Supersymmetry in Classical Mechanics E. Deotto, E. Gozzi and D. Mauro 1 0 0 Dipartimento di Fisica Teorica, Universit`a di Trieste 2 Strada Costiera 11, P.O. Box 586, Trieste, Italy n a and INFN, Sezione di Trieste J 8 1 In 1931Koopmanand vonNeumann [1]proposedanoperatorial formulationof 1 ClassicalMechanics(CM)expandingearlierworkofLiouville. Theirapproachis v basicallythefollowing: givenadynamicalsystemwithaphasespaceMlabelled 4 by coordinates ϕa = (qi,pi); a = 1,...,2n; i = 1,...,n, with Hamiltonian H 2 1 and symplectic matrix ωab, the evolution of a probability density ρ(ϕ) can be 1 given either via the Poisson brackets {, } or via the Liouville operator: 0 1 ∂ρ ={H,ρ}=−Lˆρ; L=ωab∂ H∂ (1) 0 b a ∂t / h b The evolution via the Liouville operator is basically what is called the operato- t - rial approach to CM. The natural question to ask is whether we can associate p to the operatorial formalism of CM a path integral one, like it is done in quan- e h tum mechanics. The answer is yes [2]. In fact we can describe the transition : probability P(ϕa t |ϕa t ) of being in configuration ϕ at time t if we were v (2) 2 (1) 1 (2) 2 Xi at time t1 in configuration ϕ(1) via a functional integral of the form ar P(ϕa(2)t2|ϕa(1)t1)=Z Dϕaδ[ϕa(t2)−ϕacl(t2;ϕ(1)t1)] e = DϕaDλ DcaDc¯ ei Ldt (2) a a Z R e where ϕa(t;ϕ t ) is the solution of the classical equations of motion cl (1) 1 ϕ˙a = ωab∂H and δ is a functional Dirac delta which gives weight one to the ∂ϕb classical paths and zero to the others. In the second line of (2), via some e manipulations [2], we have turned the Dirac δ into a more standard looking weight where e L=λ ϕ˙a+ic¯ c˙a−H; H=λ ωab∂ H +ic¯ ωac∂ ∂ Hcb (3) a a a b a c b The λ ,cae,c¯ are auxiliary varieables weith ca and c¯ of grassmanniancharacter. a a a The geometrical meaning of these variables has been studied in [2] and [3]. 1 It is natural at this point to make contact with the operatorial formalism of eq. (1). It is easy to prove [2] that the first piece of H in (3) is nothing else than the Liouville operator of (1). To understand the meaning of the full H is e important to notice [2] that the ca are nothing else than the basis dϕa of the e forms [4] while the c¯ are the basis of the vector fields. So we can create a a correspondence between forms (tensor fields) and polynomials of c (c¯): 1 1 F(p) = p!Fa1...apdϕa1 ∧···∧dϕap −→ F(p) ≡ p!Fa1...apca1···cap 1 b 1 V(p) = p!Va1...ap∂ϕa1 ∧···∧∂ϕap −→ V(p) ≡ p!Va1...apc¯a1···c¯ap (4) b AllthestandardoperationsoftheCartancalculus[4],liketheexteriorderivative d, the interior contraction ι , the symplectic correspondence between forms v and Hamiltonian vector fields α = (α♯)♭ can be reproduced via the graded- commutators associated to the path integral (2) in the following way: dF(p) →[Q F(p)], ι F(p) →[V,F(p)] BRS V pF(p) →[Qg,F(bp)], V♭ →[K,V]b b α♯ →[K,α], b (df)♯ →[Q ,fb] (5) BRS where b Q ≡icaλ ; Q ≡ic¯ ωabλ BRS a BRS a b 1 1 Q ≡cac¯ ; K ≡ ω cacb; K ≡ ωabc¯ c¯ (6) g a ab a b 2 2 are universally conserved charges under our H. Equipped with this formalism itistheneasytoprove[2]thatthefullHisnothingelsethantheLie-derivative e L = ι d+dι of the Hamiltonian flow and the correspondence is (dH)♯ (dH)♯ (dH)♯ e the following: L F(p) → i[H,F(p)] (7) (dH)♯ Beside the five charges in (6) also N = cea∂bH and N = c¯ ωab∂ H are H a H a b conservedunderHandasaconsequencetheseotherchargesarealsoconserved: eQ ≡Q −N ; Q ≡Q +N (8) H BRS H H BRS H They are two supersymmetry charges. In fact, while the Q and Q anti- BRS BRS commute, Q and Q close on H: H H e [Q ,Q ]=2iH (9) H H If we enlarge the base space, including two gerassmannian partners of time, θ andθ¯,wecanputtogetherallthevariablesthatappearin(3)intothefollowing superfield: Φa = ϕa +θca +θ¯ωabc¯ +iθ¯θωabλ . This superfield allows us to b b 2 connect the two hamiltonians H and H via the relation H =i dθdθ¯H[Φ] and to represent the supersymmetry charges as the following operRators acting on e e the superspace (t,θ,θ¯): Q = − ∂ −θ¯∂ , Q = ∂ +θ ∂ . This is an N=2 H ∂θ ∂t H ∂θ¯ ∂t supersymmetry. In fact bwe could combine tbhe QBRS,QBRS,NH,NH into the following two charges Q ,Q : (1) (2) Q ≡Q −N , Q ≡Q +N (1) BRS H (2) BRS H [Q ,Q ]=0 (10) (1) (2) and prove that Q2 =Q2 =−iH (11) (1) (2) As the Q of (6) is basically the exterior derievative on phase space, it would BRS benicetounderstandthegeometricalmeaningalsoofthesusychargeslikeQ (1) or Q . This was done in ref. [5]. The strategy used there was of making local (2) the susy associated to Q . The Lagrangianwith this local invariance is (1) L :=L+α(t)Q +g(t)H (12) EQ (1) where α(t) and g(t) are geauge fieelds. The physical-setate conditions associated to this gauge invariance turns out to be: H|physi=0 Q |physi=0 e(1) Π |physi=0 g Π |physi=0 (13) α where Π and Π are the momenta associated to the gauge variables α,g. Us- g α ing the correspondence (4)-(7) it is easy to translate (13) into a differential- geometric languageand provethat the states selected by (13)are in one-to-one correspondencewith the states of the so-calledequivariantcohomology[6] with respect to the Hamiltonian vector field. The equivariant cohomology w.r.t. a vector field V is defined as the set of forms |ρi which satisfy the following con- ditions: (d−ι )|ρi=0 V L |ρi=0 V |ρi=6 (d−ι )|χi V L |χi=0 (14) V This is the geometrical light we could throw on the susy charge Q . Our (1) universal symmetries, besides having a nice geometrical interpretation, should also have a dynamical meaning. This is the case for the susy invariance which seems to have some interplay with the concept ofergodicity [5]-[7]. We will not expand on it here but turn to another aspect of this susy. 3 Supersymmetry has found its most important applications in field theory where it has produced theories which have a better ultraviolet behaviour than non supersymmetric ones. With that in mind in ref. [9] an attempt was made to build the analog of H and L of eq. (3) also for field theory. Starting for example from the Hamiltonian of a ϕ4 theory H = d3x{1Π2 + 1(∂ ϕ)2+ e e ϕ4 2 ϕ 2 k 1m2ϕ2+ 1gϕ4} the associated L is R 2 4! e L = {Λ (ξ˙a−ωabδ H )+iΓ (∂ δa−ωacδ δ H )Γb}d3x (15) ϕ4 a b ϕ4 a t b c b ϕ4 Z e where ξa are made of (ϕ,Π ) and Λ (~x,t),Γa(~x,t),Γ (~x,t) are the fields anal- ϕ a a ogous to the point-particle variables λ ,ca,c¯ but, differently from these ones, a a they donotdependonlyont butalsoon~x. Likefor the pointparticle,L has ϕ4 an N=2 susy whose charges are: e Q = d3x(iΓaΛ −Γaδ H ) H Z a a ϕ4 Q = d3x(iΓ ωabΛ +Γ ωabδ H ) (16) H Z a b a b ϕ4 The same construction can be done for any field theory and even for gauge theories. For example the standard BFV Hamiltonian for Yang-Mills theories is [8]: 1 1 H = d3x πkπa+ FijFa +π ∂kAa−λa∂ πk+λaCb πkAc BFV Z (cid:26)2 a k 4 a ij a k k a ac b k +iP Pa−λaP Cb Cc−iC ∂k(∂ Ca+Ca AcCb) (17) a b ac a k bc k (cid:27) where Ak are the gauge fields, πa are their conjugate momenta, Fa is the a k ij antysimmetric tensor, λa is a Lagrange multiplier and π its conjugate mo- a mentum, (−iPa,Ca) are the BFV ghosts of the theory and (iC ,P ) the BFV a a anti-ghosts. If we indicate with ξA all the fields of the theory above, including the BFV ghosts, we have that the associated H is [9]: e → ← HBFV =Z d3x{ΛAωAB~δBHBFV(ξ)+iΓ¯AωAC δC HBFV(ξ) δB ΓB} (18) e where Λ ,ΓA,Γ are auxiliary fields. Λ has the same grassmannian parity as A A the field ξ to which it refers, while Γ and Γ have opposite grassmannianparity. It is easy to show that there are conserved BRS and anti-BRS charges of the form: Q = i d3xΓAΛ and Q = −i d3xΛ ωABΓ . The Hamiltonian of eq. A A B (18) can be wRritten as a pure BRS vaRriation in the following way: H =−i[Q,[Q,H ]] (19) BFV BFV This is not a surprise becauese it is a property of any Lie-derivative. It makes this H strongly similar to the hamiltonians of Topological Field Theories BFV e 4 [8]. The susy charges Q and Q have the same form as the ones of the ϕ4 H H theory, except for the presence of some grading factors. They are: → QH =Z d3x(iΓAΛA−(−)[ξA]ΓA δA HBFV) → QH =Z d3x(−iΛAωABΓB+ΓAωAB δB HBFV) (20) where we indicate with [ξA] the grassmannian parity of the field ξA. Their an- ticommutator produces the Hamiltonian (18): [Q ,Q ] = 2i d3xH . The H H BFV shortcoming of all this is that we have obtained a non-relatRivistic susy even e froma relativistic field theory. We feel anyhowthat it shouldbe possible to get a relativistic one. The strategy should be to start not from the Hamiltonian formalismbutfromanexplicitly Lorentzcovariantonelikethe DeDonder-Weyl approach[10]. TheHamiltonianformalismgivesaspecialroletotimeandspoils the manifest Lorentz covariance. This special role of time is what produces a non-relativistic susy in our formalism. If we succeed in getting a relativistic susy with our mechanism we can say that somehow susy is everywhere, even associated to a non-susy theory like a ϕ4-theory. We haven’t seen this susy beforebecausewehaven’tconsideredallthe othergeometricalfields(formsand vector fields) which are naturally associated with the basic fields ϕ. The susy appeared only when we did things in a coordinate indipendent fashion as the Lie-derivative does. BIBLIOGRAPHY [1] B.O. Koopman Proc.Nat.Acad.Sci. USA 17 (1931) 315; J.von Neumann Ann.Math. 33 (1932) 587; ibid. 33 (1932) 789 [2] E. Gozzi, M. Reuter and W.D. Thacker Phys.Rev.D 40 (1989) 3363 [3] E. Gozzi and M. Regini Phys.Rev.D 62 (2000) 067702[hep-th/9903136]; E. Gozzi and D. Mauro Jour.Math.Phys 41 (2000) 1916 [hep-th/9907065] [4] R. Abraham and J. Marsden ”Foundations of Mechanics”, Benjamin 1978 [5] E. Deotto and E. Gozzi hep-th/0012177 [6] H. Cartan ”Colloque de Topologie” (Espace Fibres), CBRM 15.71 1950 [7] E. Gozzi and M. Reuter Phys.Lett. 233B (1989) 383; Chaos, Solitons and Fractals2 (1992)441; V.I. Arnoldand A. Avez ”Ergodic Problems of Classical Mechanics, W.A. Benjamin Inc. 1968 [8] M. Henneaux Phys. Rep. 126 (1985) 1 E. Gozzi e M. Reuter Phys. Lett. B 240 (1990) 137 [9] P. Carta Master Thesis, Cagliari University 1994; D. Mauro Master Thesis, Trieste University 1999 [10] H.A. Kastrup Phys. Rep. 101 (1983) 1 5