Calabi—Yau manifolds & the M-theory

This post follows on from the UP hypothesis.

String theory encompasses a group of models, which advocate the concept of elementary particles as vibrating strings. In some instances, space is regarded as a fabric of woven strings, which when torn at some quantum locality, it exposes the background vacuum that represents the mass of a newly formed particle. In other instances, the theory proposes that a particle develops mass as a vibrating string, so that its mass is the exposed background vacuum maintained by its level of oscillation, which reflect its energy level. Therefore, the greater the level of oscillation, the greater the mass. Thus, energy is defined as string vibration. Strings could be open-ended or closed loops.

Despite their apparent differences, all versions of string theory are characterised by their requirement for extra dimensions within the four dimensions of space-time. Although different models of the theory demand different number of dimensions, it has been shown through the mathematical technique of compactification, that all versions of the theory are essentially the same and that their apparent differences are due to adopting different values for some parameters.

The M-theory, which epitomises all versions of string theory, espouses seven extra dimensions within the four dimensions of space-time. However, rather than strings, the M-theory describes ‘branes’— multidimensional manifolds instead of the one-dimensional stings. Those manifolds possess mass and electric charge and have a topology that harbours extra dimensions. Regardless of their description as strings or branes, manifolds on which the theory is modelled, namely the Calabi-Yau manifolds, have six spatial dimensions. Therefore, a good starting point for considering the extra dimensions in string theory is those manifolds.

The general assumption in the theory is that Calabi-Yau manifolds are smooth— i.e., elementary objects. The example often quoted by advocates of the theory is that of a pipe, which when viewed from a distance, it appears as a one-dimensional object— i.e. a line, but when viewed closely it reveals a curved surface encapsulating a three-dimensional space. In effect, a pipe could be considered a five-dimensional manifold— two as the surface and three within the cavity. However, since all dimensions are accessible from space-time, they are reducible to three spatial dimensions. Therefore, extra dimensions are required only where an object is encapsulated by a discontinuity in space-time to relate its geometry and behaviour to the global space-time.

A Calbi-Yau manifold must have a definite singularity, which makes the mass a subspace within space-time. Therefore, one can conclude that the generalised form of the Calbi-Yau manifolds, which is referred to as a conifold, represents the mass and the string element inside it. It should be noted that the electric charge interacts directly with the surrounding space-time. However, since it develops mass with low density UPs, it can be defined as 2D manifold in 3D space.

Since mass is a nonphysical subspace, it could be defined by any number of arbitrary theoretical dimensions. In fact, considering Kaluza-Klein theory, which demands four spatial dimensions to explain electromagnetism and gravity, one would immediately realise that the theory assigns only one spatial dimension to the mass. The reason is, unlike string theory, which models the mass and the string element within it, Kaluza-Klein theory is concerned only with electromagnetism and gravity. Therefore, the theory is only concerned with mass as one extra dimension, what causes and maintains mass is irrelevant to the theory. Therefore, if matter particles are modelled with one extra spatial dimension, one end up with Kaluza-Klien theory.

In fact, this concept represents a physical interpretation of the holographic principle, which is built on the realisation that information is encoded on two-dimensional surfaces from within a space of greater number of dimensions. The surface being that of the mass of a particle as a manifold, while the other dimensions are contained within the particle, namely the mass cavities, the charge and the string element.

To understand the M-theory in the context of the UP hypothesis, let us consider a matter particle in space-time in the coordinate system (X, Y, Z, T), which I shall refer to as the global coordinates system— Fig 1. According to the UP hypothesis, the mass of the particle is a discontinuity in space-time. It therefore constitutes a subspace (a singularity), which requires a different set of coordinates to define it as a manifold. Since we are concerned with defining the entire structure of a particle, we must incorporate the string element, as a dynamic 3D manifold.



As such, we must introduce a set of three spatial dimensions to the mass to locate the string— two to define the surface of the mass and one to define locations in its depth to the string element. Because mass is nonphysical and has no dynamics associated with it, it is a timeless subspace, and we can assign the spatial coordinates (x1, y1, z1) to it. I shall refer to this coordinate system as the local coordinate system of the particle.

To this end, defining the location of a point, say isolated UPs passing through the mass, requires seven dimensions, given by the coordinates (X, Y, Z, T, x1, y1, z1). Since an element of space in the mass constitutes a discontinuity, it can also be regarded as a singularity within the mass and therefore, it requires a different set of coordinates to define its topology. We can therefore assign the coordinate system (x2, y2, z2) to it, which I shall refer to as the quantum coordinate system of the particle.

Clearly, relative to an observer in the global coordinate system— i.e., in space-time, the topology of a matter particle as a manifold has six spatial dimensions, as required by the Calabi—Yau manifolds. These dimensions given by (x1, y1, z1, x2, y2, z2) are in addition to the four space-time dimensions. However, the string element in a particle is dynamic. It therefore requires a time dimensions to track its position in time. However, since time is a function of speed, and since the speed of the string element is in a different frame of reference to the observer, as special relativity confirms, a time dimension must be added to the quantum coordinate system.

Therefore, the total number of dimensions required to define a subatomic particle as a manifold, including those of space time, is eleven, (X, Y, Z, T, x1, y1, z1, x2, y2, z2, t2), as demanded by the M-theory. Note that time (t2) in the quantum realm is different from time (T) in the global coordinate reference, because it relates to different frame of reference, in line with Special Relativity.


Author: PhysicalRealityBlog

I am a structural design engineer with a passion for science and mathematics.

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