Supersymmetry and Deterministic Chaos
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Supersymmetry and Deterministic Chaos Stam Nicolis*, ** Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRS, Parc Grandmont, Tours, 37200 France *e-mail: [email protected] **e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—We show that the fluctuations of the periodic orbits of deterministically chaotic systems can be captured by supersymmetry, in the sense that they are repackaged in the contribution of the absolute value of the determinant of the noise fields, defined by the equations of motion. DOI: 10.1134/S1547477120050295
1. INTRODUCTION It is possible to interpret a set of first order ordinary differential equations, in the variables xI (t ) , dxI (1) = FI ( x), dt where I = 1, 2, … , d as describing the motion of a particle, as it probes d -dimensional space, where xI (t ) is its trajectory. The reason is that these equations can be understood as describing the minimization of the functional d
1 2 ( x
S = dt
I =1
− FI ( x)) , 2
I
(2)
where S is the Euclidian action. Their solutions saturate the bound S ≥ 0 . If FI can be expressed as the gradient of a scalar function, i.e. FI = −∂ IW , then, upon expanding the integrand, we realize that it takes the form d
d
2
d
(3) + = 1 [ xI ]2 + 1 ∂W + xI ∂W . 2 I =1 2 I =1 ∂xI ∂xI I =1 The middle term can be identified with the scalar potential, V ( x ) , while the last term is a total derivative. The canonical partition function is well-defined, upon imposing the periodic boundary conditions, that eliminate the total derivative, if it is true that V ( x) confines at infinity (that it’s bounded from below is obvious) and the fluctuations can be consistently described by the property that the “noise fields”,
(4) ηI ≡ xI − FI are Gaussian fields, with ultra-local 2-point function, ηI (t )ηJ (t ') ∝ δIJ δ(t − t ') [1–3]. Now we may ask the question of how to interpret the case, when FI has non–zero curl and, therefore,
cannot be expressed as −∂ IW (or, when it can, W doesn’t have continuous first derivatives). The answer is known: FI can be interpreted as a vector potential. More precisely, it’s known, since the work of Helmholtz, Clebsch and Monge, in the 19th century, that any vector field—under assumptions of smoothness, that ensure the existence of derivatives, of course can be written as the sum of a curl-free part (that describes the effects of a scalar potential) and of a divergence–free part (that describes the effects of a vector potential) [4]. In [5] numerical evidence has been presented, that, when FI can be written as −∂ IW , then the noise fields, in that case, do, indeed, satisfy the identities that imply that the noise fields are Gaussian, with ultralocal 2-point function. In [6] evidence was presented, that these identities also hold for target space, not only worldline, supersymmetry, when the superpotential can be globally defined. So the question t
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