Bypssing no-go theorems for consistent interctions in guge theories Simon Lykhovich Tomsk Stte University Suzdl, 4 June 2014 The tlk is bsed on the rticles D.S. Kprulin, S.L.Lykhovich nd A.A.Shrpov, Consistent interctions nd involution, 2013; S.L.Lykhovich nd A.A.Shrpov, Multiple choice of guge genertors nd consistency of interctions, 2014
Generl Pln 1. Introduction: Guge identities, symmetries nd consistent inclusion of interctions. 2. Involutivity nd consistency of interctions. Involutive closure of dynmics. Guge symmetries nd (implicit) guge identities; Involution nd control of degrees of freedom; Perturbtive inclusion of interctions, no-go theorems; Exmples: mssive spins 1 nd 2. 3. By-pssing perturbtive no-go theorems. Exmple of by-pssing the perturbtive obstruction. Multiple choice of free guge genertors nd its impct on interction. One more exmple: Topologicl grvity in d dimensions. Concluding remrks
Condensed nottion, guge symmetry nd identities The elds x i re lbel by "condensed"index i. Summtion over i implies spce-time integrtion. The derivtives i re vritionl. The EoM's re the dierentil equtions of nite order: T (x)=0 (1) In Lgrngin theory =i, nd T i (x)= i S(x). The Jcobi mtrix i T =J i (x) in Lgrngin theory, J= i j S(x) is symmetric, nd its on-shell kernel denes the guge symmetry. Left on shell kernel corresponds to guge identities between EoM's: L A J i(x) T =0 =0 L A T 0 (2) Right on shell kernel corresponds to guge symmetry of the eqs: J i (x)r i T =0 =0 δ ɛ T T =0 =0, δ ɛ x i =R i ɛ (3)
Exmple of guge identities unrelted to guge symmetries: Irreducible mssive spin 2, without uxiliry elds. For the mssive symmetric trceless tensor, the EoM's red T µν ( m 2 )h µν =0, T µ ν h µν =0. (4) The EoM's do not hve ny guge symmetry, while they possess guge identities: ν T µν ( m 2 )T µ 0 (5) Unlike spin 1, mking use solely of the irreducible eld h µν, it is impossible to equivlently represent the equtions (4) in the Lgrngin form. Introducing uxiliry sclr eld tht might be interpreted s trce of h µν, these eqution cn be mde Lgrngin.
The problem of consistent inclusion of interctions. Given: Free/liner EoM's T (0) (x)=0, or qudrtic ction S (0) (x). Find: T (x)=t (0) (x)+t int(x) or S(x)=S (0)(x)+S int (x) such tht the number of degrees of freedom does not chnge. Solution by mens of Noether procedure: The ide is to deform ction nd guge symmetry order by order S=S (0) + gs (1) + g 2 S (2) + ; R=R (0) + gr (1) + g 2 R (2) + ; (6) R (0) i i S (1) + R (1) i i S (0) =0 ; (7) R (1) i i S (1) + R (2) i i S (0) + R (0) i i S (2) =0 (8) The procedure controls the mere fct tht number of guge symmetries does not chnge. This is insucient to ensure consistency. As we will demonstrte, it is even unnecessry.
Involutive closure of guge dynmics: denitions Denitions nd terminology. The order of eq. is the mximl order of derivtives involved; The order of system is the mximl order of the eqs involved; A system of order n is sid involutive if ny dierentil consequence of the order less thn or equl to n is lredy contined in the system. Any regulr system cn be brought to involution by inclusion of the lower order dierentil consequences. Then, it is sid to be the involutive closure of the originl system. The mximl order of derivtive of guge prmeter ɛ is sid the order of guge symmetry genertor R i ; The order of guge identity is sum mximl order of the identity genertor L A nd the order of the eq. T it cts on.
Involutive closure nd implicit guge identities. Remrk 1. If the system is not involutive, it is equivlent to its involutive closure. The involutive closure hs the sme guge symmetry, while it my hve extr implicit guge identities. Remrk 2. The involutive closure of Lgrngin system is not necessrily Lgrngin. Exmple of involutive closure nd implicit identity: Proc. T µ (η µν µ ν m 2 η µν )A ν =0, ord(t µ )=2. (9) Involutive closure is got by inclusion of the rst order consequence: T µ A µ =0, ord(t )=1. (10) The involutive closure hs the third order guge identity: L T 0, =(µ, ), L=( µ,m 2 ), ord(l)=3 (11)
Involutive closure nd covrint degree of freedom count The number of physicl degrees of freedom N is understood s the number of independent Cuchy dt modul guge trnsformtions. Given the involutive system with guge symmetries nd identities, N reds: N = k(t k l k r k ). (12) k=0 t k is number of equtions of order k; l k r k is the number of guge identities of k-th order; is the number of guge symmetries of kth order Exmple - Proc: t 2 =4, t 1 =1, l 3 =1, hence N =2 4+1 1 3 1=6, tht corresponds to 3 polriztions of mssive spin 1 in d=4.
Involution nd perturbtive inclusion of consistent interctions Given the free involutive guge system, T (0) =0, L (0) A T (0) 0, R (0)i it (0) 0, (13) perturbtive inclusion of interction is deformtion of the equtions, identities nd guge symmetries by nonliner terms, T (0) T = T (0) + gt (1) + g 2 T (2) +..., (14) R (0)i L (0) A Ri =R(0)i L A =L(0) A + gr(1)i + gl (1) A + g 2 R (2)i +..., (15) + g 2 L (2) A +.... (16) Here g is coupling constnt, genertors L (1) A nd R (1)i in elds; T (1), L (2), nd R(2)i re bi-liner, etc. A Notice tht in ech order of the deformtion, the orders of equtions, identities nd symmetries cn never decrese. re liner
Involution nd perturbtive inclusion of interctions The perturbtive consistency implies tht deformed EoM's posses deformed guge symmetries nd identities in every order in g : The expnsion in g reds: R (0)i it (2) R (0)i L A T 0, R i it =U T (17) it (1) =U (1)b T (0) b R(1)i it (0), L (0) A T (1) +L(1) A T (0) =0. (18) +R(1)i it (1) +R(2)i it (0) =U (1)b T (1) b +U(2)b T (0) b, L (0) A T (2) +L(1) A T (1) +L(2) A T (0) =0, (19) The reltions (18), (19) impose restrictions on interction even if there is no guge symmetry. Resolving the reltions bove order by order one constructs ll the consistent interctions. If ny obstruction rise in some order, it is no-go theorem.
Summry of the procedure for perturbtive inclusion of interctions 1 The free system is brought to the involutive form. 2 All the guge symmetries nd identities re identied. 3 The interction vertices re itertively included to comply with three bsic requirements in every order of coupling constnt: The eld equtions hve to remin involutive; The guge lgebr of the involutive system cn be deformed, though the number of guge symmetry nd guge identity genertors remins the sme s it hs been in the free theory; The number of physicl degrees of freedom, being dened by n k,l k,r k, cnnot chnge, while ll the these numbers cn. This procedure ensures nding ll the consistent interction vertices, for ny regulr system of free eld equtions.
An exmple of by-pssing the perturbtive no-go theorem for consistent interctions. Consider the following ction in 2d Minkowski spce: ( S[φ,A]= d 2 xφ µ A µ + g A µa µ). (20) 2 The eld equtions red µ A µ + g A µa µ =0, Dµ φ=0, (21) 2 where D ± µ = µ±ga µ, nd ɛ µν D µ D ν =gɛµν µ A ν gf. Unless F 0, it is topologicl theory, s there is consequence φ=0, while the two components of A µ re subject to single eqution, so they re pure guge. In the free limit g 0, φ is still xed, while A µ should be pure guge for the sme reson s with g 0. However, the Noether procedure leds to the no-go theorem.
An exmple of by-pssing the perturbtive no-go theorem for consistent interctions. Free Lgrngin L=φ µ A µ hs n irreducible guge symmetry: δ ϱ φ=0, δ ϱ A µ =ɛ µν ν ϱ, (22) tht guges out A µ, while φ=0 shell.the free model is topologicl. The cubic vertex φa 2 is not invrint w.r.t. (22) even modulo totl divergence nd the free equtions, δ ϱ d 2 xφa 2 = 2 d 2 xφf ϱ 0. (23) It is stndrd no-go theorem for the cubic interction. The interction is consistent, however, in the sense tht it does not chnge the degree of freedom number. The explntion is tht the intercting theory hs the reducible guge symmetry with smooth limit tht diers from (22)
Multiple choice of guge symmetries nd consistency of interctions. Lgrngin L=φ( µ A µ + g 2 A2 ) enjoys guge symmetry δ ε φ=0, δ ε A µ =gε µ ɛ µν D + ν(f 1 D + λ ελ ), (24) where ε µ is guge prmeter. The guge-for-guge trnsform reds δ κ ε µ =ɛ µν D + ν κ, (25) The free limit of guge trnsformtions (24), (25) reds δ ε φ=0, δ ε A µ = ɛ µν ν (F 1 λ ε λ ), δ κ ε λ =ɛ λν ν κ. (26) These trnsformtions reproduce the irreducible free trnsformtion δ ϱ A µ =ɛ µν ν ϱ with ϱ= F 1 λ ε λ. At the free level the reducible nd irreducible trnsformtions re equivlent, s ech of them spns the on-shell kernel of the d 2 S. The reducible symmetry is comptible with interction, while the irreducible one is not.
One more exmple nd conclusions. Topologicl grvity in d>2? Consider the ction nd equtions S[φ,g]= φr gd 4 x, (27) R=0, ( µ ν g µν R µν )φ=0. (28) Linerizing over the bckground g µν =η µν one cn nd the guge symmetry tht guges out ll the degrees of freedom of g, while φ is not dynmicl. The eld φ remins non-dynmicl in generl, so d(d+1) 2 components of g µν re subject to single sclr eqution R=0, tht mens there re no physicl degrees of freedom. The linerized symmetry gin does not dmit deformtion, while the complete theory remins topologicl. It is one more less trivil exmple of bypssing the no-go theorem.
Summry of the proposed procedure for inclusion of interctions. The free system is brought to the involutive form; The generting set is chosen (the choice isn't unique) for guge symmetries nd identities of the involutive system; The deformtions re iterted for EoM's, identities nd symmetries in consistent wy with the DoF count reltion. If one generting set of symmetries nd identities obstructs interction, nother resolution cn by-pss the obstruction. Advntges in comprison with Noether procedure It controls DoF number, not just guge symmetry. All the vertices re identied once they comply with the DoF number; It pplies to Lgrngin nd non-lgrngin systems; It llows one to by-pss the no-go theorems for certin generting set by switching to nother generting set.
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