Hidden BRS invariance in classical mechanics. II

Hamiltonian superfield formalism with N supercharges

“…As a minimum requirement, any proposals for extended supersymmetry should reduce to the original N = 0 partition function via the above N = 1 path integral (1.11). Previous works [6,7] do not meet this test. For instance, the path integral of Gozzi et al [7] is confined to the classical trajectories.…”

“…Previous works [6,7] do not meet this test. For instance, the path integral of Gozzi et al [7] is confined to the classical trajectories. In their approach, the rigid N = 2 geometry has completely ironed out quantum fluctuations.…”

Supersymmetric Soliton

“…The natural question to ask is whether we can associate to the operatorial formalism of CM a path integral one, like it is done in quantum mechanics. The answer is yes [2]. In fact we can describe the transition probability P (ϕ a (2) t 2 |ϕ a (1) t 1 ) of being in configuration ϕ (2) at time t 2 if we were at time t 1 in configuration ϕ (1) via a functional integral of the form P (ϕ a (2) t 2 |ϕ a (1) t 1 ) = Dϕ a δ[ϕ a (t 2 ) − ϕ a cl (t 2 ; ϕ (1) t 1 )]…”

“…In the second line of (2), via some manipulations [2], we have turned the Dirac δ into a more standard looking weight where…”

“…= Dϕ a Dλ a Dc a Dc a e i Ldt (2) where ϕ a cl (t; ϕ (1) t 1 ) is the solution of the classical equations of motioṅ ϕ a = ω ab ∂H ∂ϕ b and δ is a functional Dirac delta which gives weight one to the classical paths and zero to the others. In the second line of (2), via some manipulations [2], we have turned the Dirac δ into a more standard looking weight where…”