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Parapsychology experiments are notoriously difficult to interpret. Many, perhaps most, fail completely, while some seem to demonstrate the passage of information across distance or time in unexplained ways. In the best circumstances, a person reports remote events as if they were immediately present. Sixty years ago those who thought about such things at Duke University tried to envision a mental or “psychological distance” that might differ markedly from the physical or temporal distance separating an observer from events. In 1961, I participated in an experiment run by John Freeman, then a research associate at Duke, which compared objects of personal significance to Zener cards in an ESP guessing game. I scored slightly better than mean chance expectation, both when I guessed about the symbols on the cards and when I guessed about my smoking pipe, my car keys, and similar items. I think other volunteers performed similarly, and the experiment ended with no clear distinction between target types. Many years later, though, J.B. Rhine held to a similar idea. We spoke about it in 1974 when he was looking for a new laboratory director following the disclosure of fraud in a long series of experiments involving animals. I argued that good science necessarily required a rather dispassionate stance by the investigator. Rhine held that most ESP experiments failed for lack of a sufficient psychological connection among all the participants and insisted upon an emotional commitment. This idea was generally shared by others working in the field at that time, but its ubiquity did not lead to a quantifiable concept of psychological distance that might help to explain why some experiments succeed while others with similar designs do not.
In 1971 the Parapsychology Association was admitted as a member of the American Association for the Advancement of Science, culminating a long battle by researchers to free themselves from charlatans and entertainers dabbling in the occult, and to gain acceptance as a legitimate field of study. More physicists were becoming interested, and new experiments of high quality were undertaken. In 1974, Russell Targ and Harold Puthoff published in the journal Nature the results of a “remote viewing” experiment carried out at SRI International and funded by the US Department of Defense. This work involved perceptual targets in Palo Alto, California, and persons located as far away as New York City. In a longer paper published subsequently in the IEEE Proceedings (1976), the experimenters attempted to quantify the information transferred. These articles brought the research to the attention of many scientists, mathematicians, and engineers who had not known about similar experiments by Sir Oliver Lodge (1922) and a few even earlier efforts. As a result, the work at SRI was subjected to a new level of rigorous scrutiny and debate. It stands as revealing phenomena in need of explanation.
When facing problems having a difficult geometry, physicists and engineers sometimes resort to a mathematical technique called conformal mapping. The method warps space or space-time in a manner that preserves essential properties such as the propagation of light waves. If a suitable mapping can be found, it may be possible to understand the problem more easily in the transformed space. Sometimes solution can be calculated there and then mapped back through an inverse transformation. In what follows we consider a conformal mapping that potentially overcomes the great distances encountered in remote viewing experiments. We do not address other difficult aspects of these experiments.
Figure 1: Mapping of concentric circles (red) and light rays (blue)
Although we will later generalize the formulation somewhat, the basic idea is captured in a transformation that is widely known in complex analysis. We let z = x + i*y denote a location in the complex plane, and we transform to a new representation denoted by w = u + i*v, where w and its constituent real and imaginary parts u and v are functions of the original coordinates x and y. The transformation
w = sqrt (z2 – 1) Eqn 1
maps two points in the complex plane, z = 1, and z = -1, together at w = 0. Points far away are not much effected by the mapping, while those located inside the unit circle or close to it are severely rearranged.
Figure 1 shows in red the transform applied to series of concentric circles. The hour-glass shape is the mapping of the unit circle, squeezed to zero for z = 1 and z = -1, and bulging out along the y axis. In addition to bringing together these two previously separated points, the transform cleaves apart those lying immediately on either side of the line joining z = -1 to z = 1. The locations just above and below z=0 are the most radically separated, being mapped to w = i, and w = -i respectively. And smaller circles are broken in two, their top halves joined in the upper half plane, and their bottom portions joined below.
Figure 2 Mapping of horizontal and vertical grid lines
We use the complex plane to represent points in space-time, taking the x-axis as a spatial dimension, and assigning to the y-axis the distance c*t where c is the speed of light and t is time. The blue traces in Figure 1 are the transforms of light rays passing through the point z=1. These cleave to their former paths along the 45-degree lines distant from the center, but they are severely bent near the origin. The ray entering at lower left must traverse the most warped region as it approaches z=1, but escapes to the upper right through a less distorted area. The ray entering from lower right is bent towards the origin as it approaches z = 1, and then exits through a highly distorted region.
Figure 2 shows the transformation of horizontal and vertical grid lines. The squeezing of points together is shown by the hour-glass shape of the vertical lines. The rending along the x-axis can be seen in the separation of horizontal lines near y=0 and in the folding of vertical lines for |x| < 1. This represents a severing of future from past, the separation being most extreme at z = 0.
As can be verified with simple algebra, the inverse transformation is
z = sqrt (w2 + 1). Eqn 2
It rips apart points lying along either side of the line connecting w = -i to w = i, squeezing those points back together at z = 0, and expanding things back to their original size along the x-axis. A plot showing the application of Equation 2 to concentric circles in w is identical to those shown in Figure 1, but rotated 90-degrees so that the hour glass image of the unit circle lies along the horizontal axis and is pinched from top to bottom.
Figure 3: Event displaced both spatially and temporally.
The transformations of Equations 1 and 2 can be factored and generalized. We imagine an event located at the point zE and an observer situated at zO. Then the transformation
w = sqrt (z - zE) * sqrt (z – zO) + (zE + zO)/2 Eqn 3
will bring the event and observer locations together at their midpoint. And it will separate points lying to either side of the line joining zE to zO, with the most radical separation occurring for those near the midpoint. When the event and observer are symmetrically displaced from the origin, as in Equations 1 and 2, then (zE + zO)/2 = 0, and the transform assumes the simple form of a product of two factors, one centered on each point. The function sqrt (z – zA) is like an androgynous coupling mechanism on offer from any point zA to all other points in space-time.
Figure 3 shows the results that follow when the observer has placed himself at zO = 0, so that time and distance are both measured from his position. The event of interest is located at a time in the past as well as at a spatial distance. The transform has been applied to a set of circles centered at zE/2, and their images are merely rotated and displaced versions of those shown in Figure 1. The transformed paths of the light rays differ more markedly, as the one entering from the lower left now transits a region of very high deformation.
We note that c = 3 * 108 meters per second is a large number, so that a distortion of continental extent in distance (3,000 kilometers = 3 * 106 meters) corresponds to a small distortion in time (1 * 10-2 seconds = 10 milliseconds). A transform bringing together San Francisco and New York at their midpoint would separate future from past by roughly 10 milliseconds in eastern Nebraska. Humans may generally be insensitive to time differences of this magnitude, but such a discontinuity would be ruinous to the operation of many electronic devices.
The same mathematics can be used to bring together an event and an observer separated by time. We might, for instance, situate an observer at z = 0, and an event a century in the past. In this case, the great speed of light causes the transform to rupture space for 100 light-years, more than 22 times the distance to the nearest star. This would effectively sever all humanly meaningful connection.
Conclusion: Rather than trying to find a psychological distance whose diminishment might enable the transfers of information seen in remote viewing experiments, we have explored a shrinking of physical space-time. A well-known functional form brings together separated points, but at cost of introducing an abrupt discontinuity in a complementary dimension.
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