Description of Vorticity by Grassmann Variables and an Extension to Supersymmetry
R. Jackiw, Typeset Latex, M. Stock, Ρv V (ρ · 2000
Hagen Kleinert’s early interest in particle physics quantum field theory served him well for his subsequent researches on statistical physics and collective phenomena. Therefore, on the occasion of a significant birthday, I offer him this essay, in which particle physics concepts are blended into a field theory for macroscopic phenomena: Fluid mechanics is enhanced by anticommuting Grassmann variables to describe vorticity, while an additional interaction for the Grassmann variables leads to supersymmetric fluid mechanics. 1 Précis of Fluid Mechanics (With No Vorticity) Let me begin with a précis of fluid mechanical equations [1]. An isentropic fluid is described by a matter density field ρ and a velocity field v, which satisfy a continuity equation involving the current j = ρv: ˙ρ + ∇ · (ρv) = 0 (1) and a force equation involving the pressure P: ˙v + v · ∇v = − 1 ∇P. (2) ρ (Over-dot denotes differentiation with respect to time.) For isentropic fields, the pressure P is a function only of the density, and the right side of (2) may also be written as −∇V ′ (ρ), where V ′ (ρ) is the enthalpy, P(ρ) = ρV ′ (ρ) − V (ρ), and √ ρV ′ ′ (ρ) = √ P ′ (ρ) is the sound speed (prime denotes differentiation with respect to argument).2 R. Jackiw Equations (1) and (2) can be obtained by bracketing the dynamical variables ρ and v with the Hamiltonian H(ρ,v)