The commutation condition [a, ,a,]x' = 0 for a diffeomorphism from xu to xi coordinates is replaced by the weaker condition x:[a, ,aV]xi = 0, which still ensures that conservation laws are covariant statements. This yields a new group, which contains the diffeomorphisms as a proper subgroup. The group is defined on a space in which paths (rather than points) are the primary elements. It determines a new geometry on this path space (just as the group of diffeomorphisms determines Riemannian geometry on a manifold). The path-space geometry is quantized via the path-integral method. In the macroscopic limit, this yields field equations which describe gravitation and electromagnetism, and which also contain terms that appear suitable for describing the weak and strong interactions.
If the relativity principle, which states that the law of propagation for light has the same form for all macroscopic observers, is extended to include quantum observers, this leads directly to the quantum unified field theory which was introduced in a previous paper. This theory appears suitable for describing all known interactions. Gravitation and electromagnetism are described by the Einstein equations G,,= + ( e , , -~, j , -~, j , ) -RK,K,, where G,, is the Einstein tensor, R is the Ricci scalar, e,, is the usual stress-energy tensor for the free electromagnetic field, and j, is the electromagnetic current. The vector K p plays a dual role. It is the electromagnetic vector potential in the covariant Lorentz gauge, and, it is also a unit timelike vector interpretable as the velocity of the observer.
The covariance group for general relativity, the diffeomorphisms, is replaced by a group of coordinate transformations which contains the diffeomorphisms as a proper subgroup. The larger group is defined by the assumption that all observers will agree whether any given quantity is conserved. Alternatively, and equivalently, it is defined by the assumption that all observers will agree that the general relativistic wave equation describes the propagation of light. Thus, the group replacement is analogous to the replacement of the Lorentz group by the diffeomorphisms that led Einstein from special relativity to general relativity, and is also consistent with the assumption of constant light velocity that led him to special relativity. The enlarged covariance group leads to a non-commutative geometry based not on a manifold, but on a nonlocal space in which paths, rather than points, are the most primitive invariant entities. This yields a theory which unifies the gravitational and electroweak interactions. The theory contains no adjustable parameters, such as those that are chosen arbitrarily in the standard model.
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