Noether’s theorem in symmetric stochastic calculus of variations

Abstract

Quite recently, a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics, which is an extension of Nelson’s stochastic mechanics. In this article a ‘‘Noether’s theorem’’ is formulated within this calculus of variations. Conservation laws of momentum, angular momentum, and energy are proved, which are related with the same laws in quantum mechanics.

Stochastic variational method as quantization scheme: Field quantization of the complex Klein–Gordon equation

“…To employ the stochastic variation, we have to specify independent degrees of freedom, to each of which independent noises are introduced. In the present case, the Lagrangian density (9) is expressed in terms of the gauge field A µ which has four components, but only two of them are independent due to the gauge invariance.…”

“…We apply the stochastic variational method to the gauge invariant Lagrangian density (9) following Ref. [29].…”

“…For example, the Navier-Stokes equation [4][5][6], the Gross-Pitaevskii equation [5,7], and the Schrödinger-Langevin (Kostin) equation [8] are formulated in SVM approach. Further aspects of SVM have also been studied, such as the Noether's theorem [9], the uncertainty relation [10], the applications to the many-body particle systems [7], and continuum media [4,5]. Although many groups [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] have investigated SVM, the applicability has not yet been well explored.…”

Unified description of classical and quantum behaviours in a variational principle

“…In the SVM quantization scheme, the physical operators are defined through Noether's theorem for the stochastic action [11]. Let us consider the spatial translation by an arbitrary time-independent spatial vector A as r(t) −→ r(t) + A.…”

The SchrÖdinger equation in rotating frames by using the stochastic variational method

“…On the other hand, if the action is invariant for the above rotation, we can show the following quantity is conserved by applying the stochastic Noether theorem [17,18],…”