6. PROBABILITY AND TRAVEL VIA A HERE-NOW

6.1. The probability of a particle's location

There is the space of finite speed and the infinitely fast point space in distance-time theory. These two things do not overlap because one happens instantaneously and the other happens over a period of time. However, they share the same coordinate system. Hence, they share the same particles in that coordinate system. In a point space, a particle is at every location, and in space, a particle has a single location only. How is this apparent paradox to be resolved? After all, these particles are in an infinitely fast point space and they are in a space that travels at speed c. (It should be noted that a space of finite speed is where an observer detects the location of a particle.) This paradox must be resolved by means of a probability. A particle has a possibility via a point space at being located at any coordinate when its location is defined in a space, as space gives a unique position to particles. However, all the other possibilities collapse once a particular position for the particle is given in a space of finite speed. There is one problem. Space continuously defines the location of a particle. Therefore, the probability is always collapsed. There is one last thing to add: elementary particles should be wave sources—not particles, as viewed classically. (In this theory, however, I often use the term “particle” interchangeably with “matter-wave”. A matter-wave is assumed to be the basic wave found in quantum theory.)

Before I proceed any further, I propose my “theory of quantum wave sources”. In this theory, I state that a particle is best understood to be a wave source, and according to Huygens' principle, an indefinite number of point-wave sources can make up a wave. (See my “Theory of Quantum Wave Sources” [11].) In quantum wave source theory, quanta are understood not to be particles but waves made up of an indefinite number of point-wave sources. Furthermore, the smaller the region that these quantum waves inhabit, the more they behave like point-wave sources. Theoretically, if a wave was restricted to a single point, this wave would be represented by a single point-wave source. However, a wave is not restricted to a single location like a particle in space rather, it always spreads out and forms a wave in space like it does in a medium.

I now make a couple of statements and I elaborate on each of them. Then I formulate the result that combining these statements allows. Statement one is that a point-wave source located in a distance-time manifold is located in a point space at a single point of time in the present. Another perspective of this statement is that a point-wave source is at all possible locations in the universe at the point of time in the present, because at this point of time, all vector coordinate locations are at a single point. At first glance, this perspective seems to contrast with an observer's measurement. Any object is not found at all points of the universe by an observer, but rather, it is found at a specific location in space when its location is measured. However, an observer never measures infinitely quickly, nor within a here-now, because a person always measures with a particle that travels at a finite speed. Statement two is that a space of finite speed gives a specific location to objects. As a result, a space of finite speed requires that a point-wave source form a wave in a specific region, and outside that region that the wave's amplitude be negligible. (For further discussion of waves sources in a hypothetical three-dimensional medium, see my “Theory of Quantum Wave Sources” [11].) According to the rules for waves in my hypothetical quantum medium, waves should have a minimum diameter of ½ a wavelength and be continuous and smooth. (I assume there are no obstructions in the medium that might constrain a wave into a smaller region.) The next question is how a point-wave source in a hypothetical medium constructs a wave. Looking at traditional waves for a clue, I know that a point-wave source can be divided to create an indefinite number of point-wave sources from which a wave is constructed. An example of this happens when a wave front emerges and spreads out from the originating point-wave source. Naturally, more of the original point-wave source would be located where this wave would have a higher amplitude, and less of the original wave source would be located where the wave had a smaller amplitude. Therefore, in a traditional medium, the amplitude of the wave at a specific point represents the amount that the original point-wave source went into that point of the wave. I now have two principles that I need to satisfy. The first is that via a here-now, a point-wave source is in contact with every vector coordinate, and it has a possibility to be at any place when its location is detected in space. The second is that point-waves sources are located in the wave they create, and this wave's location is measurable in space. Summarizing, in an infinitely fast point space, we see that point-wave source is at every location, and in a space of finite speed, a point-wave source has a location only within the region where its waving. How is this to be resolved? I resolve this through the statement that a point-wave has a possibility, via an infinitely fast point space, of being found anywhere there is amplitude for its wave in a space of finite speed. Now, I make one further statement. The amplitude of the wave at a specific point represents the amount of the original point-wave source that went into that point of the wave. I merge all previous statements into the idea that the possibility of a point-wave source to be detected at a location is related to the amplitude of the wave at that location. This hypothesis satisfies all statements. As a result, a quantum wave is constructed by point-wave sources (like traditional waves), but the amplitudes of these point-wave sources are representative of the probability of locating the original point-wave source at that spot. Of course, from this original point-wave source come all the point-wave sources that construct the quantum wave. The hypothesis I made in this paragraph agrees with quantum theory. [5, 6]

The wave in space is created by a point-wave source that has a possibility of being located, via a here-now, in different locations in a space. Hence, if a point-wave is not found in a location, there is no possibility of it being located, via a here-now, in that area. As a consequence, the possibility of it being located in that region collapses. The entire wave does not collapse, but the possibility of the point-wave source being found in that area collapses. Therefore, the wave in that area is collapsed.

In taking this new approach, I find that the probability intrinsic to a distance-time manifold is now related to the amplitude at each point of a wave, which leads me to this next question: Is there still a possibility of a point-wave source being found anywhere in the universe? The answer depends on how the wave is structured. It is possible to create a wave with essentially a minimum ½ a wavelength and with a negligible amplitude throughout the rest of the universe. Consequently, outside of that ½ wavelength of the wave, there would still be a possibility of finding the point-wave source from this wave anywhere in the universe, however unlikely.

Of course, an observer is still dependent on measuring via particles to get a more exact location of a point-wave source. This agrees with Heisenberg's uncertainty principle, which states that an observer is limited on the accuracy of his or her measurement because he or she is limited by the particles being used for measuring [6–9]. The point-wave source is located where there is an amplitude for the wave, and the measurement can be as narrow as possibly allowed by the wavelength of the particle used for the measuring and at any point of the wave. By using this new approach, I see that this theory of distance-time comes more in line with verified results of elementary quantum theory [6–9]. To be more specific, it agrees with Heisenberg's uncertainty principle, which Einstein's special relativity could never proclaim.

In distance-time theory, I do not describe the characteristics of a wave packet or any other type of wave. I am stating, however, that the idea that a matter-wave's location is given by measuring with a particle, which would not travel faster than speed c. Also, the manifold does not give a pinpoint location for the matter-wave, but via a here-now, it gives different locations for a point-wave source, which creates a wave in space because point-wave sources always combine to create a wave. In other words, for more exact information about the matter-wave, I am reliant on measuring a matter-wave with the particle. This is different from special relativity, which always gives the exact location in time and space along with an exact velocity. Since the time, space, and velocity, are given exactly without relying on particle measurement, the energy and momentum would also be exact, with no uncertainty. On the other hand, distance-time theory would be totally dependent on measuring with particles for more exact information about time position, space location, and velocity. As a consequence, energy and momentum would also be uncertain. Distance-time theory agrees with the uncertainty found in elementary quantum theory. I do not derive Heisenberg's uncertainty equations because this theory does not predict a wave. Instead, it predicts possibilities for a point-wave source's location because of the interplay of a point space and a space of finite speed. Because there is a possibility of finding the location of a matter-wave, there should be a probability and an uncertainty associated with it. I only relate the wave found in quantum theory to this probability and uncertainty.

I further delineate the relationship between a here-now and a space with the following example. Using particles with speed v ≤ c, an observer detects a point-wave source's location in a small region. Relative to the observer, who is existing at time t1, this detected point-wave source exists at a specific region in space and at a time, t0, in the past. However, when this observer also existed in the past at points of time t < t0, the point-wave source at t was located here-now relative to the observer, and this point-wave source was in contact with all the vector coordinates coexisting at t. Consequently, the point-wave source at t could make the transition from the here-now ( the here-now relative to the observer at t ) to any possible position in a space where the amplitude of the wave exists in space. Relative to the observer, this position would exist at time t, and this space could be measured via any particle with speed v ≤ c by the same observer who would then be existing between times t0 and t1. Therefore, since this point-wave source's position was not measured during this period of time between t0 and t1, the point-wave source, at time t, would possess a probabilistic location anywhere in the amplitude of the wave in space relative to the observer. However, when the point-wave source acquires a small region of amplitude, at time t0, in the space measured by the same observer who is now at time t1, this point-wave source has no other probabilistic position in that space besides that narrow region, since all positions outside that narrow region for the amplitude have collapsed. As a result, the probability of the point-wave source's location collapses to a small region relative to the observer at time t1.

The possibility of a point-wave source being found anywhere its wave amplitude existed before its location is detected in a narrow region, and the collapsing of these possibilities outside this small region after the point-wave source's location is detected, agrees with elementary quantum theory [6–9]. It is noteworthy that these results are not derived using the traditional methods of quantum theory, and they are not found in special relativity. However, they are inherent to the space and time structure within distance-time theory

Distance-time theory predicts that a point-wave source has a greater possibility of being found where the amplitude of the wave is greater. To find the exact probability of locating a point-wave source in space, I would have to use Max Born's probabilistic mathematics found in elementary quantum mechanics. I do not predict Born's probabilistic mathematics from distance-time theory. However, I do claim that a point-wave source has a probabilistic location which happens via a here-now. The consequences of this claim are discussed in the rest of the sections of chapter 6.

Finally, it is noteworthy, that the amplitude of the wave for matter exists where there is interplay between an infinitely fast space and a space of finite speed. This is true for the amplitude of a photon, too. (See chapter 5.)

6.2. Travel via a here-now

One of the more unique ideas in distance-time theory is the idea of a here-now. The here-now is the point of time that is now and is the point of space that is here. All vector coordinates (x, y, z) exist in a here-now at a given point of time. What if an object could travel via a here-now where there were no difference between locations in space at a single point of time? What would travel via a here-now look like or resemble? These are some of the questions I now address.

Within the reference frame of an observer, Joe, a here-now exists everywhere at a single point of time. Since it exists at a single point of time, it is instantaneously at every vector coordinate relative to Joe. Therefore, relative to him, travel via the here-now in his reference frame is instantaneous. However, according to section 4.7, there are different here-nows for different reference frames. The difference between Joe's here-now and the other reference frames' here-nows are the events in x(v/c2), which come from Eq. 23. (I am assuming other reference frames are moving in the positive X axis direction relative to observer Joe, and that x represents the distance from the origin. Hence, x is actually [x – 0].) These events in x(v/c2) are located here-now in the other frame but not in Joe's frame. A particle traveling through a here-now in a different reference frame from Joe's frame would travel infinitely fast in that different frame but not relative to Joe. The reason is that relative to Joe, the here-how of a frame different from his own happens across a time of x(v/c2); therefore, travel via the here-now of this different reference frame would also travel across the time, x(v/c2), relative to the observer. Since there are different reference frames, there are different here-nows from Joe's perspective. This distance-time difference between Joe’s reference frame and different frames of reference is zero along the Y and Z axes, positive in the positive X axis direction, and negative in the negative X axis direction. In other words, a particle would go back in time if it travelled via a here-now in the negative X axis direction.

Next, I discuss the range of all the possible here-nows to each reference frame relative to my observer Joe. These ranges include all the reference frames for matter of all speeds, and the range boundaries include the speed c. According to photonic distance-time in this article, a single photon possesses a single here-now within a distance-time manifold. Moreover, the photon has a speed c. This information coupled with the discussion in the preceding paragraph gives a range of times for travel via a here-now. Relative to my observer Joe, these times range from x/c all the way down to a zero time and all the way to the negative time of –x/c. As a result there are many different here-nows that a particle could traverse. Nonetheless, any particle traversing through a here-now would never travel slower than speed c. It would always travel faster than or equal to c because v is smaller than or equal to c in x(v/c2). Therefore, having faster-than-light speeds is one characteristic an observer can look for to determine if a particle went through a here-now. This is not the case within a space of finite speed. Speeds faster than speed c are not attainable in a distance-time manifold. In section 3.10, I discussed how there is no distance-time, energy, momentum, or force within a here-now. These only exist within distance-time. Therefore, barriers have no force to stop a particle. Thus, barrier penetration is another sign of travel via a here-now. According to section 6.1, there is a probability relationship between a here-now and a space of finite speed. Consequently, there should be a probability characteristic associated with travel via a here-now. All three properties of faster than speed c, barrier penetration, and a probability are not associated with any form of travel in physics beside quantum tunneling.

The velocity of a particle is independent of its position, for if this were not the case, Heisenberg's uncertainty principle would be violated. The idea of the interplay between a here-now and the finite speed of space gives the probabilistic position of a particle. As a result, traveling via a here-now has to do with the probabilistic laws governing a particle's position—not its velocity. Consequently, a particle, independent of its velocity in any reference frame, can travel via the here-now of any reference frame. It may travel across a here-now and x(v/c2) distance-time, which is the difference between its reference frame and the reference frame's here-now that the particle is traversing. In other words, a particle has a position (or moving position) in all reference frames, which allows it to be in contact with every here-now or a here-now plus x(v/c2) for each reference frame.

I never claim that I can predict exact mathematical probabilities. However, there must be some sort of probability associated with the possibility of a point-wave source's location. This probability has already been calculated in basic quantum theory. All I state is that this probability happens via a point space. Then I give the range of speeds. Of course, this method deriving probability eventually needs to be developed further. Maybe, there is a more correct way that can eventually be developed based on the method for deriving probability in this paper.

I am very limited in my discussion about travel via a here-now because distance-time theory gives a limited understanding of it. I can only discuss it based on the information I have with regard to distance-time theory.

6.3. The causality paradoxes' solution for travel via a here-now

In section 5.3, I discussed the global here-now. The global here-now possesses all events throughout all space and time. All these events in the global here-now exist here-now relative to the sum of all eventons' rules for space and time, which is the global here-now. In other words, there are no differences between any events within a global here-now. Therefore, a series of events cancel if that series of events leads to an outcome event that prevents the first event of that series from happening. This means that within a global here-now, the beginning event and the outcome event happen together. This allows the outcome event to prevent the beginning event from occurring if a causality paradox occurs in this series of events. Essentially, the outcome event cancels the first event, as they both happen together. Hence, within a global here-now, all series of events cancel that produce a causality paradox. Consequently, I will never be able to travel into the past and kill my grandfather before he marries my grandmother. In this series of events, killing my grandfather cancels my birth, which in turn cancels me from traveling into the past. All these events happen together in a global here-now. This allows the series of events beginning with me traveling into the past to be cancelled as I kill my grandfather. A path of events through a here-now that would allow me to cause a causality paradox never occurs. Travel via here-now only can happen when there is no causality paradox or any other impossible outcome. Relative to my observer Joe, it would seem that the cosmos knew ahead of time that traveling via a here-now into the past would cause a causality paradox. This resolves all potential causality paradoxes.

6.4. The speed of quantum tunneling

According to Einstein, speeds faster than light were impossible because causality paradoxes could occur. As previously discussed, no causality paradoxes occur because of faster-than-light travel via a here-now. This last point is unique to distance-time theory. At this point I delve into faster-than-light travel via a here-now. Whenever a particle travels via the here-now of a distinct reference frame between any two points in that frame, it travels across zero distance and time between these two points and relative to that specific reference frame. Since it crosses a zero period of time in this frame, relative to this frame, it travels infinitely quickly between these two points. However, relative to a different reference frame, it would travel across a quantity of distance-time.

From Figure 4, I derive Eq. (23), which gives the difference in time units, T", between the S and S' reference frames, relative to an observer in the S reference frame. Altering Eq. (23) gives

.__________________________________________________-(62)

In Eq. (62), v is the velocity of S' relative to S, and d is the distance in the S frame spanned by T" relative to S. According to Eq. (62), a particle tunneling through a barrier in the same direction as v will travel across d (distance) in T" (period of time) with velocity c2/v relative to the S frame. Since c ≥ v, the particle will always move equal to or faster than the speed of light in a vacuum. Since I do not know what determines v, I cannot predict the time it takes for a particle to tunnel through a barrier based on the width of the barrier. If the particle moves faster than light via a here-now in the opposite direction of v, it will traverse a negative period of time, −T", relative to S. For this reason, Einstein declared speeds faster than light impossible, thus preventing time travel into the past and any causality paradoxes which may arise. However, I have already give the resolution for that possibility.

6.5. What is waving?

Quantum theory only gives the rules of a quantum wave, but it does not tell us what is waving. What is it that is being disturbed that acts like a wave? Distance-time theory does not tell us either what is waving. However, distance-time theory does tell us that the wave exists in the interplay between a point space and a space of finite speed. This at least tells us something extra about the nature of the wave, assuming that distance-time theory is correct.