As I am writing this chapter there are still question marks in my head concerning the problem of nonlocality and the interpretation of the related experiments. It seems to me that both possibilities excluding each other are open namely

The first possibility is mentioned in a little known paper of H. Friedrich ⁴ He shows how the maxima of electromagnetic waves can have a faster then light motion in vacuum although the motion is obeys Maxwell equations. One can deduce from Maxwell equations directly that the propagation velocity in vacuum is equal to light velocity in vacuum. Is there a conflict? No! To transfer the mathematical argument in the paper to an easily understandable form let's consider the following experiment. Consider a set of 100 coins . Imagine that 10 of them are in a stack on a table (the top coin is numbered as 1 and the bottom coin is numbered as 10), the eleventh coin on the table touches the coin number 10 (bottom coin in the stack) and all the others placed side by side forming a chain from left to the right. Figure 16-1

Figure 16-1

Lets assume that each second we move one coin from left to the right. 1 coin/sec is our speed limit. Assume that we cannot move beyond the rightmost coin(number100). After the first second we have the situation in 16-1 b. A new stack begins to form at the right end. After 10 seconds we have a stack at right end while we have no stack at left end.(16-1 c) It seems as if the whole peak has moved from left end to the right end in 10 seconds. The distance is 90 coins so that the apparent velocity is 90coins/1 sec = 9 coins/sec. We know however that the velocity at each point of the chain was throughout the time only 1 coin/sec. As you see there is no conflict since the peak that has formed at right end is not made up of the coins that formed the initial peak at left end. Thus the motion of the peak is only apparent.

The radiation by acceleration and the radiation reaction force are consequences of Maxwell equations. Maxwell equations are however Lorenz invariant namely relativity compliant. The Dirac equation that is the relativistic version of the Schrodinger equation is also confirmed by observed energy levels in hydrogen atoms. So both fundamental equations that play a role are relativity-compliant. How can relativity compliant system of equations lead to a result conflicting with relativity. The effect of radiation reaction force is known before quantum mechanics emerged. The relativistic equations of motion of a classical particle is known since Einstein discovered them. The consideration of radiation reaction force as an additional term in relativistic equations of motion for a particle don't do any harm to the locality. So where does the nonlocality come from in quantummechanical tratment?

Lets consider the box potential as a simple example of an attractive potential for a quantum mechanical particle. The depth of the potential can be chosen so that there is only one bound state |0 > with energy E0. Lets assume that initially the particle is in a unbound energy state |1 > with energy E1 > E0 that extents from – infinity to + infinity (in real world we cannot have a wave infinitely extended in space but we can have only a very spread wave packed that is a superposition of these positive energy states . There is no limit how wide such a wave packet can be)

Lets assume that as a result of a fluctuation a contribution from the state |0 > emerges. This creates an oscillation of the magnitude of the wave function and consequently oscillation of the charge cloud. The oscillation is restricted to the region where both states overlap namely to the region where the bound state is located. (Figure 25).

Figure 25 - The magnitude of the wave function oscillating in the region where the bound state is located because of a superposition of this bound state with an unbound energyeigenstate.

It has been said that faster then light propagation of any influence suggests the possibility of traveling backwards in time according to the relativity. This is so only in Einsteinian version of relativity. In the etheristic Lorenzian version, it is the distortion of clocks because of their motion relative to the ether that slows the clocks down and not the flow of time itself. It is known that the etheristic explanation is consistent with experiment.(The Lorenz transformations -that are mathematically symmetric between reference frames regarding the clock settings and rod lengths- were initially derived by Lorenz under the assumption of a preferred frame. The symmetry is considered only as apparent in the etheristic interpretation. The clocks in the moving system are considered as asynchronous in absolute sense) It is maybe time to reconsider the Lorenzian interpretation. This doesn’t effect all the achievements of relativity. Merely the interpretation is changed in the following way: It is not the symmetry between reference frames and the geometry of space-time but an intrinsic property of ether that leads to Lorenz invariant form of all fundamental equations. ^{5, ,6,7,8,9}

 The fundamental difference between special relativistic terminology and the transition process we described above is the following: In the terminology of STR there are events. Events happen somewhere at some time. They are points in 4 dimensional space-time continuum. One can say for example that if 2 events have a time like seperation , there can be a cause effect relation between them because a signal that has a velocity smaller then the light can connect them. At a certain moment the transition described above is according to the STR not a SINGLE EVENT but infinite number of events because of the extended form of the wave function in space. These infinite number of events (the decrease of contribution of |1 > everywhere) are spacelike seperated events and cannot be causally connected. The reality is that there are in fact not infinite number of events that are in a cause effect relationship with each other but ONE SINGLE EVENT although that single event is not bound to a specific location in space.

Consider for example the process of pair creation. Initially the state of the photon may be very close to a momentum eigenstate so that it may extent over a large region in space. In a finite time period the photon disappears and for example an electron positron pair appear. The photon wave function disappears in a short period over a large region of space simultaneously. Thus during the interaction between fields (emission, absorption , pair creation) the time evolution of the wave functions cannot be judged with the criteria of STR. I think that the conditions STR imposes on the physical equations should be constricted to the following: The fundamental equations of uncoupled fields have to be Lorenz invariant.

A macroscopical object is not a single object but it is made up of large number of particles Interacting with each other and with the environment(emission and absorption of light). Because of these interactions a macroscopical object is never an opportunity to be in a coharent(“pure”) quantum state. Zureck s work shows, that the classical behavior of macroscopic systems can be understood as a consequence of decoherence that is the more effective the higher is the degrees of freedom in a system namely the larger the number of the particles and the higher the temperatures. The localized states are preferred because of decoherence¹⁰ so that wave packets never have enough time to spread.

1 Chiao 1993 on last page

2 Stenger 1995 p.140

3 Fermi 1927. Can be found also in Barut 1980 p.111 (contribution of K.Wodkiewicz)

4 Friedrich 1995

5 for derivation of Lorenz transformations from etheristic point of view see Lorenz 1952 p. 192-197

6 Bergman 1970

7 Bondi 1962

8 Oliver 1991

9 Bohm 1993 p.291

10 Zurek 1993