4. The speed of quantum tunneling

#4.1 The probability of a particle's location

#4.2 Quantum tunneling

#4.3 The speed of quantum tunneling

#4.4 A solution to a causality paradox for faster-than-light quantum tunneling

Any particle embedded in a distance-time manifold will have a probabilistic position relative to an observer who is also placed within that same manifold. Furthermore, a particle's probabilistic location will collapse when an observer measures the particle's location. These results contradict with special relativity theory, which possesses an infinitely fast space. Hence, special relativity theory predicts that a particle will always have an exact location, but never a probabilistic position.

A particle located in a distance-time manifold is located in a point universe at a single point of time in the present. Another perspective is that a particle 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. A particle 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 and within a here-now because a person always measures with a particle that travels at a finite speed. Also, relative to an observer, a space does not overlap with a here-now, and within space, all vector coordinates are distinct and separated by distance-time. (See subsection 3.11, which is about a space never overlapping a here-now.) Consequently, no particle in a space is found at all points of the universe, but rather it is only found in a distinct location. Although a space and a here-now do not overlap, they both occur in the same manifold and share the same vector coordinates. Therefore, since via a here-now, a particle is in contact with every vector coordinate, it may choose any place to be located when its location is defined in space. In distance-time theory, the definition of a particle's location in space occurs when its location is measured with another particle by an observer. Once a particle's location in space is defined, the possibility of it being found elsewhere collapses because locations in a space are distinct. Only within a here-now do we find that locations are not distinct. Since the particle's location in space is only temporarily detected, the collapse of the probability of the particle's location is also temporary.

The probability of the particle's position does not happen solely within the here-now. Neither does it happen solely within a space. It only happens between the interplay of a space and a here-now. If a particle could only be located within a here-now, it would be located at every vector coordinate and there would be no probability of its position. On the other hand, if a particle could only be embedded in a space, it would have just one position with no probability of it being found elsewhere. However, in the transition of going from a here-now to a space, a particle would have the possibility of being detected at different points of a space. Since a particle is in the state of transition before it is detected in a space, a particle will no longer possesses a probabilistic location (transitory state) after a its position is measured by an observer.

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 another particle's location. Relative to the observer, who is existing at time t₁, this detected particle exists at a specific position 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 particle at t was located here-now relative to the observer, and this particle was in contact with all the vector coordinates coexisting at t. Consequently, the particle 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. 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 particle's position was not actually measured during this period of time between t₀ and t₁, the particle, at time t, would possess a probabilistic location relative to the observer. However, when the particle acquires a position, at time t₀, in the space measured by the same observer who is now at time t₁, then this particle has no other position in that space, since all positions in any space are distinct. As a result, the probability of the particle's location collapses, relative to the observer at time t₁.

Both the possibility of a particle being found anywhere before its location is detected and the collapsing of these possibilities after a particle's location is detected agrees with elementary quantum theory. [4, 5] 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.

Although distance-time theory predicts that a particle has a possibility of being found at any position in space, the particle does not necessarily have an equal probability of being found at any position in space. To find the exact probability of locating a particle in space, I would have to use Max Born's probabilistic mathematics found in elementary quantum mechanics. I do not predict Max Born's probabilistic mathematics from distance-time theory. However, I do claim that a particle has a probabilistic location, which happens via a here-now. The consequences of this claim will be discussed in the next few subsections.

Two of the most fundamental purposes of any structure of time and space is to define the location and travel of all objects. I define travel to mean when a particle at a point in space relocates to a different point in space. Although quantum tunneling of a particle wave through an impenetrable barrier is a form of travel, according to this definition, it is not defined in terms of time and space in the traditional theories of relativity and classical space and time.

I shall now define quantum tunneling within the distance-time manifold. One of the characteristics of a particle wave in quantum mechanics is the ability to tunnel through barriers due to the fact that a particle wave can possess a probabilistic location. In distance-time theory, however, the probabilistic location of a particle occurs via a here-now. Consequently, a wave particle tunnels through a barrier via a here-now in distance-time theory. Also, forces are defined as occurring within a space of finite speed, not within a here-now. This also allows wave particles to pass through a barrier via a here-now. I shall call the tunneling of a wave particle, travel via a here-now.

A particle wave has the possibility of traveling via the here-now of a different reference frame besides the one in which it is at rest, since a particle with a velocity in a reference can still have a probabilistic location in that reference frame. (As I previously delineated, there is a distinct here-now for every reference frame. See subsection 5.7 for more information about a distinct here-now for a reference frame.) In addition, the velocity of a particle wave is independent of its position, for if this were not the case, then Heisenberg’s uncertainty principle would be violated. Furthermore, traveling via a here-now has to do with the probabilistic laws governing a particle's position-not its velocity. Consequently, a particle wave, independent of its velocity in any reference frame, can travel via the here-now of any reference frame.

According to Einstein, speeds faster-than-light were impossible because causality paradoxes could occur relative to some observers. Nevertheless, no causality paradoxes occur because of faster-than-light travel via a here-now. This last point is unique to distance-time theory. In the subsections 4.4 and 5.9 of my paper, I will explain the solutions to causality paradoxes caused by faster-than-light travel via a here-now.

At this point I will 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, then 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 derived Eq. (23). This Eq. (23) 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

In Eq. (27), 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. (27), a particle wave 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 c³ 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 wave to tunnel through a barrier based on the width of the barrier. If the particle wave 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. In the next subsection, as well as in subsection 5.9, I give distinct solutions to two different causality paradoxes.

From figure 4, I derive figure 5. In figure 5, the reference frame S' has a velocity v relative to the reference frame S. The difference in time units between S and S' relative to S is T distance-time in the positive X axis direction. A barrier is at rest in S. To the left of the barrier there is a man. To the right of the barrier there is a loaded gun. The man and the gun are at rest in S. The gun is aimed at both the barrier and the man and is connected to an electrical apparatus. This apparatus fires the gun after the man on the other side of the barrier pushes a button. However, the bullet could tunnel through the barrier and strike the man. If it travels via the S' here-now, it would travel across a -T period relative to the man and possibly kill him. Could the bullet kill the man before he pushed the button to fire the gun? If it did, a causality paradox would occur. However, this could not be the case. The period of time it takes from the point in time the man pushes the button to the point in time the gun fires could not be less than the time it takes light at speed c to travel from man to gun. However, according to Eq.27, the period of time T is limited by the slowest speed for quantum tunneling-speed c. Consequently, the bullet could not travel back in time far enough to strike the man before he pushed the button. No actual causality paradox could occur in this scenario.