An Introduction to
A Unified Field Theory in Six Dimensions
A short history of the problem of unification in physics
Developments leading to the mathematical formulation of electromagnetism by Maxwell were paralleled by a search for a comprehensible underlying mechanism. The wave-like nature of light prompted efforts to devise an ether that could propagate light as an elastic solid propagates waves. In fact, the development of the mathematical theory of elasticity by Navier and Cauchy was stimulated by the transversality of light discovered by Fresnel.
The promise of the ether theories, all but forgotten, is that they would provide a common-sense mechanism for light and other phenomena, be they electromagnetic, gravitational, or later, quantum-mechanical. But, none of the proposed ethers was able to adequately model electromagnetic phenomena. These (three-dimensional) ether theories had three degrees of freedom at each point and could not model the four-component electromagnetic potential or the six components of the electric and magnetic field vectors, let alone the gravitational potential(s). Worse, these ether theories, as a class, appeared to be incompatible with the Michelson-Morley experiment.
Relativity Theory, as introduced by Einstein in 1905, was consistent with the Michelson-Morley experiment and predicted other phenomena, most dramatically, the deflection of light by the Sun. This theory essentially denied any then-conceivable ether and proposed no alternative mechanism for electromagnetism or gravitation. Further, it replaced the traditional notion of time and space with a space-time that was "non-intuitive" if not downright mysterious. While the conclusions of the theory have been exhaustively tested, there are alternatives to the axioms of the theory, as will be seen.
The term "unified field theory", by the way, was coined by Einstein who devoted the latter part of his life to finding it, without success. To him, this meant the unification of Relativity Theory with electromagnetism since he began his quest before quantum mechanics had been developed.
Quantum Mechanics was formulated separately by Schrödinger and Heisenberg in the 1920s. Conceptually, the former had a wave approach, the latter a particle approach. These formulations were shown to be mathematically equivalent by Schrödinger and others. However, due to the dominance of Relativity Theory and its denial of a medium, the particle concepts triumphed and the wave concepts came to be viewed as a sort of clever but meaningless fiction.
Special Relativity has been directly incorporated into Quantum Mechanics: Schrödinger's equation adjusted for relativity yields Dirac's. But General Relativity and gravitation remain separate from Quantum Mechanics, stubbornly resisting all such direct attempts at unification.
String Theory, now M-Theory, was intended to address this problem. Whereas particles are conceptually points, with zero dimensions of extent, strings have one dimension of extent. And where particles moved in the void of space-time, strings move in a space-time of ten dimensions. The extra six dimensions are curled up, or circular; they are also extremely small, on the order of the Planck length. In M-Theory, more dimensions are used but in all cases a lower dimension figure moves through a higher dimensional void.
Mainstream observers rate String Theory as the most promising of competing theories. Nevertheless, it remains speculative. First, it has not been fully formulated; despite much effort it remains an idea for a theory rather than a theory. Second, the energy required to test such a theory is ridiculously large.
An untried approach
Efforts in the latter part of the nineteenth century to develop ether theory considered only three spatial dimensions. The Michelson-Morley experiment effectively killed off all such ether theories.
The first use of dimensions beyond the four of space-time in physical theory was, perhaps, in 1921 by Theodor Kaluza. This was not only after the introduction of Relativity Theory, but an extension to it. The current String Theories, in ten dimensions, and M-Theory in eleven, originated later still.
So the use of higher dimensions did not begin until work on the ether theories had been abandoned. Because of the historical sequence of events, no one, apparently, has so far attempted to formulate a higher dimensional ether theory.
Perhaps the basic objection to spatial dimensions beyond three is the simple observation that our world is obviously three-dimensional. This statement has some truth but is actually too strong. What is clear is that our world is at least three-dimensional. There is no evidence that it could not have more than three dimensions.
The proposed approach to unification
This theory is based on a higher dimensional continuous medium that fills a higher dimensional space of the same dimension. The proposed medium obeys the well-known equations of motion of an elastic solid, suitably extended to a higher dimension. As will be seen, this approach fulfills the promise of providing a common-sense mechanism for phenomena while avoiding the pitfalls that beset the three-dimensional ether theories.
This theory provides a mechanism for phenomena in the sense that the electromagnetic and gravitational potentials and sources are defined as patterns of deformation in the medium. Conventional laws, such as Maxwell's Equations, may be derived from the higher dimensional equations of motion. Forces, which may be expressed as the local interaction between a source and a field, enter this theory as the second order terms in the equations of motion. In short, everything is represented as a pattern or configuration of the medium in this theory. Particles are larger, coherent patterns of motion of the medium just as musical notes or harmonics correspond to modes of oscilation of the medium bounded by a musical instrument. For example, an electron bound to an atom is represented as the solution to a higher dimensional extension to Schrödinger's equation.
One problem with the three-dimensional ether theories was that they did not have enough degrees of freedom to model both the electromagnetic and gravitational potentials, as mentioned above. This can clearly be solved with additional dimensions. The problem of being inconsistent with the oft-repeated Michelson-Morley experiment, it turns out, can also be solved in a higher dimensional setting, as described in Relativity Theory in UFT6.
One other problem of a medium as an elastic solid is how does the Earth move through that medium frictionlessly? The answer is that the Earth and the atoms that comprise it are not objects that must push the medium out of the way as the Earth moves. The atoms of the Earth consist of a nucleus and electrons, both of which are waves in the medium. Not only may these waves occupy the same space as the medium, they require the presence of a medium in order to exist.
In the proposed theory, three additional Euclidean dimensions are postulated beyond the three common ones, for a total of six spatial dimensions. In some contexts, the extra dimensions are called "inner" dimensions. In the study of quasicrystals, the three common dimensions are called "parallel" (to our Universe) and the three added dimensions are called "perpendicular." In conventional geometric terms, of course, all six dimensions of a Euclidean space are mutually perpendicular. At the most fundamental, geometric level, all of these dimensions are equivalent. Therefore, the underlying domain of the proposed theory is six dimensional Euclidean geometry. This is in contrast to Kaluza-Klein Theory and String Theory, where the additional dimensions are circular, not Euclidean.
So why can't we see the extra dimensions, especially if they are equivalent? In String Theory, the answer is that the additional dimensions are not only circular, but their size is far too small for us to see. In the theory proposed here, the reason is quite different. The answer is that while the six dimensions are fundamentally equivalent, there is a physical mechanism that distinguishes them, as described below.
The Universe as a Wave
As postulated above, there is a six-dimensional space filled with a six-dimensional elastic solid medium. Such a medium is known to support compressional and shear waves. It is further postulated that there is a particular shear wave traveling through this medium that is widely extended in three dimensions and narrowly constrained in the other three dimensions. The direction of propagation of this wave is perpendicular to its broad extent. The wave has a torsional, soliton character that is the same everywhere throughout its extent. It is the three-dimensional extent of this wave that we know as our three-dimensional Universe. It is called the Cosmic Wave when it is desired to emphasize to its wave character..
It is this hypothesized physical wave, the Cosmic Wave, that distinguishes the three directions that are its extent from the remaining three directions. So, in the vicinity of the Cosmic Wave, the six directions, or dimensions, are not equivalent. The atoms of which we are composed and the light with which we see can move freely throughout the three-dimensional extent of the cosmic wave, but are confined, within the remaining three dimensions, to the narrowly constrained region of the Cosmic Wave. This is why we can't see or move outside of our three-dimensional Universe.
In contrast to other theories, the Universe is not the entire spatial domain of this theory. In Newtonian theory, Relativity Theory, Quantum Mechanics, and String Theory, the Universe is the entire spatial domain of theory. (In Relativity Theory and its derivatives, the spatial domain is dependent on the state of motion of the observer.) In this theory, the Universe is a phenomenon, a localized wave, in an unbounded domain. While the Universe is extended in three dimensions, it occupies a small part of the remaining three dimensions of the whole six-dimensional domain in the same sense that an atom occupies a small part of the three-dimensional Universe.
This general approach arose in an attempt to develop a mechanism for the big bang. The idea was that our 3-d Universe could be modeled as a 3-d surface of a 4-d sphere expanding in a 4-d space. Such a surface might conceivably be modeled as an object, like the membrane of a higher dimensional bubble expanding in empty space, or as a higher-dimensional wave moving through a medium. The wave representation was chosen because it seemed better able to model electromagnetic and gravitational phenomena and Schrödinger's wave function.
The overall shape of the Universe in the six-dimensional space is unspecified at this time. The Universe might be the 3-d surface of a 4-d sphere, as conjectured above. Alternatively, it might be flat with either limited or unlimited extent. If the extent were limited, the Cosmic wave might taper off at the "edges." If it were unlimited, it is too far away from us to know anything about. In any case, locally, on the scale of the visible Universe, it may be considered to be Euclidean or flat.
The Surf Wave Analogy
There is a useful analogy between an ocean surf wave and the Cosmic Wave. Both waves have a direction of propagation perpendicular to their extents. A single surf wave has a 1-d extent and moves over the 2-d surface of an ocean. The Cosmic Wave has a 3-d extent and moves through a 6-d medium. A surfer, while he is riding a surf wave, is carried forward in the direction of propagation and can move along the extent of the surf wave, left or right from the surfers vantage point. An atom is carried along as the cosmic wave propagates and can move throughout the 3-d extent of the Cosmic Wave, that is, the Universe. This analogy is not exact in the sense that the surfer is an object separate from the water medium whereas the atom is a wave, or a perturbation of the Cosmic Wave.
At any point along the extent of surf wave, the water moves in a circle in the plane including the vertical and the direction of propagation, the plane being perpendicular to the extent of the wave. At any point in the Universe, the medium rotates in a plane perpendicular to the direction of propagation and to the 3-d extent of the Cosmic Wave. This is a torsional wave with angular displacement changing along the direction of propagation.
The Unified Field Theory
Perhaps the most famous unification of physical theory was Maxwell's electromagnetic theory. The electric and magnetic fields were shown to be so intimately interconnected that a full description of one required the mention of the other. Newton is credited with the unification of terrestrial and celestial mechanics. Less recognized is his unification of gravitation and mechanics.
The key idea of a unified theory is that everything is derived from a single set of assumptions. If a separate energy or action were plugged in for electromagnetism and for gravitation, that would not lead to a unified theory. In UFT6, the axioms outlined below are the single starting point for the derivations of all laws. Here, differences between the electric and gravitational forces, such as the fact that like sources repel or attract, respectively, stem from the differences in the definitions of the electric and gravitational sources and potentials.
In common usage, a mathematical field is a function defined throughout some extended domain. In contrast, a particle has a lumped value defined only for the whole. For instance, mass density is a scalar field while the mass of the electron is defined, in the standard model, as a single lumped quantity for the whole electron. (Since the electron is considered to be infinitesimally small, the mass density must be infinitely great, a matter of some concern to "particle" physicists.) UFT6 is a true field theory where all quantities are defined throughout three-dimensional space as fields. Here, a bound electron is defined as a higher dimensional extension of Schrödinger's wave function, extending formally throughout space, while as a practical matter, it is localized in a region whose size is on the order of the Bohr radius. The bound electron still retains a particle nature, but that is not fundamental; that represents a solution to a more general partial differential equation: a higher dimensional extension of Schrödinger's wave equation.
This theory can be presented in an axiomatic manner. Already presented as axioms are the six-dimensional Euclidean domain and the six-dimensional medium. The laws of elasticity theory, extended to six dimensions are assumed to apply. The representation of the Universe as the Cosmic Wave is also an axiom. Another category of axioms is the definition of the fundamental three-dimensional physical quantities in terms of the six-dimensional displacement of the medium. Given these axioms, all the laws of electromagnetism, gravitation, Special and General Relativity and Newtonian and Wave mechanics may be derived as theorems, as will be discussed below.
Quantities: Potentials, Sources, Schrödinger's wave function
The fundamental variable of elasticity theory, and thus of this theory, is called the displacement. The displacement in UFT6 is a function whose range and domain are six-dimensional. The displacement can be thought of as the current position of a small element of the continuous medium with respect to its rest position.
It will simplify the following material to introduce coordinate names here. The three traditional dimensions are given the coordinate names x, y, and z. The direction of propagation of the Cosmic Wave is named w. The remaining two dimensions, which hold the plane of angular rotation of the Cosmic Wave are labeled u and v. The origins of u and v are at the center of the torsional Cosmic wave. These six coordinates are considered to be attached to the medium. Another coordinate, labeled W, is defined parallel to w, but its origin moves with the Cosmic Wave. In summary, the coordinates x, y, and z coincide with the Universe and the coordinates u, v, and W have their origins at the Universe and are perpendicular to the Universe.
The electric and gravitational scalar potentials are defined as displacements along W. In both cases, the displacements vary in a pattern along W. It is the pattern of variation that distinguishes the electromagnetic quantities from the gravitational quantities. The displacement for the electric potential varies as the cosine of (a constant times) W near W = 0. (The amplitude of the cosine falls to zero away from W = 0.) Therefore, the displacement along W for the electric potential is a maximum at W = 0. The displacement for the gravitational potential varies as the sine of (a different constant times) W near W = 0. Therefore, the displacement along W for the gravitational potential is zero at W = 0 and is in opposing directions above and below W = 0. The signs are such that near a massive body, where the gravitational potential is negative, the displacement is positive for positive W and the displacement is negative for negative W, so the medium is extended along W. For the electric potential, since the medium is displaced in the same direction above and below W = 0, the extension of the medium is zero at W = 0.
The cosine and sine patterns in W are strictly valid only near W = 0. The magnitude of the displacement falls to zero as the distance from W increases. The magnitude of displacement also falls to zero as the distance from u = v = 0 increases, as well.
The electric (and gravitational) vector potentials are defined as displacements along x, y, and z. The magnetic and gravitational vector potentials vary with the same cosine and sine pattern in W as the corresponding electric and gravitational scalar potentials.
The electric charge and the mass may be defined as forces along W. These six-dimensional forces should not be confused with normal three-dimensional forces. The electric current and momentum may be defined as forces along x, y, and z. As with the potentials, these six-dimensional forces corresponding to these sources vary in the same cosine and sine patterns along W as the corresponding potentials.
The similarity of the definitions of the electromagnetic quantities and the gravitational/mechanical quantities accounts for the similarity of certain laws, such as the conservation of mass and the conservation of charge The symmetric/antisymmetric or even/odd differences in the definitions can be shown to account for the differences between some laws, such as the force between the scalar sources: like electric charges repel and (like) gravitational masses attract. The reason for these differences stems from the fact that in calculating forces, the lower order terms are larger than the higher order, and with the different symmetries, different terms show up as the lowest order.
Schrödinger's wave function has a domain of all x, y, and z, and a range of two quantities. These quantities are usually expressed as a complex number with real and imaginary parts. In this theory, the two quantities are a rotational displacement of the medium (1) in the uw plane and (2) in the vw plane. As with the electromagnetic and gravitational quantities, these two quantities vary in a pattern along W, but these details are somewhat more complex than those of the other cases.
The electric and magnetic field vectors may be define in terms of the above potentials. Other quantities, such as conductivity, may be defined. But the above are the principal quantities that are related in any unified field theory. The above definitions are oversimplified in that they ignore various constants. Also, the signs of most of the quantities has not been stated.
In summary, it is important to note that all traditional physical quantities are defined in terms of a more fundamental quantity, the displacement of the underlying six-dimensional medium. To an electrical engineer, the electric and magnetic field vectors are palpable, even if undefined. The interpretation of Schrödinger's wave function is still a subject of debate and mystification (as in the many-worlds interpretation) three quarters of a century after its introduction. And concerning Relativity, its curvature of spacetime at a mass is replaced in this theory with a curvature of a medium within Euclidean space, as described by the gravitational potential. While a six-dimensional space may be hard for some to accept, it does have the benefit of providing hard definitions for these hitherto undefined quantities.
The Laws of Maxwell, Newton, Einstein, and Schrödinger
At this point, all of the axioms have been introduced. The laws of elasticity theory suitably extended to six dimensions have been assumed. The existence of the Cosmic Wave has been postulated, although a few parameters, such as its speed and torsion must be supplied. The definitions of the physical quantities have been outlined in brief.
Based on the above assumptions, the basic laws relating to these quantities can now, in principle, be derived as theorems. These include Maxwell's equations, conservation of charge and mass, the force laws for electromagnetism and gravitation, and Newton's second law relating forces to the time rate of change of momentum. Also included are clock retardation of Special and General Relativity. Also, Schrödinger's equation may be derived and the relation between Schrödinger's wave function and mass and charge can be derived. These derivations, of course, are quite tedious and exacting and beyond the scope of this Introduction.
A few words about the force laws are in order. Particle physicists, in an effort to avoid reliance on an action at a distance, would say that forces are carried by particles. In particular the electric force is supposedly carried by the photon. This force can be attractive, repulsive or even a torque for crossed conductors. How a photon particle can "carry" any of these forces and have a wavelength and frequency remains a mystery in the Standard Model.
This theory has an alternative approach to particles carrying forces. The electromagnetic and gravitational forces may be expressed as the local interaction between a source and a potential gradient. As shown above, potentials and sources have definitions in terms of the underlying six-dimensional displacements. This means that the forces are second order terms in the six-dimensional equations of motion of elasticity. These equations equate the elastic forces and the time rate of change of momentum of an infinitesimal element of the medium. That is, they are inherently local. Furthermore, these terms and equations are not some newly minted wonder for this theory. They are well known to mechanical, civil and other engineers.
Quasicrystals are substances that display ten-fold symmetry in crystallographic analysis. The existence of these patterns is evidence of long-range ordering. But this symmetry is falls outside the known 230 space groups and is thus forbidden by conventional crystallography.
There are two ways to describe quasicrystals. One depends upon rules in three-dimensional space that seem to be mathematical rather than physical. The other is based on a cubic lattice in six-dimensional space projected down to three dimensions. This approach is completely consistent with UFT6 and with traditional techniques of analysing crystals. While no experiment can prove a theory true, the simplest physical explanation of this experimental data is that the atoms are arranged in a six-dimensional lattice in six-dimensional space. Stated somewhat more loosely, quasicrystals are physical evidence of the existence of six-dimensional space.
The key features of this Unified Field Theory in Six Dimensions are
• a domain of six spatial dimensions plus a distinct time,
• a six-dimensional medium governed by the conventional equations of elasticity theory extended to six dimensions,
• the Universe represented by a torsional wave (the Cosmic Wave) whose direction of propagation and plane of rotation are perpendicular to its broad 3-d extent; we recognize the 3-d extent as our three dimensional Universe,
• conventional quantities of electromagnetism and gravitation defined as patterns of displacement of the 6-d medium,
• an electron bound in an atom defined as a pattern of displacement of the 6-d medium whose 3-d features coincide with Schrödinger's wave function,
• laws governing the above quantities, such as Maxwell's equations, are derived from the equations governing the 6-d elastic medium.
Quasicrystals are most simply explained as atoms arranged in a 6-d cubic lattice in 6-d space, providing a physical justification for the use of six dimensions.
The strongest justification for any theory is its internal consistency and the consistency between the theory and experiment and observation. Such consistency is claimed for UFT6. Beyond that, a common sense mechanism for the quantities and laws is provided in terms of 6-d elasticity theory, as a model. Since the theory covers both electromagnetism and gravitation, the term unified field theory is justified.