Minkowski to Euclidean Projection Mechanics

Minkowski to Euclidian Projection Mechanics.

An analysis of the relationship between the natural geometry of flat space-time and our methods for measuring and representing the locations of events.

28/11/2006

Contents

Abstract
Introduction
1.1 The Euclidian Representation Error
1.2 The Euclidian Representation Theorem
1.3 Minkowski to Euclidean Projection of Quantum Events
1.4 Proper Interval Locality
2 The Derivation of the Euclidean Representation Theorem
3. The World Line of a Quantum System
4. ME Projection and the Wave-function of Light
5. Secondary ME Projection
6. Diffraction Through a Small Hole.
7. Diffraction Through a Large Hole.
8. Young’s Double Slit Experiment.
9. Aspect’s Experiment.
10. Conclusion
10. References

Minkowski to Euclidian Projection Mechanics.

Absract

Based on an analysis of how we measure time and distance and how we represent the results on four dimensional inertial reference frames. An error is identified in the representation of the proper interval on space-time diagram. The Euclidean representation theorem is propounded; "On a light cone the error between the represented interval and the proper interval for the gulf between the apex and any event on the light cone is always equal to the represented interval." The theorem precludes the local representation of quantum variables relative to our inertial reference frames, a quantum variable has the same value everywhere on the light-cone. Based on the representation theorem is the principle of proper interval locality; which states “pairs of quantum system can exchange energy at events in their histories where the proper interval of separation between those events has zero magnitude.” The Euclidean representation theorem and proper interval locality form the basis of Minkowski to Euclidean projection mechanics. The new mechanics is used to describe the behaviour of light in Young’s double slit and Aspect’s experiment.

Introduction

This paper challenges the Descartean ability to fix events on four fold reference frames; arguing that Newtonian Mechanics, Hamiltonians, Lagrangeans and even the Schrödinger wave equation owe their legitimacy to statistical good fortune rather than valid argument. The post argues that given the validity of the Lorentz transformation, then analysis of the methods for measuring and representing events indicate the ability to uniquely fix an event relative to a four dimensional reference frame is precluded. For an event to possess locality it must be possible to fix it relative to a reference frame where each coordinate has a unique value. But when we examine the mathematics of the natural geometry of the world; characterised by the Minkowski metric; we find that possibility is denied to us by a mathematical theorem, which I have named the Euclidean Representation Theorem. The Euclidean representation theorem dictates that events must be projected onto four dimensional reference frames as light cones and not given a unique set of coordinates, even though the act of detecting the event will itself have a unique set of coordinates. For quantum systems the process of interference that enables classical mechanics to exist does not hide this referential non-locality thus they will exist as light cones rather than points on our reference frames. The theorem thus precludes quantum events being local relative to our inertia reference frames in as much as they are projected onto our reference frames as light cones. Any set of variables defining the state of a quantum system will also be affixed to the light cone rather than a given point say on a space-time diagram.

Although the natural geometry of space-time precludes the possibility of uniquely fixing the coordinates of any event on the history of a quantum system relative to an inertial reference frame this result should not be viewed as a complication in the description of the natural world. In fact the opposite is true and the loss of unique coordinates enables a simplification of our view of the world. Referential non-locality means that action at a distance can be explained with out having to resort to carrier particles for electromagnetism. This idea is expressed in what I have called the principle of proper interval locality; which states that pairs of quantum system can exchange energy at events in their histories where the proper interval of separation between those events has zero magnitude.

Note although the mechanism for the mediation of electro-dynamism may be completely different to the conventional phoyon descroption; the expected results in terms of say momentum exchanged between quantum systems will be unchanged. So QED may be developed equally well by both explanations for action at a distance. The advantages of the new methodology which I have termed Minkowski to Euclidean Projection Mechanics are as follows:-

1. The origins of the method go back to the outcome of the Michelson-Morley experiment and it is entirely consistent with the principles of the theory of relativity.
2. The consequences of the Euclidean Representation Theorem eliminate any contradictions between special relativity and quantum mechanics. Once the theorem is invoked then relativity permits the violation of Bell’s inequality.

3. By explaining action at a distance without invoke the need for photons simplifies the description of the natural world.

4. The experimental results that suggest the wave-particle duality of matter appear as natural consequences of the referential non-locality demanded by the Lorentz transformation.

5. Self-interference of quantum systems is natural phenomena demanded by the Euclidean representation theorem. This eliminates the need to invoke multiple universe theories.

6. The uncertainty in position demanded by referential non-locality when coupled with the foundations of quantum mechanics renders the method wholly compatible with Heisenberg’s principle of uncertainty.

The overall strength of Minkowski to Euclidean projection is that it is founded on experimentation; principally The Michelson-Morley Experiment and the work of Planck on black body radiation and it is free from arbitrary elements such as Photon’s and many worlds.

For the purposes of this post, we will use the Minkowski to Euclidean Projection to explain interference in Young’s double slit experiment and how Bell’s inequality can be violated in Aspect’s experiment without contradicting the foundations of special relativity.

1.1 The Euclidian Representation Error

We take measurements of time and distance using clocks and rules within the natural geometry of the world. When it comes to representing the results of our measurements we generally use a four fold inertial reference frame. It is common to graphically represent events on a space-time diagram which depicts the reference frame as a rectilinear grid system with the time axis orthogonal to the three spatial axes. The space-time diagram representation is essentially Euclidian in nature and the apparent separation between pairs events can be calculated by using Pythagoras’s theorem. However given the validity of the Lorentz transformation for the natural geometry of the world then the proper interval between pairs of events is calculated using Minkowski’s metric. Thus on a space-time diagram there is in general an error between the graphically represented separation and the proper interval separation between two events.

1.2 The Euclidian Representation Theorem

The following I have called the Euclidian representation theorem

"On a light cone the error between the represented interval and the proper interval for the gulf between the apex and any event on the light cone is always equal to the represented interval."

The validity of this theorem is easily calculated and is done so formally in the next section of this post.

Theorem has profound implications for how we fix events relative to our reference frame.

We must distinguish between the location of the measurement of an event and the location of event itself, the implication is we can uniquely say where we have measured the occurrence of an event but theorem precludes the possibility of uniquely fixing the event itself to that location. In fact the best we can say is the event lies on a light cone passing through the location where we measured the event to be. This is relativistic uncertainty, later we can show the uncertainty found in quantum mechanics; related to the wave-particle duality of matter; has its foundations in the uncertainty attached to our ability to fix events relative to our inertial reference frames.

For the purposes of this post we will assume that the statistical integration of quantum activity to form the macroscopic world reduces the uncertainty between the measured location and actual location to insignificance; we need only consider quantum behaviour where it is highly significant.

1.3 Minkowski to Euclidean Projection of Quantum Events

The Euclidian representation theorem implies that light cones drawn on space-time diagrams collapse to singularities in Minkowski Space-time. For example the proper interval of separation between the sun as it was eight minutes ago and ourselves, now, has zero magnitude. From our perspective we are unaware of the close proximity of our awesome neighbour. But as we develop our model we will require quantum systems to be sensitive to this immediacy. In order to develop our new form of mechanics we reverse this conclusion and deduce that a quantum event measured in Minkowski space-time is projected onto a Euclidian reference frame not as a single point location but as a light cone.

Thus ironical at quantum level special relativity precludes the possibility of local hidden variables relative to any four fold reference frame! Now Minkowski space-time is looking increasing like a suitable event arena for quantum mechanics.

1.4 Proper Interval Locality

The new mechanics requires a further deduction to make it fit with observation; this is the idea of proper interval locality. (Minkowski possesses a form of locality whose variables are projected onto the Euclidean grid system along light cones, making the variables non-local relative to our grid system. Conversely, all locations on a light cone relative to a space-time diagram are reduced to a single location in Minkowski space-time. )

Proper Interval Locality states that pairs of quantum system can exchange energy at events in their histories where the proper interval of separation between those events has zero magnitude. The zero interval paths act as causal connections between the two events and facilitate action at a distance. This implies that for instance that propagation of light is process that is facilitated by the geometry of space-time and does not require the presence of a third party carrier. Energy held by one system is directly transferred to another system, with respect to the energy the exchange is immediate and instantaneous, although with respect to our grid system the two events; the donor system giving the energy and the absorber system receiving the energy; may be separated by billions of light years.

The Euclidean representation theorem and the principle of proper interval locality form the pillars of Minkowski to Euclidian projection mechanics. We need only add Planck’s result; that the frequency associated with the exchange of energy between quantum systems is proportional to the amount energy lost or gained by the systems; to obtain a theory for the propagation of light.

The strength of this method is that it is based on experimental results and an analysis of what we are doing when we take measurements of time and distance and represent them on a four dimensional grid system. The method is free from any arbitrarily introduced elements such as photons and the true nature of the propagation of light emerges from the results of the Michelson-Morely experiment and Planck’s observations on black body radiation.

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