Behaving Like a Relativistic Quantum Gravity Theory
The Missing Link
In this section, a quaternion 3-string will be defined. By making this quantity dimensionless, I will argue that it my be involved in a relativistic quantum gravity theory, at least one consistent with current experimental tests. At the current time, this is an idea in progress, not a theory, since the equations of motion have not been determined. It is hoped that the work in the previous section on unified fields will provide that someday.
Let us revisit the difference between two quaternions squared, as worked out in the section of analysis. A quaternion has 4 degrees of freedom, so it can be represented by 4 real numbers:
Taking the difference between two quaternions is only a valid operation if they share the same basis. Work with defining the derivative with respect to a quaternion has required that a change in the scalar be equal in magnitude to the sum of changes in the 3-vector (instead of the usual parity with components). These concerns lead to the definition of the difference between two quaternions:
What type of information must e0, e1, e2, and e3 share in order to make subtraction a valid operation? There is only one basis, so the two events that make up the difference must necessarily be expressed in the same basis. If not, then the standard coordinate transformation needs to be done first. A more subtle issue is that the difference must have the same amount of intrinsic curvature for all three spatial basis vectors. If this is not the case, then it would not longer be possible to do a coordinate transformation using the typical methods. There would be a hidden bump in an otherwise smooth transformation! At this point, I do not yet understand the technical link between basis vectors and intrinsic curvature. I will propose the following relationship between basis vectors because its form suggests a link to intrinsic curvature:
If e0 = 1, this is consistent with Hamilton's system for 1, i, j, and k. The dimensions for the spatial part are 1/distance^2, the same as intrinsic curvature. This is a flat space, so -1/e1^2 is something like 1 + k. In effect, I am trying to merge the basis vectors of quaternions with tools from topology. In math, I am free to define things as I choose, and if lucky, it will prove useful later on :-)
Form the square of the difference between two quaternion events as defined above:
The scalar is the Lorentz invariant interval of special relativity if e0 = 1.
Why use a work with a powerful meaning in the current physics lexicon for the vector dt dX? A string transforms differently than a spatial 3-vector, the former flipping signs with time, the latter inert. A string will also transform differently under a Lorentz transformation.
The units for a string are time*distance. For a string between two events that have the same spatial location, dX = 0, so the string dt dX is zero. For a string between two events that are simultaneous, dt = 0 so the string is again of zero length. Only if two events happen at different times in different locations will the string be non-zero. Since a string is not invariant under a Lorentz transformation, the value of a string is
We all appreciate the critical role played by the 3-velocity, which is the ratio of dX by dt. Hopefully we can imagine another role as important for the product of these same two numbers.
Imagine some system that happens to create a periodic pattern of intervals and strings (a series of events that when you took the difference between neighboring events and squared them, the results had a periodic pattern). It could happen :-) One might be able to use a collection of sines and cosines to regenerate the pattern, since sines and cosines can do that sort of work. However, the differences would have to first be made dimensionless, since the infinite series expansion for such transcendental functions would not make sense. The first step is to get all the units to be the same, using c. Let a0 have units of time, and a1, a2, a3 have units of space. Make all components have units of time:
Now the units are time squared. Use a combination of 3 constants to do the work of making this dimensionless.
The units for the product of these three numbers are the reciprocal of time squared. This is the same as the reciprocal of the Planck time squared, and in units of seconds is 5.5x10^85s^-2. The symbols needed to make the difference between two events dimensionless are simple:
As far as the units are concerned, this is relativistic (c) quantum (h) gravity (G). Take this constants to zero or infinity, and the difference of a quaternion blows up or disappears.
Although the units suggest a possible relativistic quantum gravity, it is more important to see that it behaves like one. Since this unicorn of physics has never been seen I will present 4 cases which will show that this equation behaves like that mysterious beast!
Consider a general transformation T that brings the difference between two events dq into dq'. There are four cases for what can happen to the interval and the string between these two events under this general transformation.
This looks simple, but there is no handle on the overall sign of the 4-dimensional quaternion, a smoke signal of O(4). Quantum mechanics is constructed around dealing with phase ambiguity in a rigorous way. This issue of ambiguous phases is true for all four of these cases.
Case 2 involves conserving the Lorentz invariant interval, or special relativity. Strings change under such a transformation, and this can be used as a measure of the amount of change between inertial reference frames.
Case 3 involves conserving the quaternion string, or general relativity. Intervals change under such a transformation, and this can be used as a measure of the amount of change between non-inertial reference frames. All that is required to make this simple but radical proposal consistent with experimental tests of general relativity is the following:
The string, because it is the product of e0 e1, e0 e2, and e0 e3, will not be changed by this. The phase of the string may change here, since this involves the root of the squared basis vectors. The interval depends directly on the squares of the basis vectors (I think of this as being 1+/- the intrinsic curvature, but do not know if that is an accurate technical assessment). This particular value regenerates the Schwarzschild solution of general relativity.
In this proposal, changes in the reference frame of an inertial observer are logically independent from changing the mass density. The two effects can be measured separately. The change in the length-time of the string will involve the inertial reference frame, and the change in the interval will involve changes in the mass density.
At this time I do not know how to use the proposed unified field equations discussed earlier to generate the basis vectors shown. This will involve determining the precise relationship between intrinsic curvature and the quaternion basis vectors.
Next: Questions in Quantum Gravity
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