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V E R I C O M M sm "Quid veritas est?"
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The following article was originally published in Journal of
Parapsychology, Vol. 48 No. 4, Dec. 1984.
COMPARISON OF A TELEOLOGICAL MODEL WITH A QUANTUM COLLAPSE
MODEL OF PSI
By Helmut Schmidt
Copyright 1984 Journal of Parapsychology
ABSTRACT: This article compares two previously described
mathematical psi models. The teleological model emphasizes the
space-time independence of psi and the close relationship between
ESP and PK. But this model leads to a divergence problem in the
sense that the role of future observers seems exaggerated. The
quantum collapse model avoids this problem. But this model may
appear less attractive insofar as some of the space-time
independence is lost and the relationship between ESP and PK is
weakened. Experiments that might show the inadequacy of on or the
other model are discussed.
At an early stage of psi research, one could hope to understand
psi by a straightforward extension of physics. It seemed
reasonable to interpret telepathy in terms of "mental radio
waves" and psychokinesis (PK) in terms of some "mental force" But
this changed when precognition was discovered and, more
generally, when experiments showed the surprising insensitivity
of psi to physical parameters such as space, time, and the
complexity of the task. Psi appeared increasingly implausible,
suggesting a need for major changes in our thinking.
But, even with psi and current physics incompatible, there seems
no need yet to abandon the general mathematical physical approach
to the problem. On the contrary, physics is rather flexible and
open to changes. And the abstract mathematical method of modern
physics has been very successful in helping us to understand
phenomena that seemed at first intuitively implausible.
Several mathematical psi models have been developed by different
researchers (see Millar, 1978). None of these presently available
models can be considered as a satisfactory psi theory.
Nevertheless these models are stimulating for the design of new
experiments and the formation of new concepts.
In the following, I will review two of these models, compare
their merits and their shortcomings, and discuss experiments to
possibly show their inadequacies.
Since the first model has already been discussed in the
parapsychological literature (Schmidt, 1975; 1978), a brief
review will be sufficient. But the second model is published only
in a rather technical version (Schmidt, 1982), so that some more
detailed explanations are appropriate.
THE TELEOLOGICAL MODEL
The teleological model was originally developed to show by a
specific example that a world with precognition could "make
sense," that is, that the existence of precognition need not lead
to logical, conceptual difficulties and that the "intervention
paradox" presents no problem.
The model begins with an ensemble of all the possible world
histories with their corresponding probabilities as they would be
predicted by conventional physics. And, to introduce an element
of noncausality, the model postulates a psi law that modifies the
probabilities for the different world histories. This law is
teleological in the sense that, for example, the outcome of a
coin flip depends on its implications for the future history of
the world. This is certainly implausible, but so is precognition.
What matters is rather that this teleological law could be
formulated in a mathematically simple form (without too many ad
hoc assumptions) and that the resulting model is self-consistent.
In our case, this law is very simple, and provides for a
space-time independence of psi. At the same time, the psi law is
Lorentz-covariant, that is, it fits into the space-time frame
provided by Einstein's relativity theory.
The model does not claim to "explain" psi. It does not even try
to discuss what happens inside a subject's head. It rather
assumes subjects (the psi sources) with given abilities and then
studies how a world with such odd elements as psychics can still
be reasonable as a whole. This restricted scope of the model is
in accordance with the experimental work aimed at the physical
characteristics of psi (like its dependence on space, time, and
task complexity) rather than at its psychological or spiritual
Let me list here the most important features of the teleological
1. The universality of the psi mechanism. PK, precognition, and
other forms are integral parts of one common psi principle (given
by the teleological law). This has the practical implication
that, in a proper test arrangement, a prophet can perform PK
tasks and a successful PK subject can predict future events.
2. The "weak violation" of conventional physics. The model
changes only the probabilities of world histories that were
already possible in the frame of conventional physics. Therefore,
psi effects do not violate the established conservation laws of
physics (like symmetry laws and the laws for energy and momentum
conservation). Only statistical laws are affected.
3. Space-time independence of psi. The probability for a
particular history is affected by the activation of psi sources.
But it is irrelevant when and where this activation occurs in the
course of a world history. For a PK test, this irrelevance
implies that the PK effect is independent of the distance between
subject and random generator. It further implies that the
subject's effort does not even have to coincide in time with the
activation of the random generator.
4. The complexity independence of psi. In this model, psi appears
"goal oriented." In a PK experiment, for example, the subject
succeeds by aiming at the desired end result rather than by
working on the intermediate steps that lead to the final goal.
Accordingly, the formalism implies independence of the PK success
from the internal structure or complexity of the random
5.The vital role of feedback. To affect the probability of a
history, the psi source must be activated, that is, the subject
must receive feedback on his effort.
An interesting implication is that, in cases of delayed feedback,
the relevant feature is the subject's mental state at the time of
the feedback rather than at the time of the "test."
6. The divergence problem. The present course of the world
history can be affected by the future activation of psi sources.
The existence of such retroactive effects is emphasized by the
results of PK experiments with prerecorded targets (Schmidt,
1976). Nevertheless, the model seems to exaggerate the effect of
the future on the present, leading to a severe problem. I call
this the divergence problem in analogy to the famous divergence
problem in quantum electrodynamics. There, the problem was
finally overcome by "renormalization" procedures. But for the
divergence in our psi model, no remedy has been found yet that
would not destroy the internal simplicity and symmetry of the
The listed features seem closely linked, insofar as each one is
an integral part of our model. But if we do not want to commit
ourselves to any theoretical model, the first five features still
appear as interesting hypotheses to be studied independently.
THE QUANTUM COLLAPSE MODEL
The Observer Problem in Quantum Theory
Quantum theory describes a physical system in terms of a state
vector (or wave function). This vector can be considered as a set
of parameters specifying the state of the system. There seems to
be no question of how to use these vectors to make successful
calculations and predictions. But there is still controversy
about the interpretation of the state vectors.
Consider, for example, a binary random generator that randomly
selects a red or green lamp to be lighted, with probabilities p
and q, respectively. When the generator has been activated, but
before an observer has looked at the outcome, the state vector of
the system appears in the form
(1) |STATE> = sqrt(p)|RED> + sqrt(q)|GREEN>
In this equation, the state vectors |RED> and |GREEN> correspond
to physically reasonable, macroscopically well-defined states,
with the red or the green lamp lighted. The form of the total
state vector, |STATE>, as a superposition of two different
possibilities, however, makes one wonder whether Nature has
already decided for one outcome, or whether physical reality at
this stage is actually some intuitively implausible "ghost state"
suspended between two possibilities.
When an observer looks at the outcome, he sees either the red or
the green lamp lighted. At this stage, one feels, Nature must
have definitely decided for one outcome. In the language of
quantum mechanics, the original state vector, |STATE>, has made a
quantum jump, or has been reduced, into either the state |RED> or
the state |GREEN>. But the Schrodinger equation, which describes
the change of the state vector with time, has no provision for
such a sudden quantum jump. Then, how is this transition brought
about? Is it a physically real process or some artifact resulting
from an improper interpretation? Let me mention four different
Approach 1. The most simple, conventional explanation, accepted
by most physicists, is that the state vector describes not
"physical reality," but rather the observer's knowledge about
this reality. And obviously, this knowledge changes suddenly with
The provocative idea that the equations of physics deal not with
Nature per se, but with our knowledge about Nature, makes quantum
theory self-consistent. At the same time, this seems to imply a
very special role for the human observer, making him basically
different from, say, an observing TV camera. The situation
appears particularly puzzling when an observer tries to describe
a system that includes another observer, as in Schrodinger's cat
paradox, or in the paradox of Wigner's friend (d'Espagnat, 1976).
Approach 2. One widely explored possibility is the existence of
"hidden parameters," which would give quantum theory a (possibly
deterministic) substructure. The hidden parameters would
supplement the conventional parameters that make up the state
vector. Allowing this possibility, one could imagine new laws of
motion for the hidden parameters, such as laws describing the
collapse of a state vector in detail. Unfortunately, there are a
large number of such possible theories, all unpleasantly
complicated and based on arbitrary assumptions. And with the
original quantum formalism so elegant and practically successful,
one feels hesitant to introduce much more complicated theories.
For parapsychologists, hidden variables are tempting, because
they show some noncausal and space-independent features. If I
could change a hidden variable at my location, this might imply a
simultaneous change at some distant location. That need not
disturb physicists, because in their model hidden variables
cannot be observed and measured. But if the mind had some
nonphysical way to change and measure hidden variables, psi
effects could result. Walker and others have explored some of the
possibilities for constructing psi models based on hidden
parameters (Mattuck & Walker, 1979; Walker, 1975).
Approach 3. With the conventional interpretation of quantum
theory so attractively simple, Everett in his "Many World
Interpretation" (1957) suggests a somewhat new interpretation
that assigns "physical reality" to the state vector and relieves
the human observer from his role as a distinguished outsider. But
there is a heavy price to pay. Now the macroscopically ambiguous
"ghost states" become physical reality. When the random generator
makes a decision and the observer looks at the lighted lamp, the
real world has to split into two equally real branches, one
branch where the observer has seen "red" and one branch where the
observer has seen "green". Everett argues that the observer would
not notice his splitting into two branches and that, therefore,
the model is self-consistent.
Approach 4. Whereas the first approach opens the possibility that
the human observer might play a singular role in Nature, the new
formalism suggested by Eugene Wigner (1962) would explicitly
spell out the new role of the observer. Wigner's approach agrees
with Everett's insofar as the state vector is physically real
(not merely a measure of the observer's information), providing
maximal information about the system (no hidden parameters).
Then, the ghost state |STATE> of Eq. (1) represents a physical
reality with two macroscopically different but equally real
But as soon as an observer becomes consciously aware of the
outcome, Wigner proposes, the observer's mind induces a reduction
from the ambiguous ghost state into one of the "physically
reasonable" states |RED> or |GREEN>. Thus, by interfering with
the Schrodinger equation, the mind helps in maintaining one
Wigner had already wondered if the human mind, playing such an
active role in shaping physical reality, might not contribute
some PK effect in the process. The quantum collapse model to be
discussed pursues this idea.
The detailed mathematical description of this model has been
presented elsewhere (Schmidt, 1982). The model introduces, at the
phenomenological level, a mathematical formalism that would
provide the reduction of the state vector under an observation.
The main requirements that led to this particular formalism were
that the formalism should be mathematically as simple as possible
but at the same time quite general, applicable to all situations.
To save the reader from having to go into the details of the
quantum formalism of the original paper, I have summarized in the
Appendix some of the general ideas and mathematical results for
the simple case of a binary random decision.
The Reduction Process
In the model, the act of observation induces a gradual reduction
from the original ghost state (|GHOST> = |STATE> of Eq. 1) into
the well-defined states (|RED> or |GREEN>).
To formulate this transition mathematically, I shall introduce
the three time-dependent functions
(2): RED (t) = probability that Nature has decided for |RED>
GREEN (t) = probability that Nature has decided for |GREEN>
GHOST (t) = probability that at time t Nature is still in the
ambiguous, undecided state
At the beginning of an observation, at the time t = 0, Nature is
still completely undecided, that is to say
(3): GHOST (0) = 1
RED (0) = 0
GREEN (0) = 0
The change of these parameters with time is given by the
following equations (see also Appendix)
(4): GHOST (t) = R
RED (t) = p(1 + qf)(1 - R)
GREEN (t) = q(1 - pf)(1 - R)
where R = exp(-kt).
This shows an exponential decay of the ghost state, with the
final result, after a sufficiently long time (t > > 1/k)
(5): GHOST(END) = 0
RED(END) = p(1 + qf) = p'
GREEN(END) = q(1 - pf) = q'
In this model, the observer (and his momentary mental state) is
determined by two parameters, the alertness parameter k and the
PK coefficient f.
The value of the positive alertness parameter determines the
speed of the state vector reduction. The choice of the name for
this parameter should suggest that a highly alert observer might
produce a faster reduction than a sleepy one. But even though
state vector reduction is necessary for PK to operate, the speed
of this reduction does not determine the size of the PK effect
(perhaps, you don't have to be in an alert state to produce PK
effects). The PK effect is given by another parameter, f.
A nonvanishing value of the PK coefficient f makes p' and q'
different from the original probabilities p and q; that is, we
have a PK effect. The Appendix shows that the absolute value of f
is limited by the condition |f| > 1. This implies an upper limit
for PK success. In the case of a symmetric coin flipper (p = q =
1/2), this maximal success rate is 75%.
The Eqs. (4) and the more detailed Eqs. (A6) in the Appendix show
that PK may affect the way in which the disappearing ghost state
is redistributed among the final states |RED> and |GREEN>. As the
ghost state declines, the efficiency of the PK effort declines.
And when the reduction is completed, there is nothing left for PK
to operate on.
This mechanism implies that simultaneous PK efforts by two
subjects are rather inefficient, because each subject contributes
to the attrition of the ghost state, thus leaving less for the
other subject to work on. The Appendix shows explicitly that the
PK score produced by two subjects, working simultaneously or
consecutively, cannot be higher than the score obtained by the
better of the two subjects working alone. This is very different
from the situation in the teleological model and explains why the
quantum collapse model has no divergence problem.
Operational Definition of Consciousness
A most interesting suggestion of the quantum collapse model is
that some aspects of consciousness could be operationally defined
by its ability to collapse the state vector. Conventional physics
is not able to measure this collapse. But with the PK mechanism
serving as a measuring probe, a successful PK subject could tell
the difference between the collapsed and the noncollapsed state,
because only the noncollapsed state would respond to PK efforts.
We might even look for consciousness effects from animals. To
test whether, for example, a dog can collapse a state vector, we
would compare the outcome of two types of tests in which the
decisions made by our binary generator would or would not be
preinspected by the dog before the human subject applied his PK
effort. If the preinspection makes a difference, this would be
our evidence for dog consciousness.
The model does not specify clearly what constitutes a conscious
observation that should reduce the state vector. In the case of
the dog, it might be necessary to activate the animal's attention
by enforcing each red signal by a simultaneous food reward. And
with a human subject, a passive observation where the observer
immediately forgets the outcome, or a subliminal perception that
does not fully enter consciousness, might produce only incomplete
reduction. All these questions could be answered experimentally.
In this context, the results of two previous experiments might be
interesting. In a PK experiment with prerecorded targets
(Schmidt, 1976), the subjects listened to sequences of 256 clicks
that were randomly channeled to the right or left ear. The PK
target was to obtain more clicks on one specified side. The
clicks were generated at a rate of 10 per second so that the
subject could clearly notice the individual decisions but could
not spend much time "digesting" the information. Half of the
clicks were generated momentarily and presented once. The other
half were prerecorded and presented four times in succession. The
scoring rate on the repeatedly presented clicks was found to be
higher (at a moderate level of significance) than the rate on the
In the frame of our quantum collapse model, this effect might be
understood in the sense that, at the first presentation of the
clicks, there was not enough time for the subject to absorb the
whole information and thus to reduce the state completely.
Therefore, subsequent PK efforts could lead to a strengthening of
the observed PK effect.
Another possibly relevant result comes from a PK experiment with
prerecorded and preinspected seed numbers (Schmidt, 1981). This
experiment was performed in the following steps: (a) With the
help of radioactive decays as source of randomness, a six-digit
random number was generated and recorded. (b) This number was
carefully inspected by the experimenter. (c) The seed number was
fed into a computer "randomness" program to produce a binary
quasirandom number sequence. (d) The binary sequence was
displayed to the PK subject as a sequence of red and green
signals (or in some other manner) while the subject tried to
enforce the generation of many "red" signals.
In part of the experiment, step b was omitted (the experimenter's
observation of the six-digit random number). In this case, the PK
subject was the first person to observe the random result that
originated from the radioactive decay. But the outcome of the
experiment showed a PK effect also in the part where the
experimenter had preinspected the seed numbers. Thus, there
appeared no significant collapse, even though the experimenter
had enough information to derive from the seed numbers, in
principle, the finally displayed binary sequence.
This result might help us to a better understanding of what
constitutes a "conscious observation" that collapses the state
vector. Note that the seed numbers did not convey "meaningful
information" to the first observer, or information that he could
remember (a large block of seed numbers used for the experiment
were inspected in one sitting). Refer to Schmidt (1982) for more
COMPARISON OF THE TWO MODELS
I have listed six typical features of the teleological model. Let
us now look at the corresponding features of the quantum collapse
1. In the teleological model, all forms of psi seem intricately
linked, and there is a high degree of symmetry between PK and
precognition: A PK subject can be set up to act to predict future
events, and a prophet can be made to accomplish PK tasks. In the
quantum collapse model, however, PK seems to play a more dominant
role, and the model might not be sufficient to account for all
forms of precognition. Take as example the case where the subject
tries to predict the outcome of a future random event. If this
subject is the first to receive feedback on the results, then the
subject can still succeed (by mentally enforcing the generation
of the predicted event). But if somebody else looks at the
results first, collapsing the state vector in the process, then
the subject's efforts should be useless.
2. Both models agree insofar as only a "weak violation" of
physics occurs; that is, the effects appear only in connection
with random processes.
3. The space-time independence of psi is somewhat restricted by
the quantum collapse model. The outcome of a PK experiment is
still independent of the distance in space and time between the
subject's effort and the random event. Experiments with
prerecorded targets still work. But if two subjects make
consecutive PK efforts on a random event, it matters which
subject tries first. This feature may be an advantage of the
quantum collapse model because it helps to eliminate the
4. The complexity independence is common to both models because
the formalism makes no reference to the internal structure of the
5. Feedback is equally vital for both models.
6. There is no divergence problem in the quantum collapse model
because a complete observation reduces the state vector so that
PK effects from later observers are excluded.
For a most straightforward experimental comparison between the
two models, consider a PK experiment where two subjects, A and B,
make subsequent efforts at a total of N prerecorded binary
events. Let the arrangement be such that (a) in half of the
trials, subject A makes the first effort, and in the other half,
subject B is first; (b) in half of the trials both subjects make
an effort in the same direction, and in the other half, in
opposite directions; (c) these test situations are randomly
mixed, so that a subject never knows the test condition (i.e.,
whether the other subject has already made his effort and whether
the efforts are in the same or in opposite directions).
To evaluate the results, determine the deviations of the four
scoring rates from chance under the four different conditions
given by Table 1. Note that for opposite target directions in a
trial, the subjects have opposite scores. Therefore, the
definitions of the score deviations in the table have to specify
(last column) to which of the subjects this score applies.
From the viewpoint of the teleological model, it should not
matter whether A or B made the first effort; that is, apart from
S1 = S3; S2 =-S4 (Teleological Model)
DEVIATION OF SCORING RATES FROM CHANCE UNDER DIFFERENT CONDITIONS
Deviation of scoring Subject Target Subject who
scoring rates (S) order direction scored
S1 A,B Same A
S2 A,B Opposite A
S3 B,A Same B
S4 B,A Opposite B
Considering next the quantum collapse model, let us first assume
that the feedback from each trial provides a complete observation
with complete collapse of the state vector. Then the second
subject cannot exert an effect; that is,
S1 = S2; S3 = S4 (Quantum Collapse Model)
The predictions of the two models are compatible only if Sl = S2
= S3 = S = 0, that is, in the absence of PK. Therefore, a PK
experiment could easily distinguish between the two models.
In the quantum collapse model, the possibility of an incomplete
reduction of the states can be covered with the help of Equation
A16. But because a PK effect is necessarily accompanied by some
reduction of the state, the difference between the two models
The two models discussed in this paper are modest in their claims
insofar as they do not purport to "explain" psi phenomena.
Rather, the models try to provide a conceptually clean framework
for mathematically describing psi effects at a phenomenological
level, This approach gives a close link to laboratory experiments
concerned with the "physical" aspects of psi -- like its
relationship to space-time and causality, and its incompatibility
with currently accepted laws of physics.
For the physicist, the ultimate goal of psi research would be the
discovery of some novel microscopic law of Nature of great
mathematical simplicity and beauty, from which all psi effects
could, in principle, be derived. That law would qualify, from the
physicist's viewpoint, as an "explanation" of psi.
But the phenomenological, macroscopic approach appears as a
reasonable, and perhaps necessary, first step, as a basis for a
later, more complete understanding.
The two particular psi models were selected for the discussion
because these models are relatively simple mathematically and
because their easily testable predictions sharply disagree on a
vital question of parapsychology: the degree to which the future
may affect the present, that is, the extent of the noncausality
We have indications of such noncausality from precognition tests,
from PK tests with prerecorded targets, and from experiments that
suggest an effect of a later checker on previously collected test
results. Furthermore, this noncausality might be the source of
the uncontrollability of psi, in the sense that the future,
beyond our control, affects the results of a present experiment.
The teleological model, with its space-time-independent
structure, provides for all these noncausal effects and derives
PK, precognition, and the other forms of psi from one basic
mechanism. But this attractive high degree of symmetry leads to a
divergence problem in the sense that future observers obtain an
unreasonably high influence on the present.
The quantum collapse model drastically reduces this PK effect
from future observers. After a complete observation of a random
event, there is no more opportunity left for future observers to
affect the outcome. The model retains much of the space-time
independence of psi. And precognition and PK effects on
prerecorded targets can still occur. But some forms of
precognition seem not to work as well as they should.
Thus, the two models might be too extreme, in opposite
directions. And this makes the suggested Experiments particularly
Another difference between the two models might be emphasized.
The teleological model can be formulated completely within the
framework of classical physics. And the model makes no reference
to any concept of consciousness. Thus, there is no logical
necessity that the psi problem be related to the consciousness
problem, or to quantum theory.
The quantum collapse model, on the other hand, assumes a close
link between psi, consciousness, and quantum theory. The most
provocative implication of this model is that the effect of
consciousness in collapsing the state vector should be
measurable, with the PK effect serving as a measuring probe.
The Reduction Equation
Consider the simple case of a binary random generator that makes
a decision on the lighting of a red or green lamp, with the
associated probabilities p and q, respectively.
Before an observer has looked at the result, the status of the
system is given by a state vector
(A1): |GHOST> = sqrt(p)|RED> + sqrt(q)|GREEN>
This vector describes the quantum mechanical superposition of two
macroscopically different states |RED> and |GREEN> with the red
or the green lamp lighted, respectively.
Our model considers this "ghost state" as a physically real state
but one in which Nature has not yet decided for one or the other
possibility. Physical reality, at this stage, consists of a
coexistence of two branches of reality, one with the red lamp
lighted and one with the green lamp lighted.
After an observer has looked at the outcome of the random
decision, there is no more ambiguity because the observer clearly
sees either red or green.
The model assumes that it is the act of observation that
gradually reduces the initial |GHOST> state (Eq. (A1)) into
either the |RED> or the |GREEN> state. To describe this reduction
in a mathematical, statistical manner, let us introduce the
(A2): A(t) = RED(t)
B(t) = GREEN(t)
C(t) = GHOST(t)
(A3): 1 = A(t) + B(t) + C(t)
Here A(t) and B(t) are the probabilities that at time t Nature
has decided for |RED> or |GREEN>, respectively, and C(t) is the
probability that Nature is still in the undecided |GHOST> state.
Starting from the fully undecided state at time 0, we have
(A4): A(0) = 0, B(0) = 0, C(0) = 1
For the change of the parameters A, B, and C under an
observation, the model A gives the reduction equations:
(A6): (d/dt)C(t) = -kC(t)
(d/dt)A(t) = pk(1 + fq)C(t)
(d/dt)B(t) = qk(1 - fp)C(t)
In these equations, the parameters k and f depend on the observer
and his mental state, but are independent of the values p and q
that specify the random generator.
If k and f don't change with time, integration of the Eqs.
(A6) gives (A7): C(t) = R
A(t) = p(1 + qf)(1 - R)
B(t) = q(l - pf)(1 - R)
(A8): R = exp(-kt)
If the observation time was long enough (kt > > 1), then the
reduction is complete
(A9): C(END) = 0
A(END) = p(1 + qf) = p'
B(END) = q(l - pf) = q'
The values p' and q' give the probabilities for the observer to
see the red or the green lamp lighted, respectively. Only for f =
0 are these probabilities equal to p and q. Therefore, we call
the parameter f the PK coefficient.
The parameter k from Eq. (A8) does not appear in the result of
Eq. (A9). This parameter measures the speed of the reduction
under the observation. And since a very alert observer might be
expected to produce a faster reduction than a sleepy one, we call
k the alertness parameter.
The PK coefficient f is subject to the restriction
(A10): |f| < 1
This is easily seen from Eqs. (A9). If, for example, f were
larger than 1, then a p value sufficiently close to 1 would lead
to a negative value of B(END). But this is not admissible because
B(END) represents a probability.
The restriction of Eq.(A10) puts an upper limit to the size of
the PK effect. For a symmetric (p = q = 1/2) random generator,
Eqs.(A9) with f = 1 give the maximal success rate
(A11): p'(max) = 3/4 = 75%
PK Addition Effects
Imagine two subjects with, parameters (k,f) and (k',f'),
respectively, who observe the same event simultaneously, then the
momentary changes of A, B, and C, given by the right side of Eqs.
(A6), receive an additional contribution from the second
observer. The resulting final probability for red is
(A12): B(END) = p(1 + qf'')
(A13): f'' = (fk + f'k')/(k + k')
It is easily seen that the absolute value of f'' cannot exceed
the larger one of the absolute values of f and f'; that is, the
two subjects together cannot score higher than the better subject
It might be difficult to have two observers acting precisely at
the same time. If one starts observing only slightly earlier,
then he may already have reduced the state so that there is
nothing left for the second observer to do.
But consecutive action of two observers can be interesting if the
first observation is not complete. If, for example, the time was
so short as to give the first observer only a subliminal glimpse
at the outcome, the status after this observation would be given
(A14): C = R
A = p(1 + qf)(1 - R)
B = q(l - pf)(l - R)
with the reduction factor R somewhere in the range
(A15): 0 < = R < = 1
As an example with some potential experimental interest, consider
the case where the reduction factor and PK coefficients of the
subsequent subjects are (R,f) and (R',f'), and where any
remaining ghost state is reduced by a neutral (vanishing f value)
Simple calculation gives for the total PK effect in this case
(A16): p' = A(END) = p + pq[f(1 - R) + f'(1 - R')R]
This equation shows explicitly how the two subsequent PK efforts
are unsymmetric in the sense that a complete observation by the
first subject (R = 0) cuts out any effect from the second
For more details with a more general discussion of the reduction
mechanism, see Schmidt (1982).
D'ESPAGNAT, B. (1976). Conceptual foundations of quantum
mechanics. Reading, MA: Benjamin, Inc.
EVERETT, H. (1957). Relative state formulation of quantum
mechanics. Review of Modern Physics, 29, 454-462.
MATTUCK, R.D., and WALKER, E.H. (1979). The action of
consciousness on matter, a quantum mechanical theory of
psychokinesis. In A. Puharich (Ed.), The Iceland papers (pp.
111-159). Amherst, WI: Essentia Research Associates.
MILLAR, B. (1978). The observational theories. A primer. European
Journal of Parapsychology, 2, 304-332.
SCHMIDT, H. (1975). Toward a mathematical theory of psi. Journal
of the American Society for Psychical Research, 69, 301-319.
SCHMIDT, H. (1976). PK effects with prerecorded targets. Journal
of the American Society for Psychical Research, 70, 267-291.
SCHMIDT, H. (1978). Can an effect precede its cause? A model of a
noncausal world. Foundations of Physics, 8, 463-480.
SCHMIDT, H. (1981). PK tests with pre-recorded and pre-inspected
seed numbers. Journal of Parapsychology, 45, 87-98.
SCHMIDT, H. (1982). Collapse of the state vector and
psychokinetic effect. Foundations of Physics, 12, 565-581.
WALKER, E.H. (1975). Foundations of paraphysical and
parapsychological phenomena. In L. Oteri (Ed.), Quantum physics
and parapsychology (pp. 1-44). New York: Parapsychology
WIGNER, E.P. (1962). Remarks on the mind-body problem. In I.J.
Good (Ed.), The scientist speculates (pp. 284-302). New York:
Basic Books, Inc.
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