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Title: - Proper Interval Locality

“We are in "a period of utter confusion", said Nobel laureate David Gross, summing up the 2005 Solvay conference on the quantum structure of space and time. Einstein's relativity, which reigned supreme for a century, is a flawed basis for such a theory. Although it deals with gravity, it tells us nothing else about the nature and interactions of matter. Crucially, general relativity is incompatible with quantum theory.
There is a growing feeling that string theory has run into the sand. Gross thinks we are missing something fundamental. We need a leap in understanding, though where it will come from is not clear. Many of the greatest minds in physics were there at last week's conference, and none had an answer.”

From issue 2529 of New Scientist magazine, 10 December 2005, page 5

This site recognises there are fundamental problems at the heart of physics but disagrees that general relativity tells us nothing about the nature and interactions of matter. However, it agues that it is in elements that general relativity has in common with special relativity where we can learn most about the nature of how matter interacts.

It is therefore to special relativity, regarded by most physicists as the minor theory and all done and dusted, we must look to learn about the causal relationships between quantum interactions; how spatially separated particles of matter interact, to learn why matter exhibits characteristics of both waves and particles and how special relativity permits the violation of Bell’s inequality. This website argues that without understanding how causality works within Minkowski space-time its seems the formulation a comprehensive theory of the interactions of matter is unlikely

This site examines how causality is likely to operate in a world characterised by special and general relativity and results in model of the world where quantum mechanics and relativity are mutually consistent.

Proper interval locality is based on the assumptions that special relativity and quantum mechanics are valid theories; properly reflecting the properties of the physical world they are attempting to describe. Give the correctness of these assumptions; the two branches of physics derived from them cannot be objectively inconsistent with each other. Any perceived inconsistencies result from misinterpretation and the arbitrary inclusion of imaginary objects (photons) into our description of the world.

Proper interval locality holds fast to the Lorenz Transformation and its quadratic form the Minkowski space-time metric and uses these as the foundation for formulating causal relationships between events. The principle of Proper interval locality deviates from the standard theory in that the “principle of local causes” is replaced by the principle of proper contiguent causes; events are properly contiguous with each other if the proper interval separating them has zero magnitude. Events that are spatially remote from each other a can influence each other provided the proper interval separating them has zero magnitude. More specifically the electromagnetism can be mediated by space-time alone without the need for a third party carrier particle. Photons are unnecessary in Minkowski space-time. Their presence in standard theory is an inhibition to the development of any true understanding of the interactions of matter.

This site describes: -

1. The conditions in space-time that allow matter to interact.
2. Describes how the wave function of light develops
3. Shows how light produces interference.
4. Explains how light demonstrates characteristics of both waves and particle.
5. Demonstrates how special relativity permits the violation of Bell’s inequality with any matter or anything else travelling faster than light.

Locality in Minkowski Space-Time; Definition of the Three Classes of Contiguity.

Overt Contiguity.

This is common perception of contiguity where objects or events actually make contact with each other in space or time. Overt contiguity is the foundation of the principle of local causes, where events are regarded as independent of each other if they are spatially separated.

Primary Covert Contiguity (or direct contiguity)
This occurs when gulf between two events, which are separated from each other both spatially and temporally, has a proper interval of zero magnitude.

According to this definition; if we take any event E1in Minkowski space-time and construct a light cone with its apex at E1 then all events on the light cone are covertly contiguous with event E1. We assert that for one event to directly influence another event then the events must be contiguous, then accordingly the physical state of the world at E1 is capable of directly influencing the physical state of the world at any event on the entire light cone. “Action at a distance” may be mediated through the covert contiguity. The electromagnetic force can be propagated without a third party carrier particle to mediate interactions between remotely positioned quantum systems.

Secondary Covert Contiguity (or indirect contiguity)

This occurs when the gulf between two events, which are separated from each other both spatially and temporally, can be bridged by two or more straight paths where the proper intervals along each of the paths have zero magnitude. It is always possible to bridge the gulf between events with a pair of paths of zero magnitude proper interval.

Secondary covert contiguity allows a quantum system to influence the behaviour of other quantum systems that are spatially remote from the former even if they exist in regions of space time (“elsewhere“) where the standard interpretation of special relativity forbids the possibility of interaction.

Secondary covert contiguity is responsible the interference and diffraction of light, and the entanglement of remote events as demonstrated by Aspects experiment

Quantum Systems
Einstein wrote "But on one point we should, in my opinion, absolutely hold fast: the real factual situation of system S1 is independent of what is done with system S2, which is spatially separated from the former."

Here Einstein effectively defines the “principle of local causes”, this seemed consistent with his view of relativity.

The principle of contiguent causes could be written

The real factual situation of system S1 may be dependent on what is done with system S2, which is spatially separated from the former provided the gulf separating the two systems has a proper interval with a zero magnitude.

Locality is replaced by proper interval locality where events are covertly local to each other when separated by a proper interval of zero magnitude. Pil differs from quantum mechanics in that space-time plays an active role in the interaction of matter. It has a dual role; it acts as an event arena and mediates the elctromagnetic interactions of matter. Fulfilling the role played by the photon in the standard theory.

For our purposes we will not need to know anything of the details of the internal configuration or the mechanisms creating potentials within the system. All we need to know is the system is capable of absorbing and emitting energy does so in accordance with Planck’s equation.
E = hn ------ 1

Where E is the energy exchanged, h is Planck’s constant and n is the frequency at which the energy is exchanged

This is convenient because it allows quantum mechanics to slip neatly and seamlessly into Minkowski space-time. We may use quantum mechanics to determine the frequency and change in energy level expected during interaction. We may also regard the system as having limited spatial extension.

Minkowski Space-time and the definition of proper interval locality

To fix events in space-time we choose a rectilinear four fold inertial frame of reference against which we can graphically represent measures of space and time.

If two events E1 and E2 have coordinates on our reference system X1,Y1, Z1 and T1 and X2,Y2, Z2 and T2 respectively then the invariant proper interval S separating the is given by the metric.

S2= C2(T1-T2)2 - (X1-X2)2 -(Y1-Y2)2 - (Z1-Z2)2 ---- 2

Where C is constant (speed of light)

This quadratic form restricts the linear transformation of the coordinates X, Y, Z, T to those of the Lorentz transformations.

The condition for the proper interval between two events to have an extension of zero magnitude is: -


C2(T1-T2)2 = (X1-X2)2 -(Y1-Y2)2 - (Z1-Z2)2

And

C2 = [(X1-X2)2 - (Y1-Y2)2 - (Z1-Z2)2]/(T1-T2)2 ----- 3

This makes C a velocity and whose magnitude equals the speed of light.

For given event E1 (X1,Y1,Z1,T1,) all possible events which meet this condition form on a cone that radiates out from E1 in all directions (both into the past and into the future) with a velocity C.


All locations on the light cone have primary covert contiguity with E1.

Secondary Covert Contiguity.

Anywhere on the on the light cone radiating out from E1 we can choose an event E2 say . Radiating from E2 will be a pair of light cones, one to the future and one to the past. All points on these new light cone are directly covertly contiguous with E2. If we take an event E3 on the light from E2 then E3 is covertly contiguous with E2. But E2 is contiguous with E1therefore E1 must also be contiguous with E3. We can say E1 is indirectly contiguous with E3 via E2.

The physical state of the world at E1 according to the principle of proper interval locality may influence what happens at E2. Similarly what happens at E2 may influence the situation at E3. Thus event E1 may change the state of Event E2 which in turn influences the state at event E3. Thus event E1 may influence what happens at E3 (or vice versa). There is no end to the number of events we can select from which we draw light cones and select events from which we draw yet more light cones. Thus there are paths with zero magnitude bridging all positions, times and events in space-time. Minkowski space-time determines all things are linked and any event can influence every other event. The fact they evidently don’t do so depends purely on probability and the probability of interaction depends not only on the paths of zero contiguity but on the nature of the entities interacting; the quantum systems and their constituent parts the elementary particle of matter.

Note

See Attachment 1 Visualising Proper interval locality

It is impossible to directly graphically represent the Minkowskian relationship between space and time. In order to better visualise what is going on an attachment has been added to this post.
In the attachment the coordinate values on the gridlines (time and one space shown) of a space-time diagram transformed as follows: -

X = (S/H)x and T= (S/H)t where S is the proper interval and H is the Pythagorean hypotenuse between the origin and the coordinates.

Thus

S = c2t2 -x2 or x2 -c2t2 and H = c2t2 -x2 ----- 4

To maintain the real values for S the signs in the quadratic function reverse for space-like and time-like proper intervals.

The distance appearing on the graph between the origin and event (x t) is now proportional to the real proper interval between (0 0) and (x t). The gridline now appears as a curve in the form of a stylised lily.
Every gridline now has two events that are contiguous with the origin. A quantum system at x may exchange energy with a quantum system at the origin when the x gridline osculates with the origin.


The Development of the Wave-function of light.

For the purpose of this post the characteristics of a quantum system are:-

1. It can be regarded as having limited spatial extension; it is perceived as particle or a collection of particles.

2. It can absorb or emit energy in discrete amounts.

3. Its ability to absorb and emit energy is time dependent, following a regular sequence of high and low probability of interacting with the external world. The frequency is given by Planck’s equation


4. The changes in energy levels within occur abruptly.

5. Energy will only be exchanged when a quantum system is covertly contiguous with another system whose internal state, energy levels and probability cycle phase, is amenable to the exchange. The quantum system cannot spontaneously emit energy without a suitable receiver system to absorb the energy. There is no place for third party carrier particles (photons) in Minkowski space-time.


Given the above we can now show how the wave-function of light is causally developed by quantum systems (and not by third party intermediaries; photons) and how the wave-function influences the time and place where pairs of systems abruptly interact and the abrupt changes in the internal energies of the quantum system will create a corresponding abrupt change to their spatially extensive wave-function.

The ability of a quantum system to interact with other systems is time dependent. The exact internal processes creating the time dependency in the ability to absorb or emit energy is not relevant to the discussion, but in say a hydrogen atom it would seem to be related to the behaviour of the electron in the atoms potential field. That is to the phase and the difference in frequency of its de Broglie wave before and after the change in energy level (all measured relative to our given inertial reference frame). Quantum mechanics allows us to determine the frequency at which a quantum system will emit or absorb discrete quantities of energy. The phase will depend on the individual circumstances of the system at the time* of interaction.

* Relative to the individual quantum systems


See attachment 2a

We now have a probability distribution for the timing within the harmonic cycle for the “abrupt” donation of the discrete energy of excitation. We now must show how this evidently local feature of the “quantum system” is transformed into a spatially extensive probability wave-function when the system to exists in Minkowski space-time. And how the wave-function controls the abrupt changes in energy levels of spatially separated system.

Now lets look at attachment 2b.

Our excited quantum system is positioned relative to an inertial reference frame at the spatial coordinates x = 0 y = 0 z = 0. The world line of the system is now along the time axis. Relative to the space-time diagram we perceive the quantum system to progress in time from the bottom to the top of our diagram. As it progresses we expect the system to donate its energy of excitation to some remote quantum system; via a zero interval path. The event upon its world line when the acceptor system picks up the energy of excitation will lie on the future light cone radiating out from the event on the donating systems world line where it loses its energy of excitation. Relative to our interval diagram (see attachment one) this light cone collapses to a singularity at the point of donation; all events on the light cone are covertly contiguous with the donation event. (The two events; the donation of energy by the excited system and the absorption by the absorber system are zipped* into a single event by Minkowski space-time.)


*ZIP = zero interval path


According to the principle of proper interval locality there can be no free photons. Whenever an excited quantum system donates energy then somewhere on the on the future light cone, radiating from the donation event, there will be a corresponding quantum system accepting the energy of excitation. The timing of this event must be directly related to the timing of the donor system ejecting its energy of excitation. This implies the timing (and position) of an acceptor system receiving the energy is dependent on the probability distribution for the timing of the donation within the harmonic cycle of the donor system.

On a space-time diagram we see two* events:

1. The donor giving its energy of excitation. (event 1)
2. The acceptor receiving energy of excitation. (event 2)

The within harmonic cycle probability distribution for event 1to occur was defined for a specific position in space

x =0 y = 0 z = 0.

For the occurrence of event 2, the timing probability distribution is spatially extensive. For anywhere on light cones radiating from the donor system the probability distribution for the timing of an acceptor system, picking up the donor’s energy of excitation will vary with the donor’s timing probability distribution for it to donate energy.

It is this spatially extensive probability distribution that gives light its wave-like characteristics. (The abruptness of the interaction between two systems, gives light its apparent particle characteristic.)

The phase of the harmonic component of the probability to function will be constant over a light cone and the overall probability density will fall off with the inverse of the distance from the donor system (Providing the there is a uniform distribution of potential acceptor systems). This described a wave radiating out from the donor at the speed of light. It is not a physical wave but represents the probability distribution for where and when (within its harmonic cycle} the donor will be likely to interact with an acceptor system. Proper interval locality creates a situation where spatially remote systems touch in “space-time” (on their light cones). This allows them to sense their respective quantum states. If their states are mutual amenable for an interaction to occur, then energy will pass from the donor to the absorber system.


This seems to imply that there is a match between the” real factual situations” of the two interacting system in order for them to interact. This suggests that electromagnetic interactions between quantum systems have an element to their behaviour that is beyond the limits objective chance. However, any discussion of the must wait for another post. Character limitations mean this post must be drawn to a close.

But before doing so I would like note the above analysis of the development of the probability wave-form for light was restricted to the use of primary covert contiguity; for a full analysis of the development of the wave-function we should taken into account covert secondary contiguity. However, in open space; free from matter; those effects will cancel themselves out.

The essential example to demonstrate the influence of secondary convert contiguity; is light passing through a small hole. When the size of the hole is small compared with the wave length of the probability harmonic the likelihood of the donor finding an absorber will radiate uniformly out from the hole.
(“Relativistic uncertainty” without interference.)
But if hole is relatively large compared with the wavelength the events on the far side of the side of the hole will begin to osculate not with single moment of the donor’s time but will sense the states of the donor over an extended period of time. An absorber system will simultaneously sense the donor when it has both high and low abilities to interact. To work out the probability of interaction occurring we have to take into account total effect of the probability function for the whole period of the donor’s time sensed by the absorber. This is the basis of interference of light in Minkowski space-time as interpreted by the principle of proper interval locality.




The interpretation of the following experiments

1. Light passing through a small hole
2. The double slit experiment.
3. Multiple slit experiments
4. Aspect’s experiment.

And possible experimental verification must for later posts.





Conclusion

Proper interval locality shows the apparent wave-particle duality of light to reinforce Michelson- Morley’s experiment, supporting the idea that the geometry of our world is characterised by the Minkowski metric. Quantum mechanics and relativity are intimately intertwined. Inconsistencies result only from a misunderstanding of how causality functions within the space-time. The idea that “locality” is inconsistent with relativity may be the greatest irony of modern physics.