Klein-Stapp Dialog on Decoherence, QZE, and the Efficacy of Mental Effort. (slighly revised for added clarity) On Thu, 16 Mar 2000, Stanley Klein wrote: > Henry, that was a nice, quite brief summary of your > approach. You said that > you had four equations. I don't see the purpose of the > first two > (definition of subspace and the von Neumann process. > The full evolution of > the world seemed to be quite well described by your > last two equations > (unitary evolution and the reduction): > > Between jumps the state evolves according to: > > S(t+\Delta t)= \exp(-iH\Delta t) S(t) \exp(+iH\Delta t). > > At a jump > > S(t+0)= P S(t) P \mbox{ with probability } > Tr P S(t)/ Tr S(t) > or > S(t+0)=(1-P) S(t) (1-P) \mbox{ with probability } Tr (1-P) S(t)/Tr S(t). > > The 2nd equation (the Dirac choice) has the von Neumann choice embedded within. > > It is neat to be reminded that quantum field theory has > such a simple > expression. One hopes the reader is aware that much > is hidden in H. > > Also could you remind me how multiple minds have > independent consciousnesses. > > Stan > > The purpose of the second equation is to express the condition that a specific question must be posed. The big problem in quantum theory is to specify how the question is picked out. In Copenhagen quantum theory the human observer puts in place an apparatus that decomposes the atomic system being probed into parts such that each part produces a different experience in the mind of an observer who is observing the apparatus. The mode of transfer of the information from apparatus to experience is not really accounted for, since there is, according to quantum thinking, no actual classical world. In the von Neumann version, the interface between the physical universe and the experience of the observer is shifted to the mind-brain interface: the apparatus becomes the brain. But then the issue of picking the question gets shifted to the question of how the brain is decomposed into superposed parts that will produce, when actualized, distinct experiences. There could be an infinite number of ways in which the brain could be decomposed into such parts. The question, P, must be specified before the answer can be given. If there are, for example, continuums of projectors P, not all mutually orthogonal, that are available then it makes no sense to say that nature picks an answer to each of them. For then the total probability can be infinite. von Neumann makes the probability calculus work by insisting that at each quantum jump a specific question be posed. His rule can be simplified by taking each question be a Yes-No question represented by a projection operator P. The sum Tr PS/Tr S + Tr(1-P)S/Tr S is 1.(This over-counting problem with the probability calculus, in case nothing picks the specific question, is what kills the Everett-type theories.) I introduce the definition S_b = Tr_b S as the state of b in order to be able to speak of the state of the brain-body b of the observer. P has the form (1_a)x(P_b). where 1_a is the unit operator in the degrees of freedom other than those of b, and P_b acts on the degrees of freedom of the processor b. Thus the decomposition S--> PSP + (1-P)S(1-P) induces S_b--> P_bS_bP_b + (1_b-P_b)S_b(1_b-P_b) So the reduction has a natural image in b. Actually, I consider each reduction to be associated basically with a processor b, and the associated experience to reflect the effects of P_b on b. The effect on S is then obtained by extending P_b into the whole space by means of the extension (P_b)->(1_a)x(P_b). You ask > Also could you remind me how multiple minds have independent consciousnesses. This is closely connected to the answer to the first question. I associated each reduction with a processor b, and assume that the experience is tied to S_b-->P_bS_bP_b. This will make experiences associated with processor b dynamically connected to the effects that the P_b produce on b. Henry From stapp@thsrv.lbl.gov Sat Mar 18 13:38:57 2000 Date: Fri, 17 Mar 2000 00:08:15 -0800 (PST) Subject: Re: Paper ~ Second Cambridge Talk. On Thu, 16 Mar 2000, Stan Klein wrote: > >Hi Henry, > >I thought that: > > > S--> PSP + (1-P)S(1-P) Eq. 2 > > was a tautology since it is the sum of the projected subspace plus > all the rest. The tautology is S=PSP + (1-P)S(1-P) + PS(1-P) + (1-P)SP as simple algebra shows. This reduction, von Neumann process I, drops out the cross terms between P and (1-P). It keeps only YES (P=1) and NO (P=0). > If Eq. 2 is just a tautology then the real item that > the observer is > doing is choosing P rather than > executing Eq. 2. So there would be three steps: > > Unitary evolution > Choose P > Collapse to either P or not P. > > But this might just be quibbling about whether the > Heisenberg choice > is to be called an > equation or a selection of projection operator that is > part of the > Dirac choice. It is fine with > me to divide it into two choices. But it seems to me > that the Zeno > effect is in the Dirac part > rather than the choosing part. Am I wrong on that? > Yes. The QZE comes from S --> PSP + (1-P)S(1-P) It comes from the elimination of the cross terms. The cross term P exp(-iHT)PSP exp(+iHT)(1-P) for small T has the contribution P (+iHT) (1-P). This is linear in T, and the sum of such contributions does not drop out as one divides the time into shorter and shorter intervals. This term give the leakage from the P subspace to the (1-P)subspace > Also it isn't clear to me that all collapses are to > binary choices. > I'd prefer to see a continuum of pointer readings choices. > The transition probability to any continuum state is zero. One deals with continuous variables by , for example, specifying a nonzero interval, say the interval in x between a and b, with b>a. The corresponding projection operator P(a,b), acting on any finite function of x sets the function to zero for xb, but leaves it unchanged otherwise. One sees that P squared = P: P acting twice gives the same outcome as P acting once. > > As to my comment regarding special relativity and the > time ordering > of collapses across > observers with spacelike separations viewed by systems > in relative > motion (Pat's comment). > You replied: > > "Well, naive relativity does break down when > quantum effects come in." > > But we aren't talking quantum effects here. We are > dealing with macroscopic observations. > > Stan > The point is that all quantum predictions pertaining to observable outcomes are independent of the special frame that defines the underlying constant time surfaces, and the existence of that special frame, in which the faster-than-light transfers are instantaneous, does not permit any SIGNAL to be transmitted faster than light. So the needed observational conditions for relativistic invariance are satified, even though the underlying dynamics involves the special frame. The immediately observable relativistic features are just consequence of Schwinger-Tomonoga relativistic quantum field theory. As I have stressed, positivistic tendencies have tended to make physicists believe that since the special frame needed for the underlying dynamics apparently cannot be empirically determined within atomic physics that it does not exist. But the frame DOES seem to be determined by cosmological data, and the advantages that come from assuming that the special frame, or sequence of "nows", exists, namely the advantages for science of having a rationally coherent ontology, and of not having to renounce our ability to understand nature, outweigh the advantages of claiming that because the special frame cannot be known from atomic physics data it does not exist. Henry From stapp@thsrv.lbl.gov Sat Mar 18 14:24:15 2000 Subject: Re: Paper ~ Second Cambridge Talk. Dear Stan, On Fri, 17 Mar 2000, Stanley Klein wrote: > S=PSP + (1-P)S(1-P) + PS(1-P) + (1-P)SP > > > >as simple algebra shows. To prove this identity simply collect like terms, S, PS, SP, and PSP, and see that only S survives. > > The von Neumann process I, S-->PSP+(1-P)S(1-P), > >drops out the cross terms between P and (1-P). > >It keeps only > >YES (P=1) and NO (P=0). > > Thanks, that clears it up. I was forgetting that PS(1-P) > could be nonzero. > Let me try reviewing my picture to see if I have it right: > It seems in this picture there are two stages of reduction. > The von Neumann > reduction is the one with the main collapsing action of > converting pure states > to mixed ones. I am not commited to the idea that S started pure. Nor is the posing of the question (von Neumann process I) more "main" than nature's choosing of the answer: they go hand-in-hand. > That gets rid of the quantum cross-terms > (interference effects). > Then the Dirac reduction does the pruning of the tree, a > little like what > happens classically with the decision of coming up heads or > tails. So the > question then arizes: when does the von Neumann step take > place and when > does the Dirac one occur. I had always thought that they > were simultaneous. I am adhering to the idea that they are essentially simultaneous: the two are defined in my paper in terms of two limits to a common limit time t. > But now it seems they can be separated so that one could > have lots of von > Neumann steps before one prunes the tree. > Both steps do some pruning: the first defines the question P, by "pruning" away the cross terms PS(1-P)+(1-P)SP. Then the second one give the YES (PSP) or NO ((1-P)S)1-P)) answer. > >> Also it isn't clear to me that all collapses are to > >>binary choices. > >> I'd prefer to see a continuum of pointer readings choices. > >> > > > >The transition probability to any continuum state is zero. > >One deals with continuous variables by , for example, > >specifying a nonzero interval, say the interval in x > >between a and b, > >with b>a. > > > >The corresponding projection operator P(a,b), acting on > >any > >finite function of x sets the function to zero for x >or x>b, but > >leaves it unchanged otherwise. One sees that P squared = P: > >P acting twice gives the same outcome as P acting once. > > I thought that when I make an observation I do lots of > intervals at once: > P(a1,a2)+P(a2,a3)+P(a3,a4)... where a1-a2 is very, very small. > In setting up a set of measurements to test the theory one really is supposed to set criteria for acceptance by demanding that for YES "the pointer lies within a fixed specified interval". > As to relativity. You say: > >The frame DOES seem to be determined by cosmological data, > >and the > >advantages that assuming that it exists brings, namely > >having a simple > >ontology and not having to renouncing our ability to > >understand nature, > >outweigh the questionable advantages of denying that it > >exists. > > I thought that after 1905 we had a shift in what it means > to understand > nature. The fact that different observers could have a > different view of > simultaneity became a blessing rather than a curse. That is, > we learned to > live with it. We learned to live with it, as long as classical determinism prevailed. With the whole spacetime structure all laid out it is pretty clear that choosing coordinates is question of convenience. But when indeterministic quantum theory replaced deterministic classical theory the issue was reopened. Quantum theory, when measurement issues are involved, is not fully reconcilable with the relativistic idea that "now" has no meaning: one needs a "now" to define the quantum state, at least in the Schroedinger picture. (The Heisenberg picture has associated troubles) > Nature's having this relativity seemed > elegant and nonrigid. > It seems to me that nature has gone out of her way to make > the basic laws > maintain their relativistic nature. Copenhagen preserves > that relativity. At the stiff price of renouncing our ability to understand the world about us. > Although your aesthetic is that a fixed frame with unique > time ordering > seems more elegant It is not a matter of elegance or inelegance. It is a matter of not renouncing the broader aims of science. It is a matter of not accepting the claim that science is restricted to making statistical predictions about outcomes of human observations. > and has a clearer 'ability to understand > nature', my > asthetic is that it is prettier to have a relativistic > framework where one > observer's frame is as good as the next observer. Prettiness is in the mind of the beholder: Science ought to try to provide an idea of the aspects of reality that do not depend on our sundry human viewpoints. > There > are indeed initial > condition issues (big bang) where one frame is selected > as being more > fundamental, but that doesn't change the prettiness of > Einstein's > relativity. The fact that all the faster-than-light signals > can't transfer > information is further evidence of the lengths to which > nature has gone to > protect relativity. > The notion that nature is going to great lengths to preserve some idea is suspect. The ideas generated by the theory of relativity are partially true. Relativistic quantum field theory displays and accounts for these truths. But it also displays explicity in its structure the need for the concept "now", and reveals no way to avoid the requirement that information sometimes be transmitted faster than light. It is a grotesque perversion of our gift of rationality that scientists became so taken with the idea that they could make sense of an idea that defied common sense---namely that nature did not really "unfold"--- that they continued to cling to it even when the deterministic theory that made this possible was shown to be false, and the replacement seemed incompatible with the relativism that the false theory suggested. In the language that physicists use, "signals" cannot travel faster than light, but "information" apparently must. My proof of nonlocality is evidence that SOMETHING is wrong with SOME of the ideas that WE HAVE EXTRACTED from the theory of relativity. So let us not be blinded by notions of beauty and elegance that arose in 1905, when we knew so much less. To renounce our quest for rational understanding of the world about us in order to preserve the scientifically unsupported philosophy of relativism is not my idea of good science. > Let me try asking it differently. Would your von Neumann > picture work just > as well in a moving frame. It is just that the story of the > time ordering > would change. No predictions would change. The only > casualty would be the > uniqueness of the ontology. Is that correct? > Can ontology be relative? Not in my vocabulary. The truth that science seeks is not relative to the human individual. Relativity theory in its 1905 form and quantum theory in its 1927 form were both incomplete, and failure to recognize their incompleteness led to an unwarranted deviation of science from the path of rational inquiry into the nature of nature. Henry From stapp@thsrv.lbl.gov Sat Mar 18 14:25:52 2000 Dear All: I should fill in a few details about QZE. Suppose that P is posed twice in succession. Then the state is S(t+T) = P exp(-iHT) [PS(t)P+(1-P)S(t)(1-P)] exp (+iHT)P +(1-P)exp(-iHT)[PS(t)P+(1-P)S(t)(1-P)] exp (+iHT)(1-P) If T is sufficiently small then we can replace exp(siHT) by 1 + (siHT) + (1/2)(isHT)squared + ... Because (1-P)1(P) = 0 we see that the terms involving PH(1-P) or (1-P)HP enter with a factor T squared! But if one divides some time interval into n steps, with the question P posed at each step, one sees that as n tends to infinity only the linear term in time survives. This is the QZE. It entails in our case that only PHP+(1-P)H(1-P) survives the large n limit: transitions between the two subspaces are blocked. Notice that one gets this result from just the "posing of the questions" alone. This is crucial. The further (Dirac) choice is random, and is not used. This is important because the Dirac choice is random, and not under control of the processor. If we want to compute the "statistical" net effect of the quantum dynamics, averaging over the Dirac choices, appropriately weighted, then the effect is computed by imposing the Heisenberg choices of questions, but NOT the Dirac choices on the part of Nature: we must ADD the contributions from the different Dirac choices, and this amounts to just ignoring the Dirac reductions. So to see the PREDICTED EFFECTS OF QUANTUM THEORY on individual behaviour or evolution of species one includes in the dynamics only the effects of posing the questions, not the effects of answering them. Henry From stapp@thsrv.lbl.gov Sat Mar 18 14:27:27 2000 Subject: Re: Paper ~ Second Cambridge Talk. On Fri, 17 Mar 2000, Stanley Klein wrote: > Henry, I like the two steps of the reduction. Could you clarify further > their timing. For normal conscious activity, about how many von Neumann > reductions are there per second and how many Dirac reductions? Are they > one-to-one or can there be multiple von Neumann reductions for every yes/no > reduction? > I have assumed the each posed question is answered immediately. I do not know the timing, but is should be such the key process upon which P acts does not generally evolve much between questions, so that QZE can be effective. Henry From stapp@thsrv.lbl.gov Sat Mar 18 14:28:01 2000 Subject: Re: extra baggage? On Sat, 18 Mar 2000, Stanley Klein wrote: > [Stan previous] > Henry, I like the two steps of the reduction. Could you > clarify further > their timing. For normal conscious activity, about how > many von Neuman > reductions are there per second and how many Dirac > reductions? Are they > one-to-one or can there be multiple von Neumann reductions > for every yes/no > reduction? > >> > [Henry] > >I have assumed the each posed question is answered > >immediately. I do not know the timing, but is should be > >such the key process upon which P acts does not generally > >evolve much between questions, so that QZE can be effective. > > > > [Stan] > If each question is immediately answered then the ontology > has no need for > the mixed state [PSP + (1-P)S(1-P)]. Why not have a simpler > ontology of a > pure state that evolves, then a P is chosen, then the > von Neumann/Dirac > reduction. Can there be any experimental consequence of > combining the two > choices into one (as von Neumann originally did, I presume)? > Actually, von Neumann used ONLY the von Neumann reduction (i.e.,von Neumann process I, which I call "posing the question", or the Heisenberg choice on the part of the experimenter=processor) That is, von Neumann spoke ONLY of S--> PSP +(1-P)S(1-P) and NEVER of the final Dirac choice on the part of nature of one term OR the other. He had good reason to do this. He regarded quantum theory as a statistical theory: a theory that makes statistical predictions about outcomes (i.e., predictions about probabilities of outcomes, not about outcomes.) Once the state was represented as a mixture his job was done: this mixed state could be interpreted as a classical mixture. But he never mentioned the final Dirac choice. WIGNER, in explaining what he said was von Neumann's theory, did speak of the final step. That is one reason why I generally call the theory von Neumann-Wigner quantum theory, and include the final step that von Neumann did not explicity mention. > but I > think it makes for a cleaner ontology to avoid the > unnecessary mixed state. As far as the statistical predictions of quantum theory are concerned it is ONLY the von Neumann process I (which leads to the (explicitly) mixed state) that matters: the final Dirac choice on the part of nature is irrelevant, as far as the statistical predictions of quantum theory are concerned. So it is absolutely crucial for my purposes that the QZE be a consequence of the von Neumann process I alone. It permits the choice that is under the jurisdiction of the processor to control the behavior of the processor, in the statistical sense that quantum theory can predict. This control causes the behavior to be, statistically, different from what it would be if the posing of questions did not occur. Thus a strictly quantum effect, which is not weakened by decoherence effects, and which comes from merely asking questions, without regard to how nature answers these questions, controls the statistical description of behavior, both for individuals and species. This gives individual minds power over individual behavior, and also allows the nature of the questioning by members of a species to be shaped by survival dynamics. > If on the other hand there were some reason to have lots > of vN choices for > every Dirac choice then the situation would be different. As I have stressed above, it is ONLY the von Neumann process of posing the question that enters into the statistical description of behavior: it does not matter whether the questions are ever answered or not. The statistical description of behavior that quantum theory provides is not affected by the answering of the questions, provided the answers, when and if they do come, satisfy the quantum statistical rules. > And, of course, > there may well be didactic reasons to have mixed states. > As you point out, > the QZE is easiest to derive using the mixed states. It is not a matter of "ease". The point is that it is ONLY the von Neumann processes that enter into the statistical description of behavior, so to determine what quantum theory predicts one must compute the effects of these von Neumann processes on the statistical description of behavior that quantum theory predicts. > But > here I'm talking > about ontology. Why introduce extra baggage? > I hope you see now that the von Neumann process I is the essential process, not extra baggage. > To be precise, the simpler ontology would be: > > S(t) evolves > > Then a conscious event occurs and one has: > PSP or (1-P)S(1-P) > But if one does not know which outcome occurs, or wants to determine WHAT QUANTUM THEORY SAYS about what will happen, then the correct description of the state is PSP + (1-P)S(1-P). > Incidentally, I am presuming that there are about 10-100 > choices per second > per awake (or dreaming) humanoid in the universe > (different rates for > different conscious creatures). Is that about right? > I have not reached any firm conclusion on that. Henry From augustyn@erim-int.com Sun Mar 19 14:21:48 2000 Date: Sun, 19 Mar 2000 13:51:08 -0500 From: augustyn@erim-int.com To: hpstapp@lbl.gov Subject: Re: extra baggage? (fwd) At the end of this discussion, by saying you associate the vN question with the local processor, I think you mean that the local processor "serves up the question" but that it is an extra-physical act on the part of the self to allow the served-up question to be put to Nature. The idea that the physical world is not causally closed but is in fact open to such extra-physical influence is what is altogether new and somewhat difficult to grasp. stapp@thsrv.lbl.gov on 03/19/2000 12:54:09 AM Subject: Re: extra baggage? (fwd) ---------- Forwarded message ---------- Date: Sat, 18 Mar 2000 22:52:13 -0800 (PST) From: stapp@thsrv.lbl.gov To: Stanley Klein Subject: Re: extra baggage? On Sat, 18 Mar 2000, Stanley Klein wrote: > [Henry] > But if one does not know which outcome occurs, > or wants to determine WHAT QUANTUM THEORY SAYS > about what will happen, then the correct description > of the state is PSP + (1-P)S(1-P). > > [Stan] > Maybe I'm being dense here but I still don't get it. > I fully appreciate that the critical step is the > von Neumann choice where the interference terms are > eliminated. That is what converts the cat from being > dead AND alive to dead OR alive. And that is the step > that does the QZE. > That is well understood. > > What I don't get is why don't you do the trimming > right away so that you > avoid the splitting universe extra baggage that > is reminiscent of Everrett. > I am aware that having PSP OR (1-P)S(1-P) isn't > that bad since that is what we have classically. > But > it still seems cleaner to me to do the Dirac choice at > the same time as the von Neumann choice. You even said > that you think of it happening right away. > First, let me mention the historical situation. There the experimenter sets up the experiment that would in, due course, cause the apparatus to separate into the two superposed state, one of which would be experienced by the human observer. Exactly what happens in the brain that corresponds to this is not known. But I have for mathematical neatness collapsed the two choices so that they occur in tandem. von Neumann did not speak of the second choice, and Wigner did not go into this detail. But the main point is that quantum theory does not specify the individual outcomes. So if on wants to compute only that which quantum theory specifies, then one will not include the final Dirac choice. The full content of what quantum theory predicts is given by the transformation S-->PSP + (1-P)S(1-P). This contains all the information that quantum theory provides. Let me elaborate upon this. Suppose one says that we also know that one OR the other outcome will occur. The we could say that the state is either PSP or (1-P)S(1-P). But we do not know which. But we do know something else! We DO know the probability of each of the two possibilities. So if we include that information we will be adding something else that quantum theory tells us. To incorporate this extra knowledge provided by quantum theory we should use for our computation the properly weighted sum of computations for the two individual alternative possibilities. This we should use for computations that incorporate ALL that quantun theory can tell us is the normalized state S/TrS =(PSP/TrPSP)(TrPSP/Tr S) + ((1-P)S(1-P)/Tr(1-P)S(1-P))(Tr(1-P)S(1-P)/TrS). The first factor, (PSP/TrPSP), is the normalized state that would be used if we knew that the state was PSP, and the second factor, (TrPSP/TrS), is the probability that the outcome of question P is Yes. The second term is analogous. But then cancellations yield S= PSP + (1-P)S(1-P). So to get ALL the information that quantum theory supplies one should simply replace S by PSP+(1-P)S(1-P) each time the question P is posed and answered. Quantum theory does not tell us whether the answer is P=1 or P=0, so that information is not included. This is the reason that von Neumann did not even mention the final Dirac choice. > So let me ask it again. Is there any problem (other than > possible > mathematical inelegance) with having the Dirac choice > happen simultaneous > with the vN choice? I don't think you answered that. > I do assume that the Dirac choice happens immediately after the vN choice: I assume that the two always appear in tandem. But what enters into the computation is only the vN process S-->PSP +(1-P)S(1-P). This process is determined by, and determines, the choice of P. This determination is not dependent upon whether the Dirac choice is Yes or No. I suppose one could say that the Dirac choice chooses both the question and the answer. But I believe that that is not the right way to look at it. The reason is that the von Neumann process S --> PSP + (1-P)S(1-P) is specified by just fixing question, not the outcome. But fixing the question is a local operation: it corresponds to the experimenter in a region fixing his local apparatus in some particular way. But the choice of the answer has global effects. It is central to my entire endeavor to distinguish what is under the control of the local processor and what is global. I do this by associating the von Neumann process with the local processor, but the Dirac choice with some global process. > I have another quibble over language. You call the vN > choice the 'posing of > the question'. That seems pretty benign. But isn't the > vN choice the > critical one in actually doing the measurement. Isn't the > most important > and critical aspect of measurement the erasing of the > cross terms. The > words 'posing the question' seems too mild for what is > happening. But that > is just a semantic problem and not a fundamental one at all. > The von Neumann process is indeed critical. It corresponds to the choosing by the experimenter of the experimental set up. This is often described as choosing what aspect of the probed system is going to be probed: choosing what question is going to be asked. This is normal terminology, and it accurately captures what is going on, since the two possible outcomes, P=1 or P=0 do correspond to the two possible outcomes to the query: Does the system have the property charactized by P? Choosing P chooses the question that is being posed. So this is not just a matter of arbitrary choice of terminology: the chosen terminology is both the one normally used, and the one that accurately portrays what is going on. Henry From stapp@thsrv.lbl.gov Sun Mar 19 14:23:19 2000 Date: Sun, 19 Mar 2000 14:12:56 -0800 (PST) From: stapp@thsrv.lbl.gov Subject: Re: Epistomology vs ontology On Sat, 18 Mar 2000, Stanley Klein wrote: > At 5:53 PM -0800 3/18/00, Stanley Klein wrote: > >[Henry] > >But if one does not know which outcome occurs, > >or wants to determine WHAT QUANTUM THEORY SAYS > >about what will happen, then the correct description > >of the state is PSP + (1-P)S(1-P). > > > > Let me try again with a little fancier language. > It seems to me that the measurement process (or the conscious awareness > process) reduces the ontological state to one of the two possibilities. I > would think this is the reduction of S to either PSP or to (1-P)S(1-P). > However, from an epistomological point of view the observer might not know > which state was selected so it makes sense to keep the density matrix with > both possibilities: PSP + (1-P)S(1-P). > > Isn't the following a possible interpretation of what's going on? > > Ontology: PSP OR (1-P)S(1-P) > Epistomology: PSP + (1-P)S(1-P) > > Stan > Yes. But what is "known" to one processor, because its "own" knowing has registered a P=1, may be unknown to another processor. So the formalism accomodates nicely the distinction between subjective knowings and objective knowledge, the later being the total knowledge of all the processor, up to the present. This is what the state S(t) represents. Predictions are computed from the formula = Tr PS(t)/Tr S(t), where is the absolute or objective probability that the answer to the question P asked at time t will be Yes. But no individual processor "knows" the absolute or objective state S(t), which could be written S(t)=P(t_n)U(t_n,t_(n-1))P(t_(n-1)...P(t_1)S(0)P(t_1)...P(t_n). where P(t_i) is the question answered Yes at time t_i, and U(t_i,t_j) = exp(-iH(t_i - t_j)). If some processor knows only part of the information needed to construct S(t) then he must in principle ADD the contributions corresponding to the various possibilities compatible with what he knows. Thus if he knows that question P(t_i) was posed at time t_i, but does not know that the answer was Yes, then he must add to the actual S(t) the similar expression with P(t_i) replaced by (1-P(t_i)). This gives a personal S(t) that represents his own knowledge. It can be used in place of the objective S(t) in the formula for to give the probability that is entailed by his knowledge. Of course, no processor knows even all the questions that have been posed over the course of history. But one can make prediction about small subsystems that have been prepared in a known way, and are known to be isolated from interacions with the rest of nature between preparation and subsequent measurement. In such a case the space can be divided into the part built on the degrees of freedom of the isolated system and the rest, so that the matrix elements of S(t) are . After preparation the system and its environment evolve independently up until the measurements. von Neumann's theory explains how information about the preparation and the measurements are conveyed into the brain of the observer. If P_0 is the image in the space of the system of the knowledge of the preparation, and P is the image in the space of the system of the knowledge corresponding to a possible outcome then the probability of P given P_0 and a state S(t) at the time of the preparation is = Tr PU(t',t)P_0 S(t) P_0 U(t,t')P/Tr P_0 S(t), where U(t',t) =exp -iH(t'-t) is the inverse of U(t,t'). The definition of Tr X gives Tr X = Sum over all s, Sum over all r of . The U(t',t) and U(t,t') break into a product of two independent factors, = , due to the isolation of the system during the interval in question. Then the fact that P and P_0 act as unit operators in the r variables, entails that the r-dependent factors separate out, and cancel out of the quotient, leaving = Tr_(s) PU'(t',t)P_0 S_s P_0 U'(t,t') P/Tr_(s) P_0 S_s where Tr_(s) means trace over the s variables, and S_s = Tr_(r) S = Tr_s S, [according to our notation, which makes Tr_s the trace over all variables "other" than those associated with the system, which, in an abuse of language I call s: I use s also to denote the indices that label the basis vectors of the system s). In an ideal preparation the initial state of s is completely fixed by P_0: S_s(t) = a multiple of the identity. Then = Eq. 1 Tr_s P U'(t',t) P_0 U(t,t') /Tr_s P_0, where I have repeatedly used PP=P, U(t',t)U(t,t')=1, and Tr_(s) AB = Tr_(s) BA. Eq. 1 is the Copenhagen formula. Suppose P_1 is measured in region 1, and P_2 is measure at the same time in the spacelike separated region 2. This means P=(P_1)(P_2), and (P_1)(P_2)= (P_2)(P_1). Then the general formula gives = Tr_(s) P_1 P_2 U(t',t) P_0 U(t,t')/ Tr_(s) P_0. Similarly, = Tr_(s) P_1 (1-P_2) U(t',t) P_0 U(t,t')/ Tr_(s) P_0. If it is not known whether the outcome in region 2 is P_2 or (1-P_2) then one must add the two probabilities to get the probability that P_1 is 1. This gives = Tr_(s) P_1 U(t', t)P_0 U(t',t)/ Tr_(s) P_0, which is just the formula for the case in which no measurement is made in region 2. It also shows that the probability for P_1=1 does not depend on which measurement, (P_2) or some other (P_2)', is performed in region 2, if one does not know the outcome. This is the basis for saying that no SIGNAL can be transmitted faster than light: the probability for P_1 =1 (or for P_1=0) does not depend on which measurment is performed in the other region, or on whether any measurement at all is performed there. I have included here these more technical details so that the interested readers can actually see how the theory works. One sees, in particular, the close connection between ontology and epistemology: the ontological structure really seems to be, insofar is it is captured by quantum theory, a theory of knowledge: the objective "physical" structure, S(t), represent objective knowledge, and represents it in a way that naturally accomodates also subjective knowledge. Henry From stapp@thsrv.lbl.gov Sun Mar 19 16:23:53 2000 Date: Sun, 19 Mar 2000 16:09:09 -0800 (PST) From: stapp@thsrv.lbl.gov Reply-To: hpstapp@lbl.gov Subject: Re: epistemology vs ontology On Sun, 19 Mar 2000, Stanley Klein wrote: > [Henry] > First, let me mention the historical situation. > There the experimenter set up the experiment that would > in, due course, cause the apparatus to separate into > the two superposed state, one of which would be experienced by the human > observer. > > [Stan] > 1) Ahh, I think we are getting close to clarifying my > confusion. Your use > of the words 'in due course' is critical. Are you > saying that setting up > the orientation of the polarizers ('posing the question') > is really > eliminating the cross terms, way before the particle ever > comes near the > polarizers? No! I have specified that the posing and answering are in tandem as different limits to a common time t. I mentioned the historical situation only to bring in the idea that the choice as to which question will be asked is a local question: it is basically the experimenter/processor's choice as to which question will be asked: which aspect will be probed. We are considering now going over to the limit in which the processor is both the apparatus, the system being probed, and the memory device that stores the record of the outcome given to it by nature. But I retain the idea from the Copenhagen interpretation that the experimenter (Processor) decides what question will be asked, and stores the answer that then comes immediately. > > 2) I'm looking forward to your comments on my distinction > between ontology > and epistemology. That is, the combined vN/Dirac choice > is what is actually > happening in that nature chooses one or the other > (ontology). But we might > not know which choice it was (epistemology). In terms of > normalized states: > > Epistemology: S -> PSP + (1-P)S(1-P) > > Ontology: S* -> PSP* with probability TrPSP/Tr S > or S* -> (1-P)S(1-P)* with probability Tr(1-P)S(1-P)/TrS > > where S* = S/Tr S. > > > > 3) You say that eliminating the cross terms (posing > the question) is a > local operation. I agree that the experiment itself is > local. But is the > outcome of the vN choice really local? Doesn't the > elimination of the cross > terms have global impact? If eliminating the cross terms > has no global > implication then I have indeed learned something quite > i0mportant and this > has been very educational. If the Dirac choice is the > only step that does > the nonlocal transfer of information then my point 2 about > epistemology and > ontology is plain wrong. > In my answer to "epistemology and ontology" I showed that just posing the question (i.e., S--> PSP + (1-P)S(1-P)) does not affect expectation values in regions spacelike separated from the region where the question was posed. (Actually, I showed it only for the case in which the two measurements were simultaneous, but in the relativistic version that I am considering that conclusion holds insofar as the two are spacelike separated.) It is only when the answer is taken into account that the far-away expectation value is affected. This is the no-faster-than-light signalling result. It is, of course, completely to be expected that what the far-away experimenter freely chooses to measure, and then does measure, cannot affect what the theory predicts about what happens here, if no account is taken of the outcome of that faraway measurement. This non-dependence upon the vN process itself is what makes reasonable the notion that the vN process is a local process. And this process both determines what the question P is, and is fixed by the fixing of the question P. Henry From stapp@thsrv.lbl.gov Mon Mar 20 10:04:56 2000 Date: Sun, 19 Mar 2000 18:02:50 -0800 (PST) From: stapp@thsrv.lbl.gov Reply-To: hpstapp@lbl.gov Subject: Re: epistemology vs ontology (fwd) Dear Stan, Let me get right to the point. I break the jump process into two steps: 1) The von Neumann process I, S-->PSP +(1-P)S(1-P), which I call "posing the question", followed immediately by 2) Picking the answer PSP or (1-P)S(1-P). The first step entails that there will immediately be a jump to either PSP or (1-P)S(1-P), but it does not specify which. The first is a local process, for if P_1 and P_2 act locally in two spatially separated regions then the expectation value of P_1 is unaffected by the process S--> [P_2 S P_2 + (1-P_2) S (1-P_2)]: = 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/Tr S = . The middle step is proved immediately by just using, repeatedly, PP=P, and Tr XY =Tr YX , for any (bounded) X and Y, and P_1 P_2 = P_2 P_1, which is a consequence of the space-like separation. I do not want this simple basic point to become obscured by the discussion. Henry From stapp@thsrv.lbl.gov Tue Mar 21 07:56:19 2000 Date: Tue, 21 Mar 2000 06:15:21 -0800 (PST) From: stapp@thsrv.lbl.gov Reply-To: hpstapp@lbl.gov Subject: vN process summary I put down in writing many things during this discussion with Stan, and I will not try to mention all of them. But let me briefly summarize the situation as regards the vN process I (the processors choice)and the Dirac process (nature's choice). In a sense all that happens ontologically is the sequence of quantum jumps S(t+0)=P(t)S(t-O)P(t) at a sequence of times. But vN never mentioned these jumps. He mentioned only the jumps S-->PSP + (1-P)S(1-P), which he called process I, and which I call vN process I, and process II, which is the unitary evolution between jumps. Indeed, it is not wholly clear that vN really was making an ontology: he may have gone along, at least in part, with the Copenhagen philosophy that the objective was to provide a theory of computation. But he also made a lot of statements that can be construed as supporting an ontological stance. Wigner seemed a little more clearly ontological. But since vN talked only about the vN process, not the Dirac process I felt I should, to be true to him, bring it in. Wigner added to the von Neumann approach the further selection of one alternative or the other . I already mentioned that von Neumann's likely reason for bringing in only the vN jump, and not the final selection, was that he considered the theory to be a statistical theory that makes only statistical prediction: it makes no prediction about individual outcomes. (Well, sometimes the statistical prediction happens to have probability one, but this is still a statistical prediction, strictly speaking). I made good use of von Neumann's process I. Because this process is local, in the sense that it does not affect the predicted probabilities for outcomes of far away experiments, I could associate it with the processor, and allow just the final "Dirac" choice "on the part of nature" to contain the global effect. Since S--> PSP + (1-P)S(1-P) does define P, and P is associated with "the question" that has two possible answers, P=1 and P=0, I called this vN process "posing the question" or "asking the question" or "choosing the question". It can be construed as the image in the physical world of a choice made by the processor. Stan wanted to say that the choice of P could be considered as different from the associated vN process I, which is, ontologically, a change in the physical state, not just a mere choosing of a question that would be put to nature. That is a possible point of view. But I am trying to keep things as physical as possible, and so preferred to give a physical representation of this choice, not a more nebulous choice that has no effect until the final jump to PSP or (1-P)S(1-P). Representing the choice in the physical way, as the vN process I allowed me to keep things physical, stay true to von Neumann, and identify a process that could be attributed to the processor. This step S-->PSP + (1-P)S(1-P) poses a definite yes-no question: it allows nature to choose between two alternative possibilities, where these alternative possibilities have been specified by a local processor. Henry