2006-10-26T00:53:00.000-07:00

Can a Quantum Event Be Fixed Uniquely on a Space-Time Diagram?

All our measurements of distance and time are measured in the real physical world, whose geometry, where flat, is characterised by the Minkowski Metric. To be able to represent the location an event in the world we must use a four-fold inertial reference frame, where one direction is chosen as time and the other three become spatial dimensions. We then can graphically represent this reference frame as a space-time diagram, where normally we can suppress one or two spatial dimensions. This representation is Euclidian; its geometry differs from the geometry of the real physical world where we must make our measurements with clocks and rules.

Is this difference between the geometry of the event arena where we measure the location of events and the geometry of the graphical representation of the events important?.

Let us consider two events E1 (X₁, Y₁, Z₁, T₁) and E2 (X₂, Y₂, Z₂, T₂).

The proper interval separating the events is given by the Minkowski metric

S_M = ((X₂ –X₁)² + (Y₂-Y₁)² + (Z₂-Z₁)² – c²(T₂-T₁)²)^1/2

The perceived graphical representation of the interval is given by Pythagoras’s theorem.

S_E = ((X₂ –X₁)² + (Y₂-Y₁)² + (Z₂-Z₁)² + c²(T₂-T₁)²)^1/2

This gives an error in the graphical representation of: -

S_E-S_M = ((X₂ –X₁)² + (Y₂-Y₁)² + (Z₂-Z₁)² + c²(T₂-T₁)²)^{1/2 -}((X₂ –X₁)² + (Y₂-Y₁)² + (Z₂-Z₁)² - c²(T₂-T₁)²)^1/2

We see that the representational error becomes equal to the Pythagorean representation of the interval when.

S_M = ((X₂ –X₁)² + (Y₂-Y₁)² + (Z₂-Z₁)² – c²(T₂-T₁)²)^1/2= 0

This is the condition where it would require an observer to travel at the speed of light to get from event E1 to event E2.

If we fix the location E1 and then consider all possible events that meet this condition, then

((X –X₁)² + (Y-Y₁)² + (Z-Z₁)² = c²(T-T₁)²)^1/2

This equation now represents a light cone radiating out from eventE1.

The representational error in the interval from an event on the light cone to event E1 is equal to the proper interval separating the events.

We can state this as the following 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. (Let’s call this the Euclidian representational theorem.)

This theorem has important consequences for when we try to represent the location of a quantum event on a space-time diagram.

We are use to uncertainty in quantum mechanics; we know because of the apparent wave-particle duality of quantum objects, it is impossible to simultaneously know the position and momentum of the object. But what has not been considered is the possibility that the geometry of the world and its relationship with the way we measure and represent events relative to our inertial reference frames introduces uncertainty to the true location of an event. The Euclidian representation theorem implies that the light cone when expressed in terms of the proper interval between events collapses to a singularity. Thus in Minkowski space-time all events on a light cone are connected to its apex via zero interval paths. The converse of this is that an event E₁ in Minkowski space-time cannot be represented on a Euclidian space-time diagram as a single unique location but must be considered to be projected onto the space-time diagram as a light cone with its apex at E_1.

This implies that when we detect a quantum event we can uniquely identify the coordinate values on a space-time diagram for the act of detection but cannot say that the event itself is uniquely located at those coordinates. We are limited to knowing that the quantum event lies on a light cone passing through those coordinates.

Special relativity thus precludes the possibility that an event experienced by a quantum object can have a unique location on an inertial reference frame. The event is projected onto as space-time diagram as a light-cone.

Conclusion

Given the validity of special relativity and the Lorentz transformation; an event experienced by a quantum object cannot be uniquely fixed on a space-time diagram! Quantum events are projected onto space-time diagrams as light cones. This result must have profound implications for the nature of causation in the physical world and must play an essential role in our understanding the underlying nature of quantum mechanics.

Developments from the Euclidean Projection Theorem

The recognition of the Euclidean Projection Theorem creates the opportunity to develop a new form of mechanics lets call it Minkowski to Euclidean projection mechanics (or proper interval locality). MEP mechanics offers a new understanding of the nature of causation, provides new insights into how “action” at a distance is achieved, explains the development of the wave-function, clarifying why quantum objects display characteristics of both waves and particles and removes any contradictions that seemingly exist between quantum mechanics and relativity by predicting the violation of Bell’s inequality.

Euclidean Representation Theorem