IB: You seem to jump to many different items. There's no crisis.

IB: You seem to jump to many different items. There's no crisis.

Speaking of jumping. Quantum mechanics arises in this model precisely because of the jumps of the ring on the surface of a closed manifold.

When we consider the Markov process of random walk on the surface of a torus stretched over a sphere with punctured poles, we obtain the generalized Schrödinger equation

i∂ψ(x,t)/∂t=-e-tβ∂2ψ(x,t)/4π∂x2 (page 27)

which in the case of β=0 becomes the usual Schrödinger equation.

Suggest you do normal QM in 3D plus time, ie. just particle in torus, before you try to ask things like this.

Juan Weisz If I understand you correctly, you suggest considering a 3-dimensional torus as a model space of quantum mechanics. However, in my opinion, do not rush, you should first deal with the one-dimensional model of quantum mechanics.

Well the problem can be anything. 1 2 or 3 D. It is just that I was not interpreting your 4 D model.

!D is the easiest. Guess you do a torus by separating longitudinal and transverse modes.

Juan Weisz I consider waves on the torus as probability waves of angular action. You did not look at the source — all the misunderstandings come from here.

If I seriously have to comment on what is said, will say as follows.

First the title. Quantum theory can be adapted to either open or closed systems, so none of the two is favored by QM.

Clearly you can have mass in either case.

Angular velocity cannot be mass, if only from simple dimensional reasons. Similar coments for the rest.

@Juan Weisz The thing is that the angular velocity in the extended mathematical model (which is considered in the above source) is specified in space with additional dimensions (space-time + 4-momentum), therefore dimensional considerations are not an argument in this case. Imagine a space in which each space-time coordinate corresponds to a 4-momentum coordinate and each such pair of coordinates wraps around a corresponding torus. This is where ( S3×S3×T2 ) the particle-ring rotates.

An excellent question! It may be shown (see LP et al, PRA92, 022122, 2015, and references therein) that

1. Quantum field theory is a response theory in disguise, with input and output of the system formulated Kubo- and Glauber-style, respectively, and

2. Information contained in the Heisenberg field operators defined without external Kubo-style sources suffices to describe response properties of the system with respect to the said sources. The simplest example of this mathematical property of QFT is Kubo's renowned formula expressing the linear response of a system in terms of the commutator of operators defined without sources.

Due to these exact mathematical properties of quantum field theory a formulation of a system as a closed one already contains all its response properties, necessary to reformulate it as an open one.

Igor

My use of dimension refers to the physical units you use. 1/sec not the same as Kg.

Juan Weisz L. I. Plimak There really shouldn't be a contradiction between physics and metaphysics. I understand that there are open and closed statistical systems, that the unit of angular velocity and the unit of mass are different things, but you also understand that metaphysics is simply a deeper look at physics. The essence of the proposed metaphysical concept lies in the principle of least action applied to rings in a closed manifold. From the point of view of a physical observer, this metaphysical principle turns into a Feynman formulation of quantum mechanics.

The Feyman formulation does not use rigorous math and also

is full of divergences.

Just what exactly do you propose with least action on closed rings. You must be specific.