CC BY
24
UDC 530.1; 539.1
On general Lagrangian formulations for arbitrary mixed-symmetric higher-spin fermionic fields on Minkowski backgrounds
A. A. Reshetnyak
Laboratory of Non-Linear Media Physics, institute of Strength Physics and Materials Science, Tomsk, 634021 , Russia.
E-mail: reshet@ispms.tsc.ru, reshet@tspu.edu.ru
The details of unconstrained Lagrangian formulations (being continuation of earlier developed research for Bose particles in NPB 862 (2012) 270, [arXiv:1110.5044[hep-th]], Phys. of Part, and Nucl. 43 (2012) 689, [arXiv:1202.4710 [hep-th]]) are reviewed for Fermi particles propagated on an arbitrary dimensional Minkowski space-time and described by the unitary irreducible half-integer higher-spin representations of the Poincare group subject to Young tableaux Y(si,...,sk) with k rows. The procedure is based on the construction of the Verma modules and finding auxiliary oscillator realizations for the orthosymplectic osp(k|2k) superalgebra which encodes the second-class operator constraints subsystem in the HS symmetry superalgebra. Applying of an universal BRST-BFV approach permit to reproduce gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive fermionic fields of any spin with appropriate number of gauge and Stukelberg fields. The general construction possesses by the obvious possibility to derive Lagrangians with only holonomic constraints.
Keywords: higher spins, BRST symmetry, Lagrangian formulation, Verma module, gauge invariance.
1 Introduction
The interest to higher-spin (HS) field theory is based on the hopes to reconsider the problems of an unique description of variety of elementary particles and all known interactions, in particular, due to recent success with relating to finding of Higgs boson on LHC [1]. One should remind, that it waits, in addition, both the proof of supersymmetry display, and probably a new insight on origin of Dark Matter ( [2]). Due to close interrelation of HS field theory to superstring theory, which operates with an infinite tower of HS fields with integer and half-integer spins it can be viewed as an method to study a superstring theory structure. On current state of HS field theory one may know from the reviews [3-6]. The paper considers the results of constructing Lagrangian formulations (LFs) for free half-integer both massless and massive mixed-symmetry spin-tensor HS fields on fiat R1>d-1-space-time subject to arbitrary Young tableaux (YT) Y(n1,..., nk) for s1 = n1 + 2,..., sk = nk + 2 in Frons-dal metric-like formalism on a base of BFV-BRST approach [7], and precesses the results which appear soon in [9] (as continuation of the research for arbitrary HS fields with integer spin made in [8]).
We know that for higher then d = 4 space-time dimensions, there appear, in addition to totally symmetric irreducible representations of Poincare or (Anti)-de-Sitter ((A)dS) algebras the mixed-symmetry representations determined by more than one spin-like parameters [10], [11]. Whereas for the former ones the LFs both for massless and massive free higher-spin fields is well enough developed [12-16], as well as on base of
BFV-BRST approach, e.g. in [17]- [19], for the latter the problem of their field-theoretic description has not yet solved. So, the main result within the problem of constrained LF for arbitrary massless mixed-symmetry spin-tensor HS fields on a Minkowski space-time was obtained in [20] in so-called "frame-like" formulation (in AdS space in [21]), whereas in the "metric-like" formulation corresponding Lagrangians were derived in closed manner for only reducible Poincare group ISO(1,d — 1) representations in [22].
We use, first, the conventions for the metric tensor = diag(+, —,..., — ), with Lorentz indices ^, v =
0,1,..., d — 1, second, the relations |ym, yv} = for Dirac matrices ym, third, the notation e(A), gh(A) for the respective values of Grassmann parity and ghost number of a quantity A, and denote by [A, B} the supercommutator of quantities A, B, which for theirs
[A, B}
= AB — (—1)e(A)e(B)BA.
2 Half-Integer HS Symmetry Algebra for Fermionic fields
A massless half-integer spin Poincare group ir-rep in R1>d-1 is described by rank Y^k> 1 n* spintensor field ,(M2)n2,...,(Mkw = ^Mi...M^jM2...M22,...,
Mfc...Mfc A (x) with generalized spin s = (n1 + 2 , n2 +
2, ...,nfc + 2), (n1 > n2 > ... > nk > 0, k < [(d - 1)/2]) subject to a YT with k rows of lengths n1,n2, ...,nk and suppressed Dirac index A,
(M1 )nj ?(M2)n2 ?...?(Mk)n
Mi ■
MÏ Mn2
Mk uk Mnfc
7Mii t(„1L (m2)
(M1)ni ,(M2)n2 ,...,(Mfc)nfc = °,
(m1 ) n i ,...,{ (m^ )ni 7 ..., ... , l-..^)!
|*)
CO ni
e e ■
ni =0 n2 = 0
xnn ai
i=1 (¿ = 1
)|^) =0, fori < j; *i <ji, (6)
where (t0,ti,ti1 jl) = (—¿7MdM, 7Ma^, a^1+ajlM).
The set of (2k(k + 1) + 1) primary constraints
(6), {oa} = {i0,i®, tiljl}, with additional condition, g01^) = (n + f)|^) for number particles operators, g0 = —a*+aMi + |, are equivalent to Eqs. (2)-(4) for given spin s.
The fermionic nature of equations (2), (3) and the bosonic one of the primary constraint operators i0,i® with respect to the standard Lorentz-like Grassmann parity, e(i0) = e(i®) = 0 are in contradiction and resolve the problem of, (7)2 = 2ymyva*aV + aVa* = a*a* = with new "traceless" operator, we equivalently transform above operators into fermionic ones. Following to Ref. [18], [19] we introduce a set of (d +1) Grassmann-odd gamma-matrix-like objects 7* 7,
{y m,Yv } = 2nMV, {7 *,7} =0, Y2 = —1, (7)
which are related to the conventional gamma-matrices as: ym = 7mY-
;|°),
Therefore, the odd constraints,
to = — *7MdM , t® = 7MaM,
(8)
(1)
The spin-tensor is symmetric with respect to the permutations of each type of indices m® and obeys to the Dirac (2), gamma-traceless (3) and mixed-symmetry equations (4) [for i, j = 1,..., k; = 1, ...,nj]:
*7MdM^(M1)ni ,(M2)n2-...-(Mfc )nk =°, (2)
(3)
..k, =0, W
for i < j, 1 < j < nj and where the bracket below denote that the indices in it do not include in sym-metrization.
Joint description of all half-integer spin 1SO(1, d —
1)
auxiliary Fock space H, generated by k pairs of bosonic creation a* (x) and annihilation 0+ (x) operators (in symmetric basis), i,j = 1,..., k, ^, vj = 0,1...,d — 1: [a* i, aV+ ] = — ^ij and a set of constraints for an
arbitrary string-like (so called basic) vector |^) G H, being as well Dirac spinor, nk- l
• E ^(Ml)nl ,(* )n2 ,...,(Mfc )nk
nk = 0
k ni
+*L
related to the operators (6) as: (t0,ti) = 7(t0,t®), solve the problem above.
Finding of Lagrangian as, L ~ (^|Q|^), implies the Hermiticity of BFV-BRST operator Q, Q = Caoa+ ..., that means the extension of the set {oa} up to one of {oj} = {oa,1j,1jj; o+,/+,/+■; g0}> f°r divergent and gradient operators (/*,/+) = — *(afdM, af+dM) and for
/+ = 2 af+a*+, i < j, which is closed with respect
[, }
conjugation related to odd scalar product on H,
/^,(n)fc-l TO,(p)fc_l fc;pj 3-
dfx £ £ (0| H aj
(n)fc = 0 (p)fc =0 j'=1;mj = 2
X ^(vl)Pl ,(v2 )p2 ,...,(vk)Pk 70^(Ml )nl ,(*2)n2 >...>(Mfc)nfc
k;ni
n ».+'ii|°)
(9)
i=1;1i=1
fr\r V^O,(n)k-i — v-^ni
iU1 Z^(n)k = 0 — 2^ ni=^Z^ n2 = 0
Enfc-1
nfc = 0’
,Pj e N0.
Operators o/ satisfy to the Lie-superalgebra commutation relations, [o/, oj} = f/JoK, for structure constants fKJ = — ( —1)e(oi)e(°J)f/J, to be determined by anticommutators,
(5)
{t0,t0} = —2Z0, {t0,ti} = 2Z®, {tj,tj}
{t®7j} = 2 0—g0 + tji + tij j )7
4Z.
ijj
(10)
(with Heaviside 0-symbol 0ij) and from the multiplication table 1 with only commutators.
The products ,Ai2j2 ’iljl, Filjl’i, Li2j2’iljl in
the table 1 are given by the relations,
Bi2j2 iljl = № — gj2 )^i2 ^j'2 + (tjlj2 0j2 jl (ii)
+tj2 +l »jlj2 )<2 — «i2 ^i2 il + ti2 il ^ili2 )^j12,
A*2 j2,iljl = £ilj2 Ji2jl — ¿i2jl jilj2 Fi2j2 ,i = ¿[i2j2 j ]i
Li2j2 ’il jl = 2f^i2il ¿j2 jl [2g02 ^i2 j2 + g02 + gj2]
—(<5j2{il[tjl}i20i2jl> + ti2jl>+0jl>i2]+(j2 ^ ¿2))}.
oj
integer higher-spin symmetry algebra in Minkowski
k
(Y(k), R1’f-1). It appears as the superextension of integer higher-spin symmetry algebra A(Y(k), R1’f-1) introduced in [8] for bosonic fields.
Hamiltonian analysis of the topological dynamical system of the operators {oJ} permits to classify 2k(k +1) operators {o0} = {ti,/ij,tiljl ,i+,/((,t+jl} as second-class and 2(k + 1) raes {t0, /0, /i, /+} as first-
k
X
[ 4, t^Ui t+ . *171 lo l* l*+ l*1 J1 lni^ So
to 0 0 0 0 0 0 0 0
t*2 —tj1¿*2*1 t• ¿*2 • 6*10 .1 0 0 —^¿^ 0 — 11{*1 + ¿71 }*2 t*2¿*2*
t*2 + tii+5i2ji t+ ¿i, *2 71 *1 0 too*2* 0 2t{*1 ¿7'1 }*2 0 — ^ + ¿*2*
t*2J2 A*2 72 ’*171 R*2J2 - • B *171 0 72 ¿*2* — li2 + ¿j2i l{7172 ¿*1 }*2 — li2{i1+¿j1}j2 F*2 72 ’*
t+ . *2.72 _ Ri1J1 . . B *2 72 A+ *1 71 ’*2 .72 0 l- ¿* l*2 ¿72 —1+ ¿* 72 *2 l- {ji ¿*1} l*2 ¿72 -7 {jl+¿il} l72 ¿*2 _F. . *+ F *2 72
lo 0 0 0 0 0 0 0 0
7 —lj1¿*17 —1*1 ¿7 *1 j1 0 0 W7* 0 — 1 l{i1+¿j1}j 7 ¿7
7 + 1*1+7' l+ ¿7 j1 *1 0 —lo¿ji 0 1l{*1¿j1}j 0 —lj+¿ij
1*2.2 lj1 { j2 ¿*2 }*1 —1*1 {*2 +072 } - *1 j1 0 0 — 11{*2 ¿j2}i 0 ¿*2 72 ’*1J1 li{i2 ¿J2}i
1*2.2 + l*1 {i2 + 5j2}j1 * ¿* + 2 { 0 |l{*2 +¿ij2} 0 — ¿*171 ’*2 72 0 li{i2 + ¿j2}i
_F*1 j1 7 F ■ 7+ F*1J1 0 — M7 li+¿ij — l7 { * 1 ¿71 }7 lJ{i1 + ¿71 }J 0
Table 1: Even-even and odd-even parts of HS symmetry superalgebra Af (Y(k),R1,d 1).
class constraints subsystems whereas k elements form supermatrix Aab(go) in [oa,ob} ~ Aab.
The subsystem of the second-class constraints {oa} together with } forms the subalgebra in A(Y(k), R1>d-1) to be isomorphic, due to Howe duality, to orthosymplectic osp(k|2k) algebra (the details, see in
[9]). The HS symmetry superalgebra (Y(k),R1>d-1) itself can not permit to construct BRST operator with respect to oj due to second-class constraints {oa} presence in it. Therefore we should to convert orthosymplectic algebra osp(k|2k) of {oa,go} into enlarged set of operators OJ with only first-class constraints.
3 Scalar Oscillator realization for osp(k|2k)
We consider an additive conversion procedure developed within BRST method, (see e.g. [17]), implying the enlarging of oJ to OJ = oJ + oj, with additional parts oj supercommuting with all oj and determined on a new Fock space H'. Now, the elements Oj are given on H <8> H' so that a condition for Oj, [Oj, Oj] ~ Ok, leads to the same algebraic relations for Oj mid oj as oj
Not going into details of Verma module construction for the superalgebra osp(k|2k) of new operators oj considered in [9] and for the case of its sp(k|2k) subalgebra in [8], we present here theirs explicit oscillator form in terms of new 2k(k + 1) creation and annihilation operators (B+,Bd;) = (/+,&+•, d+s; /¿Aj ,drs), i, j, r, s = 1,..., k; i < j; r < s as follows (for ko = l)
g'j = /+/ + E + ¿im)
1<m
i — 1
• = = /+ + 2b+ /i + 4 £ b+ /i, (13)
1 = 1
1 — 1 k
= d+n - E - E(1 + ^"1 )b++mb1n , (14)
n=1 n=1
1 — 1 k
= — — (1 + n=1 n=1
m—1 —1 m—1 m—1 p
+ E E ... E Ckpm(d+,d) n dkj-1kj p=0 ki =1 + 1 kp = 1+p j=1
s—1
+ [4 b+n/n + (2b+r i”r — /+ )]/s. (15)
n=r+1
Note, first, that Bc, B+ satisfy to the standard commutation relations, [Bc,B+} = £cd, for instance, {/i,/+} = ^ij f°r °dd /i,/+• Second, the arbitrary parameters hi in (12) need to reproduce correct LF for HS field with given spin s, whereas the form of the rest elements ti, Ij, for i < j, to be expressed by means of the operators C1m(d+,d),1 < m, as well as the property of Hermiticity for them may be found in [8], [9]1.
4 BRST-BFV operator and Lagrangian formulations
Due to algebra of Oj under consideration is a Lie superalgebra (Y(k),R1>d—1) the BFV-BRST opera-
tor Q' may be constructed in the standard way as
Q' = OjC j + 1 CjC J /Jj Pj (-1)£(°k )+e(°i) (16)
with the constants /Jj from the table 1, constraints Oj = (To,Ti+, Lo,L+, Li, Lij, L+,Trs, T+,G‘), fermionic [bosonic] ghost fields and conjugated to them momenta (Cj, Pj) = ((no, Po); (n\ P+); (n+, Pj);
+ £d+sdrs(^is - ¿ir) + hi, (12)
r<s
1The case of massive HS fields whose system of 2nd-class constraints contains additionally to elements of osp(fc|2fc) superalgebra the constraints of isometry subalgebra of Minkowski space to,l*,l+,lo may be treated by dimensional reduction of the algebra Af (Y(fc), RM) for massless HS fields to one Af (Y(fc), R1^-1) for massive HS fields, (see [9]). Now, the Dirac equation in (2) is changed on massive equation corresponding to the constraint to (to = — t7MdM + 7m) acting on the same basic vector |^) (5).
(nij, P+); (n+, Pij );(^rs,A+s ); (^r+s,Ars);(nG, P g)),
[(qo,po), (q+,Pi), (qi,p+)] with the properties
(nij,^rs) = (nji,^sr), Rs, A+} = <Srt<U
P+} = ¿гi¿Jm, {Pj,n+} = [qi,P+] = % (17)
and non-vanishing (anti)commutators {no, Po} =
[qo,po] = *, {nG, Pj} = *^ij for zero-mode ghosts2.
To construct LF for fermionic HS fields in a R1,d—1 we partially follow the algorithm of [23], being a particular case of our construction for n3 = 0. First, we extract the dependence of Q' (16) on the ghosts nG, PG> to obtain generalized spin operator ai and the BRST Q
constraints {Oj} \ {Go} on appropriate Hilbert subspaces:
Q' = Q + nG (a + hi) + B^G, with so me Bi, (18) Q = (1 qoTo + 2 noLo + q+T X+ Li + E i<m nL^
(Q|xo),^|xo),...,^|A(r—1)^ = (0,Q|Ao),...,Q|A(r))), [ai + hi](|xo), |Ao),..., |A(r))) =0, (23)
with r = k=1 m + k(k — 1)/2 — 1 being the stage of reducibility both for massless and for the massive fermionic HS field. Resolution the spectral problem from the Eqs.(23) yields the eigenvectors of the operators a1: |xo)(n)fc, |Ao)(„)fc, ..., |Ar)(„)fc, m > m > ... nk > 0 and corresponding eigenvalues of the parameters hi (for massless HS fields and i = 1,..,k),
—hj = mj + d 24i , mi, ..., mfc_i G Z, G Nq .
(24)
+ El<m <„T + h.C.) + 2C1 CJ JPK,
a® = gq — hj — + n+ P® + q®p+ + q+Pi
+ E(1 + )(n+mP'm — +n)
(19)
+ Ei<J^+— ] — Ei< I^+ — A+ ] , (20) of the vectors |x
Q
the subspaces determined by the solution for the Eq. (23), second, to construct Lagrangian for the field corresponding to a definite YT (1) we must put mi = ni, and, third, one should substitute hi corresponding to the chosen ni (24) into Q (18) and relations (23).
To get the Lagrangian formulation with only first order derivatives, we, because of the functional dependence of the operator Lo on fermionic one To, Lo = —To2, Lo, no
Q
|A(s
/(n)fc
To do so, we extract
where {Cj, pj} = {Cj, Pj} \ {nG, PG}• Next, we choose a representation of Hi0t: (qi,pi, ni, nij ^rs, Po> po,Pi,Pij Ars,PG)|0) = 0, and suppose that the field vectors |x) as well as the gauge parameters |A) do not depend on ghosts nG: k
the zero-mode ghosts from the operator Q as: Q = qoTo + noLo + *(n+ qi — n*q+ )pq — *(q0 — n+ ni)P 0 + AQ
(25)
AQ
Eqs. (18), (25) and
IX = E n (/+)ni(6i+')”ij (d+s)
)prs qnt0 nn/0
To = To — 2q+p j — 2qjp+
Tq0 = — Lo.
(26)
n 1,i<j,r<s
I[(q+)n'
e,g,i,j,1<m,n<o
(P+O)npno n
(p+)nbg (n+)nfi (P+)npj
(n+m)n
r<s,t<u
(^+s)n/rs (A+l)nAtu X
We also expand the state vector and gauge param-
s=
0,. .., ^k=1 n0 + k(k — 1)/2 — 1, m = 0,1:
(21) |x) = Eqo(|Xo) + no|x1)),gh(|xm)) = —(m + ¿G (27)
I*(«+)
+ \ nb0nf0 ;(n)ae(n)bg(n)/i(n)pj(n)/Zm(n)pno(n)/rs(n)Atu
(n)0 (n)ij (p)r
/
We denote by |xk) the state (21) satisfying to gh(|xk)) = —k. Thus, the physical state having the ghost number zero is |xQ)j the gauge parameters |A) having the ghost number —Ms |A0) and so on. The vector |x0) must contain physical string-like vector
|^) = |^(a+)(0)fo...(0)rs ).
|X0) = |*) + |ŸA) : |ŸA)| [b+=ci =p =q] = 0. (22)
Independence of the vectors (21) on nG transforms the equation for the physical state Q'|x0) = 0 and the BRST complex of the reducible gauge transformations, %) = Q'|A0), ¿|A°) = Q'|A1) .., ¿|A(r-1)) = Q'|A(r)), to the relations:
i>0
|A(s)) = E qo 1=0
(28)
Now, we may gauge away of all the fields and gauge parameters by means of the equations of motion and set of the gauge transformations (23) except two, |xo), |xo) for the fields and |A(s)o), for l = 0,1 Mid s = 0,..., r, for the gauge parameters. To do so, we use in part the procedure described in [18], [23].
As the result, the first-order equations of motion corresponding to the field with given spin (n1 + 2,..., nk + 1) have the form in terms of the matrix notations,
(0 \ w
f To AQ
\AQ 2{To,n+ni},
|xQ
2The ghosts possess the standard ghost number distribution, gh(CJ) — —gh(Pj) — 1
3The brackets (n)f*, (ra)pj, (ra)*^^ ^rans, e.g., for (ra)*^^ ^^^f indices (ran,
taken over ra-bo, raa^ ra^^ ra^j prs ^^d ^^rning from 0 to infinity, and over the rest ra’s from ^o 1.
Sh(Q') = 1.
, nik, ...,rafci, ...,rafcfc). The sum above is
Q
(massive) HS fields. General action (30) gives, in principle, a straight recept to obtain the Lagrangian for any component field from general vectors |x0)(n)k-
5 Conclusion
Thus, we have constructed a gauge-invariant unconstrained Lagrangian description of free half-integer HS fields belonging to an irreducible representation of the Poincare group with the arbitrary YT having k rows in the “metric-like" formulation. The results of this study are the general and obtained on the base of universal method which is applied by the unique way to both massive and massless bosonic HS fields with a mixed symmetry in a Minkowski space of any dimension.
Acknowledgement
The author is grateful to V. Krykhtin, E. Skvortsov, K. Stepanyantz, Yu. Zinoviev for valuable comments and to I.L. Buchbinder for collaboration on many stages of the work. The work was supported by the
RFBR grant, project Nr. 12-02-00121 and by LRSS grant Nr.224.2012.2.
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They are Lagrangian ones and can be deduced from the Lagrangian action for fixed spin (m)k = (n)k, (being standardly defined up to an overall factor and with omitting the subscript (n)k)
S(-»=^<x°')<Qi{Wn,})(|XS)) (30)
where the standard odd scalar product for the creation and annihilation operators in = H & H' & is assumed and non-degenerate operator K = K(„)fc provides reality of the action following from modifying Hermiticity for oI in Section 3. The action (30) is invariant with respect to the gauge transformations, following from the tower of the Eqs. (23) with omitting (n)fc,
|л08)°)\ (To aq A m0s+1)r
Iaq 2{W^y Uos+1)1b
for s = —1,0,..., Ek=1 no + k(k — 1)/2 — 1, and
|Ao~1)1) = |xo)
Concluding, one can prove the action (30) indeed reproduces the basic conditions (2)-(4) for massless
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Received 01.10.2012
А. А. Решетняк
ОБ ОБЩИХ ЛАГРАНЖЕВЫХ ФОРМУЛИРОВКАХ ДЛЯ ПРОИЗВОЛЬНЫХ СМЕШАННО-СИММЕТРИЧНЫХ ФЕРМИОННЫХ ПОЛЕЙ ВЫСШИХ СПИНОВ НА
ФОНЕ ПРОСТРАНСТВ МИНКОВСКОГО
Выполнено обозрение деталей лагранжевых формулировок без связей для Ферми частиц, распространяющихся на пространстве-времени Минковского произвольной размерности и описывающимися унитарными неприводимыми представлениями группы Пуанкаре с полуцелым высшим спином подчиненными диаграммам Юнга Y (si,...,sk) с k строками (являющееся продолжением исследования ранее проведенного для Бозе частиц в [NPB 862 (2012) 270, [arXiv:1110.5044[hep-th]], Phys. of Part, and Nucl. 43 (2012) 689, arXiv:1202.4710 [hep-th]]). Процедура основана на построении модулей Верма и нахождении вспомогательных осцилляторных реализаций для ортосимплектиче-ской osp(k|2k) супералгебры, кодирующей подсистему операторных связей второго рода в супералгебре симметрии полей высших спинов. Применение универсального БРСТ-БФВ подхода позволяет воспроизвести калибровочноинвариантные лагранжианы с приводимыми калибровочными симметриями, которые описывают свободную динамику как безмассовых, так и массивных фермионных полей любого спина с подходящим числом вспомогательных калибровочных и штюкельберговых полей. Общая конструкция обладает очевидной возможностью вывести лагранжианы с только голономными связями.
Ключевые слова: высшие спины, БРСТ симметрия, лагрансясева формулировка, модуль Берма, калибровочная инвариантность.
Решетняк А. А., кандидат физико-математических наук.
Институт физики прочности и материаловедения СО РАН.
Пр. Академический, 2/4, Томск, Россия, 634021.
E-mail: reshet@ispms.tsc.ru
