Dear Vasil,

I tend to believe that is in quantum mechanics.

The following publication fully covers the topic of concern:

On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature

Article in Physica D Nonlinear Phenomena 34(1-2):42-89 · January 1989

DOI: 10.1016/0167-2789(89)90228-5

Abstract

We consider geodesic motion on three-dimensional Riemannian manifolds of constant negative curvature, topologically equivalent to S × ]0, 1[, S a compact surface of genus two. To those trajectories which are bounded and recurrent in both directions of the time evolution t → + ∞, t → − ∞ a fractal limit set is associated whose Hausdorff dimension is intimately connected with the quantum mechanical energy ground state, determined by the Schrödinger operator on the manifold.We give a rather detailed and pictorial description of the hyperbolic spaces we have in mind, discuss various aspects of classical and quantum mechanical motion on them as far as they are needed to establish the connection between energy ground state and Hausdorff dimension and give finally some examples of ground state calculations in terms of Hausdorff dimensions of limit sets of classical trajectories

https://www.researchgate.net/publication/222437724_On_the_calculation_of_quantum_mechanical_ground_states_from_classical_geodesic_motion_on_certain_spaces_of_constant_negative_curvature

Hoping this will be helpful,

Rafik

Article On the calculation of quantum mechanical ground states from ...

It seems so, but I guess not much work has been done on this. Feynman had pointed this out as well.

A quotation by Feynman would be in favor of me. Might you cite Feynman's paper exactly?

@ Vasil, see attached file, pages from chapter 7 of Feyman's book  "Quantum Mechanics and Path Integrals"

In the terms used by you:

1) Does any space supplied by Hausdorff topology (and thus metrizable) admit to be complemented by a symplectic structure?

2) Can a symplectic structure to be introduced in a manifold, the topology of which is not Hausdorff? 

(The direction is further the following: from symplectic geometry to Riemannian geometry)

Thank you, Vikash. Indeed, "the paths are nondifferentiable" in Feynman's text and his argument for that is very interesting since it is anyway metrical though negative. I will think of that. 

1) It seems to me that the Schroedinger equation is an example for that generalization, which allows of the definition of differenciability on a discrete topology. The discrete topology is substituted by its mapping into a separable complex Hilbert space and that mapping being already differentiable can be defined as a differentiable structure on the initial discrete topology.

Furthermore, I hesitate whether nonstandard analysis does not generate a direct general method for constructing differentiable structures on discrete topologies.

2) I do not know how a cotangent bundle might be defined for a manifold which is not smooth. If the manifold is anyway smooth, this implies its Hausdorff topology according to me.   

Vasil,

About eighty years ago Garret Birkhoff and John von Neumann discovered a structure that is suitable as foundation of quantum physics. They called it "quantum logic". It immediately leads to a separable Hilbert space. Every infinite dimensional separable Hilbert space has a non-separable companion that supports continuums as eigenspaces of a category of its operators. It is possible to define a Hausdorff topology inside these continuums. You can use the reverse bra-ket method in order to relate the eigenspaces with equations of motion.

http://arxiv.org/abs/1101.5690

http://vixra.org/abs/1511.0266

That the paths of Feynman be differentiable or not is really the least we could care. The paths integral is nothing more than a reformulation of the wave equation, which is a partial differential equation, and thus already requires a differentiable structure. There have been nothing new in that matter since de Broglie and Schrödinger.

While the statement that a topological space is a Hausdorff space is unambiguous, e.g. http://www.maths.qmul.ac.uk/~bill/topchapter3.pdf , absent a statement that defines what ``motion'' is supposed to mean, the question is meaningless. 

The answers don't take this into account and are, correspondingly, off-topic, with regard to the question. 

In any case, one shouldn't confuse phase space and spacetime-which is a quite common mistake. 

The path integral is in fact a summation of the contributions of the hops in a stochastic hopping path of which the landing locations form a coherent swarm that can be described by a continuous location density distribution. This function corresponds to the squared modulus of the wave function of the hopping point-like object. The location density distribution has a Fourier transform and as a consequence the swarm owns a displacement generator. It means that at first approximation the swarm can be considered as to move as one unit. It travels along its own path. What happens can be described by a sequence of up and down jumps between configuration space and Fourier space, where the displacement in configuration space is replaced by a multiplication in Fourier space. The multiplication terms are all very close to unity and in this way the multiplication also can be approximated by a summation.

A careful notation of what happens explains the otherwise incomprehensible path "integral". See the corresponding section about the path integral  in the following link:

http://vixra.org/abs/1603.0021

The graphic in the Feynman paper is a bit misleading. The hopping locations form a swarm which moves rather slowly when this is compared with the hopping steps. As a consequence the swarm is constituted of a huge number of hop landing points.The swarm has well defined statistical characteristics and the location density distribution is a very smooth function that does not show a sign of a singularity. If the distribution is approaching a Gaussian density distribution then the smoothed location density description has the form of an ERF(r)/r function. This function can then be considered as part of the local gravitation potential. At some distance r it behaves as 1/r. Due to its spatial spread this potential has at short ranges a much stronger binding effect than the potential of a single point-like object. This makes gravitation much more effective than is expected at first sight.

Indeed, I do not know what mechanical motion is. In fact, it is a term utilized in a series of physical theories. Even more, it seems to me that all those theories do not define 'mechanical motion', too. Still even more, I am almost convinced that they used the term ambiguously or incommensurably to each other though maybe consistently within each of them. For example, mechanical motion might be define both in phase space and space-tine eventually differently.    

Indeed the concept of Hausdorff space being mathematical seems to be much more clearly and unambiguously defined. So the meaning of my question is whether and as far the concept of mechanical motion as both implicit common term and set of possibly different notions in a series of physical theories might be or not defined as or by means of 'Hausdorff topology".

Meaning that, all answers are within the implicit topic of the question according to me. I am thankful for them expecting many new ones. :)