Supersymmetry in Classical Mechanics

5 Pages·0.09 MB·English

by  E. Deotto

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.
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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
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