing constraints on that for each case. It is as follows for massless fields in flat space-time.
Totally anti-sym. Mixed anti-sym.
Bosonic dßd ß aßdß dßdß 2aißaß a'ßdß , a+ßaß
Fermionic Yßdß, Y ßaß Yßdß, a^ a1ßa2
(1)
to construct Fock space and consider suitable vanishing constraints on that for each case. For example, for massless fields in flat space-time, it is as follows.
These approaches correspond column based classification of associated Young-tableau. While totally anti-symmetric case corresponds one column Young-tableau, mixed anti-symmetric case corresponds Young-tableau with more than one column.
In fact, for the antisymmetric case, there was no clear explanation about the fact that above condition gives irreducible representation of Poincare algebra. Therefore in the next section we will explain, by considering Casimir operators, how irreducible representation of Poincare algebra and the vanishing constraints are related.
xThis talk is based on the collaboration with J. L. Buchbinder and A. A. Reshetnyak [1]
2 Irreducibility condition One can easily see that if the following constraints
In this section, we consider the meaning of the to W — ^ W — ¿2 W — gi2
supplementary condition in terms of the irreducibil- — (gu + si — |)|^} — 0, (12)
ity. To consider general type of tensor indexes we can
TTi-i-n, The second Casimir operator C2 can be written by
use symmetric base or anti-symmetric case. Unlike the r i
, • , ,-j , , l, • j -u-r-t the Pauli-Lubanski vector Wu 1101
symmetric base we did not know how irreducibility con- M L J
1
2
dition for the general type of tensor can be written 1
in the anti-symmetric base. We study it for massive C2 — WmWm, Wm — 2P sp . (13)
bosonic tensor in four space-time dimension as the simplest case. Consider bosonic tensor - (x
) By defining
with constraint on indexes as
(x) = • • • ls1 ],[vi-Vs2](X)> (3)
Cij = W;Wjl, = 1 e^vpapv, (14)
the second Casimir can be written as C2 — WlMWf + where square bracket describe anti-symmetrization. W2mW2m + 2W1mW2m. In the following we will show that We are going to explain that the following conditions: C2 are multiples of the identity operator on the vector.
Note that these Cij can be directly checked to com) — 0 (4) mute to the generators. By inputting expressions of
(□ — m2)$• 1 • • • • vs2 (x) — 0 (5) eqs (10) we get the following expression
dM1 • • • Ms1,vr• • S12(x) — 0 (6) Cij = {- (a+i • a+')(. • a') + (a'+ • ai)(a+i • a') + (.+' • a')
dV1 .„ . V1..,V2(x) — 0 (7) d2 , ...
+ (a+i • a+')(. • d)(a' • d)- Sij (a+i • d)(a' • d) +h.c.
can be understood as the condition that Casimir operator C2 is a multiple of the identity operator. For When ^g above conditions eq. (11), it lead to the Poincare algebra we adopt oscillator representation on final result
space |$}: C2|$} — m2 {si(si — 3) + S2(S2 — 5)}|$}. (16)
ai • • • «i a2 • • • «2 • • M s 1 ,V1 • • • vs 2
(x)
Thus we have shown that the second Casimir operator
C2 is a multiple of the identity operator on the vector
with two sets of anti-commuting oscillators which satisfies supplementary conditions eq. (11), that
r a +,,, ,,,, c . . , _ is a necessary condition of irreducibility of the repre-
{af,a+v} — —Sij,«,7 — 1,2. (9) ,
{ j } 1 w sentation.
It is found that the representations of generators of
Poincare algebra can be written as [9]. 3 HS Symmetry Superalgebra for mixed-
antisymmetric fermionic fields
Mlv = Llv + £MV, In this section, we consider a massless half-integer
Llv = —¿(xMdv — xvdM), (10) spin irreducible representation of Poincare group in a
£MV = + Y?lv, Minkowski space which is described by a tensor field
j = i(j+aV — aV+ajJ,j = 1, 2 • • is 1 ],[vr• • vs2](x) to be corresponding to a Young
tableaux with 2 columns of height si, s2, respectively
Supplementary conditions (4)-(8) are also rewritten as2
[ft • • • is 1 ],[vr • • vs 2 ^
Mi V1
VS2
(17)
gi2|$) — lol^} — li |$} — ¿12^} — (gjj + Si — 2)|$} — 0 (11)
where we have used definitions (pM — —)
We call this field as spin [sl, s2] field, that is antisym-10 — —p2 + m2, 1i — afpM, 112 — 1 alMa2, metric with respect to the permutations of each type
gij — a+jaj' + 2Sij, 1+ — aM+pM, 1+2 — 1 «2+«+M. of Lorentz indices v. This is realized in the space
2The first condition (3) is automatically satisfied.
3 General formula for arbitrary Young tableaux in anti-symmetric base and in symmetric base are similary given as C2|$) =
m2 colums ^ — 2j — 1)|$) and C2= — m2 rows (^j — 2j + 3)|$), respectively. It can be easily proved that
these two formulas are equal to each other for given fixed Young tableaux, independently from the choice of the base as expected.
Pv
of fermionic tensor field ],[V1...V ] (x) satisfying We call the vector (21) as basic vector. The fields
the following constraints
],H-vS2 ](x) =
^[.1-"2s1 ], [ Vi ••• V s 2 ] (x) =
Y 1 ], [ Vi *** v s 2 ] (x) =
^MlM2 - [Ms1 ,V1V1- Vs2] (x) =
(18)
K, a+v} = -nMVSij, nMV = diag(+, -, ■
{w,n1}
Y a+.^A a+2n1a+.v.\ a+Vn2 ,{v}„2 (x).
{n1,n2} = {0,0}
(21)
{.}n1 ,{v}r
(x) are the coefficient functions of the vec-
those can be naturally guessed from the bosonic version (4)-(8) in the previous section. Our purpose is to construct a Lagrangian which reproduces these constraints as the consequences of the equations of motion. It is convenient to introduce grassmann-odd creation and annihilation operators with space-time indices v = 0,1,...,d — 1) and column indices (i,j = 1, 2)
tor |^), where the simplified notation {^}n is used because its symmetry properties are stipulated by the properties of the product of the creation operators.
Let us define a set of operators being quadratic in powers of all oscillators, af(+), odd gamma-matrices, 7f, momenta pf (with notation pf = — idf) and requirement to have af, af+, pf in the products
to = ti = 7.af, t+ = a.+Y.,
lo = li
1+ = a.+
.
_ + 2 , d r
gij = aiuaj + 2 °ij,
2 a1.a2 ,
112 _
/+2 = 1 a2 a
1.,
(22)
One can easily see that if the following constraints
• -) (19)
We follow [6] and introduce a set of (d +1) Grassmann-odd gamma-matrix-like objects ym, 7, subject to the conditions
{Yf, 7V} = 2nfv, {7f, 7} = {y7, a(+)v} = 0,
Y2 = —1. (20)
An arbitrary vector in this Fock space has the form
to |tf> = ti|tf> = t2|^> = g12^> = (gii + Si - f )|tf> =0,
(23)
are satisfied for |^> then each component ,{v}n2
of (21) obeys the conditions (18)-(18) Let us turn to the algebra generated by operators t0, ti, g12 and gii, (i = 1, 2). To get a real Lagrangian we need a Hermitian BRST operator. Therefore, we should add operators t+(i = 1,2) and g21. By taking super-
commutators we get operators 10, 1(+)(i = 1, 2),1(+). As a result we need all of operators defined in (22). Our task is to find a closed algebra including these operators. After simple calculation, we find the following closed algebra using these operators.
[4, to lk l+ lo tk t+ l12 l+ l12 gki
to -2lo 0 0 0 2li —2l+ 0 0 0
li 0 0 Sik lo 0 0 —Sik to 0 —Si[2l+ Skill
l+ 0 Sik lo 0 0 Sikto 0 Si[2l1] 0 Slil +
lo 0 0 0 0 0 0 0 0 0
ti -2li 0 -Sikto 0 — 4li2«ik —2gki 0 —Si[1t+ —Sik tl
t+ 2l+ Sik to 0 0 2gik 4l+2£ik Si[1t2] 0 Silt+
112 0 0 Sk[2l1] 0 0 —Sk[1t2] 0 1 (gn + g22 ) —Skll12
l+ l12 0 Sk[21 + 0 0 Sk[1t+ 0 — 4 (gn + g22) 0 Skll+2
9ij 0 Skilj -Skj l+ 0 Skitj —Skj t+ Sijl12 _S. J+ Sijl12 Sugkj — Sjkgu
HS superalgebra for massless fermionic field corresponds
In this table, the first and the second arguments of the super-commutators are listed in the first column and the first row, respectively. The algebra corresponding to this table is a base for massless halfinteger higher spin field Lagrangian construction corresponding Young tableaux with 2 columns in flat space. We should emphasize some points. First, there are four hermitian operators 10,t0,gjj for i = 1, 2. Second, to, h and 1+ are grassmann-odd operators and others are grassmann-even. Third, a straightforward use of BRST-BFV construction as if all the operators are the first class constraints doesn't lead to the proper
to two columns Young-tableaux. (e^ = —£fci> £12 = 1.)
equations (23) for any spin [si,s2] (see e.g. [4,6]). In fact, in the table, there are second-class constraints caused by super-commutators among eight operators: t(+)(i = 1, 2), 1(+), gi2 and g21. Thus we must somehow get rid of these second class constraints. Method of elaboration of the second class constraints consists in constructing new representations of the superalgebra so that the hermitian operators g11 and g22 will be modified with constant parameters to be controlling possible values of spin [s1, s2].
4 Conversion of HS Symmetry Superalgebra
In this section, to solve the problem of modification of tj, ,li2,l+2, gi2 and #21 we describe the method of auxiliary representation construction for the superalgebra with second-class constraints, in terms of new creation and annihilation operators. It implies the enlarging of o/ = {to, lj, /+, Zq, tj, t+, li2, /+2, gjj} to 0/ = o/ + o/, where additional parts o/ are given on a new Fock space with requirement,
[o/, o'j} = 0, [o/, o'j} = fKjo'K, [O/, Oj} = //jOk
with the same structure constants /J of the superalgebra in the table. Because of only sub-algebra of left-bottom part of the table has the second-class constraints, it is enough to get new operator realization form them. We will solve this problem by means of special procedure known as Verma module construction [11] for the latter algebra.
Corresponding to four pair of bosonic operators t1, t2,l12 and #12 and their conjugations, we introduce the same number of pair of auxiliary bosonic oscillators [bj, b++] = Sij , (i, j = 1, • • • , 4). Owing to this conversion two arbitrary constants h1 and h2 can be introduced in the system. They, in fact, control the size of spin [s1, s2]. Detailed calculation and explicit form are written in [1].
As explained in the beginning of this section, new representation of the algebra is
Ti = ti + ti
T+ = t+ +t+'
L
12
1l2 + l
12
L12 = 1 1+2 + 1 1+2
2
11 2
Gij 9ij + 9ij
It is obvious that these operators with capital letter form the same algebra to the table.
5 BRST operator and fixed spin theory
In order to construct BRST operator one may use prescription [5]. It is as follows. Corresponding to the each generators 0/'s of the algebra having only the first class constraints, we introduce a pair of creation -annihilation operator C/ and P/ satisfying to the canonical commutation relations, [C/, PS} = Sj, whose Grassmann parity is opposite to the one of 0/. We call C/ ghost coordinate and P/ ghost momentum . Then nilpotent BRST-BFV operator Q' are given as
1
Q' = C1 Oi + 1CICJ fK Pk (-1)£(°k ),
2
(24)
corresponding to the structure of given Higher spin algebra
[Oi, Oj} = fKjOk, fKj = -(-1)e(Oi)e(Oj)fKi, (25)
where e(O) describe Grassmann parity of O. Following to this prescription, we introduce ghost coordinate
and ghost momenta and define Q' corresponding our superalgebra of the table. Including oscillators and creation operators of ghosts, Hilbert space is now extended from of eq. (21) to that we will write as |x). As usual prescription of BRST approach, one need specially treat Hermitian ghosts parts, which originate from hermite operators to and Gjj. The detail is in [1]. Generalized spin number operators <7j + hj are defined in Q' as the coefficient of the ghost coordinate that corresponds to gjj. As a result for spin [si,s2] model, we fix parameters hj to be
hi = si - ( d + 1 + (-1)
(26)
Thus, we now study spin fixed theory.
6 Unconstrained Gauge-invariant Lagrangian
Nilpotent BRST operator Q' gives us equations of motion and gauge transformations in the space |x). After partially fixing gauge and using equation of motion those do not give any new constraints, eq. of motion and gauge transformations in reduced subspace are given as
¿|X("-1)'0}=AQ|X(")'0} + 1 {T0, q+qi}|x(n)'1}, (27) ¿|x("-1)'1}=AQ|x(n)'1} + T0 |x(n)'°), (28)
where n = 0 giving equations of motion (l.h.s= 0) and n = 1, 2, • • • 2=1 si + 3 giving reducible gauge transformations. AQ is defined as terms that independent of any Hermitian ghosts in Q' of (24). Tq is also defined there as coefficient of the Hermitian ghost coordinate that corresponds to T0. and qi are ghost coordinates corresponding to Li and L+ respectively. Two independent vectors |x(n)'"0}, (n0 = 0,1) are supposed to be independent of any Hermitian ghosts and to have ghost number —(n + n0), where ghost number +1 is assigned to the ghost coordinates and -1 to the ghost momenta. We can easily find Lagrangian that derive above equations of motion as
£
(/jKTO |x°> + - № {T0 ,q+qi}|x1)
+ <X0|K AQjx1) + (x1|K AQ|x°>,
(29)
where |xno) = |x(0)'"°} and operator K has been introduced in order to make Lagrangian real [1]. This (29) is invariant under gauge transformations (27) and (28) with finite stage of reducibility, and is an unconstraint Lagrangian of massless free half-integer HS fields corresponding Young tableaux having two column in any space-time dimension. These are our final results.
7 Conclusion
In the present work, we have constructed a gauge-invariant Lagrangian description of massless free half-integer HS fields belonging to an irreducible representation of the Poincare group with the corresponding Young tableaux having two column in the metric-like formulation in a Minkowski space of any space-time dimension..
By using oscillator representation of Poincare algebra in mixed anti-symmetric base, an explicit explanation has been given, through the Casimir constraint (16), about initial conditions of BRST construction for massive bosonic fields in four space-time dimension.
We have started from initial condition (23) or (18)
supposed to give an irreducible Poincare-group representation for massless fermionic field with corresponding two column Young-tableaux (17).
These constraints generate a closed Lie superalgebra of HS symmetry as in the table, whose representation can be additively converted to have only first class constraints.
The nilpotent BRST operator Q' (24) are defined and in its reduced subspace equations of motion, gauge transformations with finite stage of reducibility (27)-(28) and unconstraint lagrangian (29) for massless free half-integer HS fields corresponding Young tableaux having two column in any space-time dimension are given.
References
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Received 24.11.2014
X. Таката
КАЛИБРОВОЧНО-ИНВАРИАНТНАЯ ЛАГРАНЖЕВА ФОРМУЛИРОВКА ДЛЯ СМЕШАННЫХ АНТИСИММЕТРИЧНЫХ ФЕРМИОННЫХ ПОЛЕЙ С ВЫСШИМИ
СПИНАМИ
Представлена лагранжева формулировка неприводимого представления алгебры Пуанкаре с полуцелым высшим спином, основанная на методе БРСТ. Исходя из связей Казимира в осцилляторном представлении алгебры Пуанкаре, мы находим замкнутую супералгебру высших спинов. С целью преобразования всех связей к первому классу, мы вводим четыре вспомогательных осциллятора и используем метод Верма. Чтобы получить нильпотентные БРСТ операторы мы вводим духи. Используя связи на спиновое число и духи, мы конструируем калибровочно-инвариатный лагранжиан.
Ключевые слова: поля высших спинов, калибровочные теории, метод БРСТ, лагранжева формулировка, когерентность.
Таката X., доктор.
Томский государственный педагогический университет.
Ул. Киевская, 60, 634061 Томск, Россия. E-mail: takata@tspu.edu.ru
