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Interpretation of wave function for free particle is suggested as a description of microscopic distortion of the space-time geometry, namely, as some closed topological 4-manifold. Such geometrical object looks in three-dimensional Euclidean space as its topological defect having stochastic and wave-corpuscular properties of quantum particle. All possible deformations (homeomorphisms) of closed topological manifold play the role of “hidden variables”, responsible for statistical character of the theory.

This investigation started long ago as an attempt to extend Einstein’s idea of the geometrization of the theory of gravity to a possible geometrization of quantum theory. In the Einstein’s general theory of relativity, gravitation is considered as a result of macroscopic distortion of the space-time geometry [

Let’s consider the free neutral particle with mass m and spin 0. It will be shown that wave function of such particle can be interpreted as a mathematical description of some geometrical object. This scalar wave function is the solution of the Klein-Fock-Gordon equation, and it has the form [

Ψ = const ⋅ exp ( − i ℏ ( E t − p r ) ) . (1)

This function describes within existing interpretation the particle’s state with definite energy E and definite momentum p. The particle’s position before measurements is unknown—it may be observed in any point with equal probability. This fact reflects statistical character of quantum mechanics—unusual property within classical representations. Another unusual property—wave-corpuscular dualism of quantum particles that is defined by phase of the wave function and by wave length and frequency, connecting with the particle’s energy and momentum by known relations [

λ i = ℏ p i , ω = E ℏ , i = x , y , z . (2)

Substituting (2) in (1), we have

Ψ = const ⋅ exp ( − i ω t + i k r ) , k i = 2 π λ i (3)

This type of functions (plane wave) is often used in classical physics (for example, for description of plane running sound wave). Within existing interpretation of quantum mechanics the origin of periodical dependence of wave function is not discussing.

Let us rewrite the function (1) not with space coordinates x, y, z and, separately with time coordinate t, but with only space coordinates x^{1}, x^{2},x^{3},x^{4} of the specific space—the space of events of the special theory of relativity—four dimensional pseudo Euclidean space of index 1 (the Minkowski space [

Ψ = const ⋅ exp ( − i x μ p μ ) . (4)

Here and later relativistic units are used where ℏ = c = 1 . Summation over repeating indexes is suggested in (4) with signature ( + − − − ). In relativistic case [

p 1 2 − p 2 2 − p 3 2 − p 4 2 = m 2 (5)

where m—the particle’s mass. Let’s write down (4) in such a way that it contains only values with dimensionality of length

Ψ = const ⋅ exp ( − 2 π i x μ λ μ − 1 ) , (6)

where

λ 1 − 2 − λ 2 − 2 − λ 3 − 2 − λ 4 − 2 = λ m − 2 , λ μ = 2 π p μ − 1 , λ m = 2 π m − 1 . (7)

In contrast to (1, 3) function (6) does not look as a plane wave—it represents periodical function of four space coordinates in the Minkowsri space.

Function (6) may be considered as a function realizing representation of the group whose elements are discrete translations along four coordinates axes in the Minkowski space. Indeed, function (6) goes into itself at translations

x μ → x μ ′ + n μ λ μ , (8)

where n μ —integers (μ = 1, 2, 3, 4). This group is isomorphic to the group ℤ 4 , whose elements are products of integers n μ In turn, the group ℤ 4 is isomorphic to the fundamental group of closed 4-manifold that is homeomorphic to the foir dimensional torus T^{4} [

Representation of particle as a closed manifold means that this particle before measurement may be considered as a “mixture” of its all possible geometrical representations (homeomorphisms), and only interaction with device fixes one of them. This means that wave function describes not an individual particle, but statistical ensemble of all its possible geometrical representations, and this explains statistical character of quantum mechanics. Thus, ensemble of all possible homeomorphisms plays the role of “hidden variables,” responsible for stochastic behavior of particles.

Let’s proceed to decoding of the representation of quantum particle as a closed 4-manifold, that is let’s show how such object looks from the point of view of the observer in Euclidean space. But the important notice should be made before going to this problem. The geometry of four dimensional closed manifolds is now under development: the full recognition algorithm is not now known even for three dimensional closed manifolds [^{1}—a circle. Such torus may be considered in three dimensional Euclidean space as a surface obtained by rotation of a circle around vertical axis lying in the plain of this circle (

The particle’s positions in Euclidean (one dimensional) space are defined by pounts of its intersection with the circle, corresponding to the only one of the torus possible homeomorphisms. Accounting for all possible homeomorphisms leads, obviously, to “blurring” of this circle and so leads to transformation of the one intersection point in finite region of Euclidean space (this region is indicated at

ensemble of its possible positions, and this explains statistical character of quantum mechanics. It is obvious that all possible homeomorphisms of the closed manifold, representing this particle, play the role of “hidden variables”, responsible for the particle’s stochastic behavior: each homeomorphism corresponds to the one particle’s possible position in Euclidean space. The points of the intersection region have different velocities. This means that the intersection region at

The fact that the particle can be represented in physical Euclidean space as a part of topological defect allows to explain the particle’s wave properties. It is sufficient for this to suppose that the defect’s position in the external five dimensional Euclidean space relative to the three dimensional space changes according to periodical low described by wave function (1) (a rigorous proof of this assumption is not possible within the framework of low dimensional analogy). It can be said that the phase of the defect’s periodical movement is an additional degree of freedom on which the effect of the particle on the device depends. The particle’s corpuscular properties (4-momentum) are defined through parameters of above periodical movement of defect by relations

p μ = 2 π λ − 1 . (9)

These relations are identical to the definition (2) of the particle’s wave length through its momentum within existing interpretation [

The wave function plays a dual role within suggested interpretation. First, it is a function, realizing the representation of the fundamental group for a closed 4-manifold, representing a free particle. Second, this function describes periodical movement of topological defect in the external space, and intersection of this defect with physical space defines the possible particle’s positions. These properties of the wave function make it possible to explain the stochastic behavior of the particle and its wave-corpuscular dualism. The role of “hidden variables”, responsible for the particle’s stochastic behavior, is played by all possible homeomorphisms of the closed 4-manifold, representing the particle. Notice in conclusion that relation (7) defines geometrical interpretation of the particle’s mass as a characteristic of some fundamental length λ m . Geometrical interpretation of elementary electrical charge and the particle’s spin and possibility of application of geometrical approach to the quantum field theory is now under consideration. After that, the advantages of the proposed approach will become clear.

The author declares no conflicts of interest regarding the publication of this paper.

Olkhov, O.A. (2021) Possibility of Geometrical Interpretation of Quantum Mechanics and Geometrical Meaning of “Hidden Variables”. Journal of Modern Physics, 12, 353-360. https://doi.org/10.4236/jmp.2021.123025