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         MindNet Journal - Vol. 2, No. 3
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         V E R I C O M M sm                 "Quid veritas est?"
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    The views and opinions expressed below are not necessarily the
    views and opinions of VERICOMM or the editors, unless otherwise
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    The following is reproduced here with the express permission of
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    Editor's Note:

    The following article was originally published in Journal of
    Parapsychology, Vol. 48 No. 4, Dec. 1984.

    From: http://www.fourmilab.ch/rpkp/teleo-quant.html

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    COMPARISON OF A TELEOLOGICAL MODEL WITH A QUANTUM COLLAPSE
    MODEL OF PSI

    By Helmut Schmidt

    December 1984

    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
    aspects.

    Let me list here the most important features of the teleological
    model:

    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
    generator.

    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
    model.

    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
    approaches:

    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
    an observation.

    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
    branches.

    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
    single reality.

    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
    only-once-presented events.

    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
    information.

    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
    model.

    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
    divergence problem.

    4. The complexity independence is common to both models because
    the formalism makes no reference to the internal structure of the
    random generators.

    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
    statistical fluctuations,

    S1 = S3; S2 =-S4 (Teleological Model)

    TABLE 1

    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
    remains observable.

    CONCLUSION

    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
    of psi.

    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
    interesting.

    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.

    APPENDIX

    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
    following parameters

    (A2): A(t) = RED(t)

          B(t) = GREEN(t)

          C(t) = GHOST(t)

    with

    (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)

    with

    (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'')

    with

    (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
    alone.

    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
    by

    (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)
    final observer.

    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
    subject.

    For more details with a more general discussion of the reduction
    mechanism, see Schmidt (1982).

    REFERENCES

    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
    Foundation.

    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.

    Helmut Schmidt
    8301 Broadway, Suite 100 San Antonio, TX 78209

    Copyright 1984, Journal of Parapsychology. Reproduced with
    permission. To purchase a complete issue write Journal of
    Parapsychology, 402 N. Buchanan Blvd., Durham, NC 27701, USA
    (journal@rhine.org). For copies of complete articles, write The
    Genuine Article, ISI, 3501 Market St., Philadelphia, PA 19104,
    USA.

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