=Tr PS(t)/TrS(t). No means S(t+)=(1-P)S(t) = (1-P)S(t)(1-P) The probability of No is <1-P>=Tr (1-P)S(t)/TrS(t) III. Between jumps S(t') evolves via S(t')=exp(-iH(t'-t)) S(t) exp(-iH(t-t')) For "small" (t'-t), S(t') ~= (1-iH(t'-t)) S(t) (1-iH(t-t')) -------------------------------------------------------------- A key property is LOCALITY. Suppose there are participants in two well separated regions. Suppose the first one asks the question represented by P_1, and the second one asks, at the same time, the question represented by P_2. Then the rules of (relativistic) quantum field theory demand P_1 P_2 = P_2 P_1. A KEY LOCALITY RESULT: The probabilty for outcome Yes to the question represented by P_1 is not affected/changed by just "ASKING THE QUESTION P_2". PROOF: = Tr P_1[P_2 S P_2 +(1-P_2)S(1-P_2)]/ Tr [P_2 S P_2 + (1-P_2)S(1-P_2)] = Tr P_1[P_2 S + (1-P_2)S]/Tr[P_2 S + (1-P_2)S] = Tr P_1 S/Tr S. {For each term use Tr AB = Tr BA, P_2 P_1 = P_1 P_2, and P_2 P_2=P_2 (which entails (1-P_2)(1-P_2)=(1-P_2)).} The first line represents the probability of P_1 under the condition that P_2 is asked, but no information about the answer (to that question in region 2) is included. The third line represents what the probability of P_1 would have been if NO QUESTION had been asked in region 2. The purpose of the computation is to show that JUST ASKING the question in region 2 (i.e., asking the question, and ADDING the properly weighted contributions of the two possible immediately given answers) does not affect the PROBABILITY OF RECEIVING THE ANSWER Yes to question P_1 in region 1. This is the KEY LOCALITY RESULT: I have derived it from scratch. ---------------------------------------------------------- The SECOND KEY RESULT. THE QUANTUM ZENO EFFECT. QZE is caused by Rapid Repetitive asking of same question P; Consider S(t') after two closely spaced "Asking question P", with the answers unknown, but folded in with proper statistical weight: S(t')~= P (1-iH(t'-t)) [PS(t)P + (1-P)S(t)(1-P)] (1-iH(t-t')) P + (1-P) (1-iH(t'-t)) [PS(t)P + (1-P)S(t)(1-P)] (1-iH(t-t')) (1-P) Breaking this formula into a sum of terms one sees that H enters in four ways: PHP, (1-P)H(1-P), PH(1-P), & (1-P)HP. Look at the "cross" terms containing PH(1-P) or (1-P)HP. PP=P means that P(1-P)=(1-P)P =0. Thus P(1-iH(t'-t))(1-P)= -iPH(1-P)(t'-t) etc. Consequently, each of these "cross" terms appears with AT LEAST TWO powers of (t'-t). On the other hand, because P 1 P = P (not zero), the PHP and (1-P)H(1-P) appear (sometimes) with a SINGLE power of (t'-t). Consider a long interval T. Divide it into n intervals of length T/n. The contributions LINEAR in T/n give n contribution T/n: n/n is zeroth order in n. But n contributions of SECOND ORDER in T/n give n/nn: As n-->large n/nn --> Zero Thus the cross terms PH(1-P) and (1-P)HP drop out in limit: The Effect: H-->PHP + (1-P)H(1-P). No P <--> (1-P) transitions. The system gets trapped either in the Yes subspace PSP, or in the No subspace (1-P)S(1-P) This is the Quantum Zeno Effect. ------------------------------------------------------- **I have derived it from scratch. **QZE works just as well for a "decoherent mixture", S(t)= Sum of "classical" contributions, as for a pure state S(t). **QZE keeps the state from "diffusing" in the way that it would without the "ASKING OF QUESTIONS" by the participant. **The effect comes from just ASKING the question, and then ADDING the weighted contributions from the two possible answers. ----------------------------------------------------- Added Explanation. The environmental decoherence reduces the state S(t) to effectively a statistical mixture of almost-classical states. This mixture, if it evolved according to the rules of classical mechanics that would evolve (at least to first order) according to S(t') = exp -iH(t'-t) S(t) exp -iH(t-t') BUT THEN THERE IS NO SUPPRESSION of the PH(1-P) and (1-P)HP contributions to the evolution of this mixture. There is no confinement: no focussing of attention. For there are no quantum jumps in the classical mechananics case, and it was these jumps, which are a quantum effect, that PRODUCED THE DEVIATION from the classical spreading. _______________________________________________________ MAXIMALLY PREDICTIVE FORM OF THE THEORY. To make the theory maximally predictive we must maximally curtail the freedom of the participating knower. I assume that each participating knower has at any time just a certain definite set {P}(t) of P's that he/she/it can ask. These P's act on the physical degrees of freedom of the participant. For a given participant let P(t) be defined as follows: P(t)= The P in {P}(t) that maximizes Tr P S(t). Suppose the participant at time t is allowed only to consent to or to veto P(t). And suppose the timing and rapidity of accepting questions is controlled by "mental effort": increasing mental effort shortens the time interval between the "askings". And suppose that, for each processor, each P in the set of allowed P's tends to create, if PSP is actualized, a subsequent P that is very similar to P. If the rapidity of asking P is sufficient then QZE will kick in, and the system will tend to remain confined to the subspace P for a long time, as a consequence of the continued focus of mental attention on the question P. In this scheme the brain does most of the work, but the mind can influence brain activity via QZE by applying consent-veto and mental effort. ------------------------------------------------------------ Tie-in to Psychology. Wm. James. in "Psychology: The Briefer Course" Ch. Attention: Sect. Attention and Free Will. I have spoken as if our attention were wholly determined by neural conditions. I believe that the array of {\it things} we can attend to is so determined. No object can {\it catch} our attention except by the neural machinery. But the {\it amount} of the attention which an object receives after it has caught our attention is another question. It often takes effort to keep mind upon it. We feel that we can make more or less of the effort as we choose. If this feeling be not deceptive, if our effort be a spiritual force, and an indeterminant one, then of course it contributes coequally with the cerebral conditions to the result. Though it introduce no new idea, it will deepen and prolong the stay in consciousness of innumerable ideas which else would fade more quickly away. The delay thus gained might not be more than a second in duration--- but that second may be critical; for in the rising and falling considerations in the mind, where two associated systems of them are nearly in equilibrium it is often a matter of but a second more or less of attention at the outset, whether one system shall gain force to occupy the field and develop itself and exclude the other, or be excluded itself by the other. When developed it may make us act, and that act may seal our doom. When we come to the chapter on the Will we shall see that the whole drama of the voluntary life hinges on the attention, slightly more or slightly less, which rival motor ideas may receive. ...'' Posing a question is the act of attending. In the chapter on Will, in the section entitled Volitional effort is effort of attention'' James writes: Thus we find that {\it we reach the heart of our inquiry into volition when we ask by what process is it that the thought of any given action comes to prevail stably in the mind.}'' and later {\it The essential achievement of the will, in short, when it is most voluntary,' is to attend to a difficult object and hold it fast before the mind. ... Effort of attention is thus the essential phenomenon of will.''} Still later, James says: {\it Consent to the idea's undivided presence, this is effort's sole achievement.''} ...`Everywhere, then, the function of effort is the same: to keep affirming and adopting the thought which, if left to itself, would slip away.'' The vN/W theory, with the Quantum Zeno Effect incorporated, explains the mind-brain features that are the basis of James's conception of the action of human volition on brain process. ------------------------------------------------------------- EXAMPLE: How does your mental action raise your arm? The set or sequence of P(t)'s is being offered to the mental aspect of the participant-observer by his body-brain. If raising the arm might be appropriate in the current circumstance then the question "Am I experiencing an intention to raise my arm?" is likely to be put forth by the brain. If consented to, not vetoed, this question will be asked. If effort is applied, the same question will be asked in rapid succession. Attention will become focussed on the idea of raising the arm. According to James's ideo-motor theory, the action of raising the arm will be initiated (in the brain) by this dwelling of attention on the idea. The answer given by nature might be No! Then there will be some more searching. But if raising the arm is by far the most appropriate action under the circumstance, then the probability for a Yes answer--- with this probability ground out by the mechanical (survival-honed) evolution ---should be high. If the initial answer is No, and all other alternatives are "bad", on the basis of the mechanically conditioned criteria, then the question should get posed again soon. Maybe this time it will get caught by QZE. ------------------------------------------------------------------- Thus von Neumann-Wigner quantum theory provides the rudiments of a neat and simple framework for studying mind-brain interaction. It has significant explanatory power: it explains immediately the essential features of volition described by Wm. James. ------------------------------------------------------------ What does this theory of mind-brain interaction say about paranormal phenomena? One immediate conclusion is the locality result: CONCLUSION I. (LOCALITY) Insofar as normal relativistic quantum field theory holds. PROBABILITIES FOR FARAWAY PERCEPTIONS cannot be influenced by mental effort here. I am assuming here, on the basis of the claimed evidence for NO FALL-OFF AT ALL with either distance or time displacement, that NO DIRECT PHYSICAL CONNECTION is responsible for the purported effects. _____________________________________________________________ Example: Suppose P_1 and P_2 are correlated at time t: S(t) = a(t)(P_1)(P_2) + b(t)(1-P_1)(1-P_2). C = ((P_1)-(1-P_1))((P_2)-(1-P_2)) = Correlation Operator = Tr C S(t)/Tr S(t) = 1. Suppose HS(t') = (H_1 + H_2)S(t') t'>t and S(t')H = S(t')(H_1 + H_2) t'>t Suppose S--> P_2 S P_2 + (1-P_2)S(1-P_2) ~ mental effort by processor 2 for t'>t. This cannot effect (t') for t'>t ----------------------------------------------------------------- CONCLUSION II: Distant effects can be obtained by modifying the theory slightly: Mental Effort: S--> [(1+s)PSP + (1-P)S(1-P)] , small s This violates normal quantum theory. It would permit SIGNALS to be sent faster-than-light, in violation of ideas from the theory of relativity. But if experiments really do conflict with this conclusion, then the modification proposed above seems to be the simplest way out. It would mean that mental process can influence not only which questions are asked, and when they are asked [This appears to be all that is needed to explain the NORMAL effects of mind on matter] but also can BIAS THE PROBABILITIES away from what quantum theory specifies. That would be a direct violation of present-day quantum theory, but would at least be a well defined and calculable small violation of the normal rules. _________________________________________________________________ CONCLUSION III. A less drastic possibility would be to suppose that the measuring devices are extentions of the human participating knower, as more or less suggested by the Copenhagen approach. This would allow, in principle, the P to act not only upon certain high-level degrees of freedom of the brain, but also on some degrees of freedom of the device. This would perhaps allow paranormal effects to come in without actually directly violating the basic quantum rules. One would be exploiting, instead, the looseness of quantum theory pertaining to the choice of P in the vN process S-->PSP + (1-P)S(1-P). This suggestion departs from what I consider to be a main suggestion of von Neumann and Wigner, which is to move the mind-matter interface into the brain of the "observer". But if the purported paranormal phenomena are to be modelled then their idea might need to be slightly relaxed.