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THE TECHNIQUE OF SIGNIFICABLES
A proposed complete scientific theory of the world counts as empirically adequate if it makes the right predictions about everything observable.
Putting things that way, however, suggests that in order to settle the question of whether or not some particular proposed complete scientific theory of the world is empirically adequate, we must first (among other things) settle the question of what the observable features of the world are.
And that isnt right. Its a sufficient condition of the empirical adequacy of any complete scientific account of the world (as a matter of fact) that it make the right predictions, under all physically possible circumstances, about the positions of golf-balls. And (by the same token) it is a sufficient condition of the experimental indistinguishability of any two proposed complete scientific accounts of the world, that they both make the same predictions, under all physically possible circumstances, about the positions of golf-balls.
The argument runs like this: Suppose that there is some complete scientific account of the world that makes the right predictions, under all physically possible circumstances, about the positions of golf-balls. And suppose that this account is in accord with our everyday pre-scientific empirical experience of golf-balls - suppose (more particularly) that this account endorses our conviction that we can observe the positions of golf-balls, and that we can put golf-balls more or less where we want them. And suppose that this account makes the wrong predictions, under some physically possible circumstances, about certain observable features of the world other than the positions of golf-balls. And suppose that we were to measure the values of those other observables, under those circumstances. And suppose that we were to record the outcomes of those measurements in the macroscopic configurations of golf-balls. In that case, the account in question would have to get the predictions about the golf-balls wrong too. And that (of course) precisely contradicts the hypothesis with which we started out.
In worlds like ours, then, every observable feature of nature either is a configuration of golf-balls, or can be encoded as, can be correlated with, a configuration of golf-balls. And so - in worlds like ours - a thoroughgoing empirical adequacy vis-a-vis the positions of golf-balls is necessarily also a thoroughgoing empirical adequacy simpliciter.
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Or it is (rather) subject to the following disclaimer:
The above argument takes it for granted that we can observe the positions of golf-balls, and that we can put them where we want them, under all physically possible circumstances. And that cant possibly quite be true. What (for example) about precisely those circumstances in which golf-balls are absent, or those human brain-states in which the very sight of a golf-ball immediately results in paralyzing horror or disgust, or those cultural or societal circumstances in which the manipulation of golf-balls amounts to a mortal sin? Surely a proposed complete fundamental scientific account of the world might get everything right, under all physically possible circumstances, about the positions of golf-balls, and yet get the behaviors of other observable features of the world, under some of the above circumstances, wrong. And this is of course perfectly true - and this (come to think of it) is precisely the ubiquitous old-fashioned skeptical worry about everything suddenly becoming altogether different when we turn our backs, or fall asleep, or leave the room - and this sort of worry can plainly have no final or general cure. But it seems very unlikely ever to amount to an epistemically serious sort of worry, either. Worrying seriously about any of this - after all - is usually thought to require positive and explicit and particular reasons for supposing (for example) that whatever claims T makes about those other observable features of the world, in the presence of golf balls, will somehow collapse if the golf balls are removed.
And so a somewhat more careful and more precise way of putting the conclusion we reached above might run as follows: If the world endorses our intuitive conviction that we can observe the positions of golf-balls, and if the world endorses our intuitive conviction that we can put golf-balls more or less where we want them, and if the world is not otherwise exceedingly strange, then a thoroughgoing empirical adequacy vis-a-vis the arrangements of golf-balls is necessarily also a thoroughgoing empirical adequacy simpliciter.
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And all this is plainly not unique to golf balls. Any physical variable of the world whose value we can both observe and control, any physical variable of the world (that is) that can be put to work as a mark or a pointer or a symbol, any physical variable of the world that can carry the burden of language, will serve just as well. And the sort of flexibility this observation affords will sometimes turn out to be important - as (for example) in the context of relationalist accounts of space-time, or in the context of many-worlds or many-minds interpretations of quantum mechanics, all of which deny, in very different ways, that there are ever any determinate matters of fact about the positions of golf-balls.
Lets put the conclusion (then) in a form which takes this flexibility explicitly into account. Let T be some proposed complete and fundamental theory of the world. And call V a significable of the world described by T if V is any physical variable of the world described by T, if V is any physical degree of freedom of the world described by T, whose value - according to T - we can both observe and control. The upshot of the preceding argument (then) is that it suffices for the thoroughgoing empirical adequacy of T that it gets things right, under all physically possible circumstances, about V.
And the point to be rubbed in here is that any significable whatsoever will do. All we need is an explicit formulation of the physical theory in question, and enough of an idea of who we are to point to just one physical variable of the world - the distances between oranges, say, or the shapes of pipe-cleaners, or the number of marbles in a box, or whatever - whose value, as a more or less unassailable matter of our experience of being in the world, we can both observe and control, and were in business.
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And note that precisely this kind of trick can easily and usefully be applied to a host of very different sorts of questions. There is a fairly general and fairly powerful technique implicit in all this, a technique for teasing all sorts of information about ourselves, about what we can possibly learn and what we can possibly do, out of the fundamental laws of nature.
Consider - for example - the following:
Every standard textbook presentation of the fundamental principles of quantum mechanics includes a rule about what particular mathematical features of the world count as observables. And Shelly Goldstein and Tim Maudlin have repeatedly and eloquently made the point that that is not at all the sort of rule that we ought to expect to find among the fundamental physical principles of the world. A satisfactory set of fundamental physical principles of the world ought (rather) to consist exclusively of stipulations about what there fundamentally is, and of laws about how what there fundamentally is behaves. The facts about what is or is not observable ought to follow from those fundamental principles as theorems, just as the facts about tables and chairs and mosquitoes and grocery stores ought to. And we ought not be surprised - since facts about what is or is not observable are facts about the behaviors of complicated macroscopic measuring-instruments, or about the capacities of sentient biological organisms - if the business of actually deriving those facts from any set of genuinely fundamental physical principles turns out to be immensely difficult.
And all of this is transparently and importantly true, except that the business of deriving facts about what is and (more particularly) what is not observable - in the particular case of quantum mechanics - turns out not to be even remotely as difficult as these sorts of considerations initially make it seem. There turns out to be a crisp and rigorous and exact way of deriving, from a properly fundamental set of quantum-mechanical first principles, that every observable feature of the world is necessarily going to be connected - in precisely the way all the textbooks report - with a Hermetian operator on the Hilbert space.
Heres the idea:
Suppose that everything there is to say about the world supervenes on some single universal quantum-mechanical wave-function.1 And suppose (in particular) that there is a rule for reading off the positions of golf-balls from that wave-function. And suppose that that rule, together with the fundamental laws of physics, endorses our everyday convictions to the effect that we can both observe and manipulate those positions. And suppose that it follows from that rule that any universal wave-function on which any particular golf-ball is located in any particular spatial region A is orthogonal - or very nearly orthogonal - to any universal wave-function on which the golf-ball in question is located in any other, non-overlapping, macroscopically different spatial region B.
Divide the world into three systems - a system S whose properties we are interested in measuring, and a golf-ball G, and the rest of the universe R. And consider two distinct possible physical conditions and of S. And note that it will follow very straightforwardly from the sorts of considerations we have just been through that and can only be observationally distinguished from one another if there is at least one physically possible initial condition of G+R on which G ends up located in some particular macroscopic region R if S is initially in condition and on which G ends up located in some other, non-overlapping, macroscopically different region R if S is initially in condition .
And it follows from the unitarity of the fundamental quantum-mechanical equations of motion - since (again) the wave-function of a golf-ball located in R must be orthogonal to the wave-function of a golf-ball located in R - that the counterfactual dependence described in the previous sentence can only obtain if the quantum-mechanical wave-function associated with condition is orthogonal to the quantum-mechanical wave-function associated with . And it is a theorem of complex linear algebras that any two orthogonal wave-functions in the same Hilbert space are necessarily both eigenfunctions, with different eigenvalues, of some single Hermetian operator on that space. And that is why the business of observationally distinguishing between any two possible physical conditions of the world must invariably come down to distinguishing between two different eigenvalues of some Hermetian operator, just as the textbooks say.
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This way of arguing must certainly have occurred, in passing, to lots of people. But it seems almost never to have been the object of any explicit or sustained sort of attention, and I suspect that opportunities to exploit it often get overlooked or mishandled, and I guess a part of my motivation here is just to drag it a little further out into the open, to make it more available, to put it more directly to hand, by means of a more detailed and more surprising example.
There is (then) an almost diabolically subtle and beautiful paper of John Bells, from 1975, called The Theory of Local Beables. The paper is mostly taken up with an analysis of the idea of locality - an analysis which is astonishing for its clarity and its generality and its abstractness - and with an account of the Quantum-Mechanical violation of locality which Bell had discovered some years earlier.
Let me quote a short chapter of that paper - a chapter called messages - in its entirety:
Suppose that we are finally obliged to accept the existence of these correlations at long range, and the gross non-locality of nature in the sense of this analysis. Can we then signal faster than light? To answer this we need at least a schematic theory of what we can do, a fragment of a theory of human beings. Suppose we can control variables like a and b above, but not those like A and B. [The above here refers to earlier sections of Bells paper, where he had used lower-case as and bs to represent - among other things - the settings of various measuring devices, and upper-case As and Bs to represent - among other things - the outcomes of various experiments.] I do not quite know what like means here, but suppose that beables somehow fall into two classes, controllables and uncontrollables. The latter are no use for sending signals, but can be used for reception. Suppose that to A corresponds a quantum-mechanical observable, an operator A. Then if
A /b 0
we could signal between the corresponding space-time regions, using a change in b to induce a change in the expectation value of A or of some function of A.
Suppose next that what we do when we change b is to change the quantum mechanical Hamiltonian H (say by changing some external fields) so that
Idt H = B b
where B is again an observable (i.e., an operator) localized in the region 2 of b. Then in is an exercise in quantum mechanics to show that if in a given reference system region (2) is entirely later in time than region (1)
A /b = 0
while if the reverse is true
A /b = [A, -(1/h)B]
which is again zero (for spacelike separation) in quantum field theory by the usual local commutativity condition.
So if the ordinary quantum field theory is embedded in this way in a theory of beables, it implies that faster than light signaling is not possible. In this human sense relativistic quantum mechanics is locally causal.
This passage is remarkable for its hesitation.
The suggestion that answering the question about whether we can signal faster than light might require some fragment of a theory of human beings is particularly disturbing. What Bell seems to think is that in order to settle the question of whether or not some proposed complete and fundamental physical theory of the world allows for that sort of signaling, we are first going to need to settle what seem bound to be hopelessly amorphous and difficult questions of which particular physical variables of the world can in principle be subjected to our intentional control - on the theory in question - and which can not. Indeed, in conversation, Bell would sometimes go so far as to worry aloud that the status of the "law" which forbids any super-luminal transmission of an intelligible message might imaginably turn out to be more or less akin to the status of the second law of thermodynamics - not a strict law at all, but a law for all practical purposes.
And I want to take up the invitation implicit in all this, and suggest another approach.
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Something further needs to be said, to begin with, about what it means, about what it amounts to, to be in a position to send a message.
The possibility of transmitting a message from space-time region 1 to space-time region 2 is obviously going to require that the value some physical variable of space-time region 2 - call it the output-variable - can somehow be made to counterfactually depend on the value of some other physical variable - the input-variable - of space-time region 1.
But not just any such dependence is going to do.
Consider (for example) the outcomes of measurements of the x-spins of a pair of electrons in a singlet state - one of which is carried out in space-time region 1 and the other of which is carried out in space-time region 2. It seems reasonable enough, in circumstances like those, to speak of the outcome of the measurement in region 2 as in some sense counterfactually dependant on the outcome of the measurement in region 1 - it seems reasonable enough (that is), in circumstances like those, to say that if the outcome of the measurement in region 1 had been different, then the outcome of the measurement in region 2 would have been different as well. But that sort of a dependence is manifestly going to be of no use at all for the purpose of sending a message. The sort of counterfactual dependence we are dealing with here (after all) is only going to obtain in the event that the value of the input-variable is selected in a very particular way. The sort of counterfactual dependence we are dealing with here is only going to obtain in the event that the value of the input-variable is a record of the outcome of a measurement of the x-spin of the electron in region 1 - but not (say) in the event that the input-variable is a record of my shoe size, or of what Max wants for desert, or of whether or not the court intends to grant a stay of execution, or any of an infinity of other imaginable topics of conversation. And its part and parcel of what it means to be in a position to send a message from space-time region 1 to space-time region 2 that the content of the message in question can be determined in absolutely any way you please.
What the possibility of sending a message from space-time region 1 to space-time region 2 is going to require, then, is that the value of the output variable in region 2 can be made to counterfactually depend on the value of the input-variable in region 1 completely irrespective of how it is that the value of the input variable in region 1 is selected. Whats going to be required is that the value of the output variable in region 2 can be made to counterfactually depend on the value of the input-variable in region 1 whether that latter value is the result of free dynamical evolution, or of intervention by some external physical system, or of the imposition of some external field, or of the outcome of a game of chance, or of an imaginary act of free will, or whatever. And this will be worth making up a name for. Whats going to be required (then) is that the value of the output variable in region 2 can be made to counterfactually depend on the outcome of a free selection among the possible values of the input variable in region 1.
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Now, the point that I want to draw attention to here, the point that simplifies everything infinitely, the point that Bell seems to have overlooked, is that it will involve no loss whatsoever in the generality of these considerations to suppose that the input variable in space-time region 1 and the output variable in space-time region 2 are both (say) positions of golf-balls. I want to argue (more particularly) that any proposed complete and fundamental physical theory of the world T will allow for the transmission of messages from space-time region 1 to space-time region 2 if and only if it allows for the existence a contraption whereby the position of a golf-ball in region 2 can be made to counterfactually depend, in some knowable and particular way, in any knowable and particular way, on the outcome of a free selection among the possible positions of some other golf-ball in region 1.
The argument - which the reader will no doubt already have been able to construct for herself - runs like this: If T allows for the existence of the sort of contraption described above, then, plainly, messages can be transmitted - all you do is encode them in the position of golf-ball 1 and then read them off of the position of golf-ball 2. If (on the other hand) T does not allow for the existence of contraptions like that, then messages can not be transmitted by the method just described, and they cannot be transmitted by any other method, either. Heres why: Suppose that there were some other method - any other method - of transmitting intelligible messages between regions 1 and 2. And let there be an experimenter in region 1 who is resolved to transmit a report from region 1 to region 2 - by the method in question - about the position of golf-ball 1. And let there be an experimenter in region 2 who is resolved to encode the content of the report she receives in the position of golf-ball 2. Then whatever physical contraption it is that instantiates that other method, together with the two experimenters, each equipped with their respective resolutions, will amount to precisely the golf-ball contraption described above.
We have no need, then, of any fragment of a theory of human beings - or rather, we have need of only the most trivial imaginable fragment of a theory like that - in order to decide whether or not this or that proposed fundamental physical theory allows for the transmission of information from region 1 to region 2. All we need (once again) is an explicit formulation of the physical theory in question, and a single, unassailable, empirical fact of our experience - the fact that we can both measure and control the spatial positions of golf-balls. What that latter fact turns out to entail, all by itself, is that the physical theory in question will allow for the transmission of intelligible messages from region 1 to region 2 if and only if it allows a golf-ball transmitter of the sort described above. The more general question, the more difficult question, of which particular physical variables of the world are in principle subject to our intentional control and which are not, need never even be taken up. The result is exact and principled and rigorous and universal.
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Lets have a look at how this diagnostic applies in the contexts of two very different strategies for solving the quantum-mechanical measurement problem.
The sorts of questions that are going to interest us - again - are questions of whether or not this or that proposed complete fundamental scientific theory of the world allows for the controlled transmission of intelligible information between space-like separated regions of a relativistic Minkowski space-time. But the only examples we have of completely worked-out solutions to the measurement problem, as yet, are non-relativistic ones. And so we are going to need to start out by slightly changing the subject - as everybody always does in these sorts of discussions - in such a way as to allow the it to come up in the context of the versions of quantum mechanics that we currently have.
The question at the core of our business here is whether quantum-mechanical non-locality can somehow be parlayed into a controlled transmission of intelligible information between two separate physical systems, even in circumstances where the Hamiltonain of interaction between those two systems, and between the two of those systems and any third one, is zero. If the answer to that question, on a certain non-relativistic version of quantum mechanics, is yes, then there would seem to be nothing standing in the way of our exploiting that non-locality - in the context of an appropriate relativistic generalization of that version of quantum mechanics - to transmit messages between space-like separated regions of a relativistic Minkowski space-time. And if the answer to that question, on a certain non-relativistic version of quantum mechanics, is no, then there would seem to be no reason for suspecting that messages can be transmitted across such separations - in the context of an appropriate relativistic generalization of that version of quantum mechanics - at all.
Consider (then) two golf-balls, GA and GB. And let there be a system A in the vicinity of GA, and another system, a disjoint system, B, in the vicinity of GB. And let A and B each contain absolutely anything you like - tables, chairs, oceans, measuring-instruments, scientists, cultural institutions, whatever. And stipulate that the Hamiltonian of interaction between (A + GA) and (B + GB) is identically zero. And stipulate that for times t$t0, the Hamiltonian of interaction between (A+B+GB) and the rest of the world is zero as well. And allow the Hamiltonian of interaction between GA and the rest of the world, at any time, be anything you like2.
And what we now want to know, what the question at issue now boils down to, is whether (A+B) can somehow be fashioned into a contraption whereby the position of GB at t2 can be made to counterfactually depend, in some knowable and particular way, in any knowable and particular way, on the outcome of a free selection among the possible positions of GA at t1, where t0 < t1 < t2. And note it will suffice to show that (A+B) can not be fashioned into a contraption like that - and this is precisely how the demonstration is in fact going to get done - if it can be shown that it is a general law of whatever particular fundamental theory of the world is under consideration, a law that makes no reference at all to the nature or the constitution or even the existence of A and B, that there can be no statistical correlations at all - under the circumstances described in the previous paragraph - between the position of GB at t2 and the outcome of a free selection among the possible positions of GA at t1.
Lets start with the GRW theory.
On GRW, everything there is to say about the physical situation of the world will supervene on a single universal wave-function. And the rule for reading the positions of the centers-of-masses of golf-balls off of that wave-function is presumably going to run something like this: the center-of-mass of a golf-ball G is located in the region Z if and only if
I(Z) Tr[[x><x]G]dx ( 1, (1)
where G is the reduced density matrix of the center of mass of the golf-ball in question, and the integral (as indicated) is evaluated over the region Z.3 And note that a rule like that, together with the fundamental laws of the GRW theory, is going to count the positions of golf-balls as signifigables - note (that is) that a rule like that, together with the fundamental laws of the GRW theory, is going to endorse our everyday conviction to the effect that the positions of golf-balls are the sorts of things that we can both observe and manipulate.
Now, it happens that we have a proof from Bell himself that the GRW theory, together with the above-mentioned restrictions on the Hamiltonians of interaction between the various systems in our example, entails that the probabilities of spontaneous localizations in GB at and around t2 will be completely independent of the strength and the direction and the time-dependence and the physical character of whatever external fields GA may happen to encounter at and around t1. And note that on the GRW theory, the probabilities of the various possible positions of GB at and around t2 will be fully determined by the probabilities of spontaneous localizations in GB at and around t2. And so it follows that the probabilities of the various possible positions of GB at and around t2 will be completely independent - on the GRW theory - of whatever external fields GA may happen to encounter at and around t1. And the imposition of external fields can of course be put to work as a means of moving GA around, the imposition of external fields can of course be put to work as a means of selecting a position for GA, at and around t1. And so it is a theorem of GRW, given the above-mentioned restrictions on the Hamiltonians of interaction between the various systems in our example, that the position GB at and around t2 is statistically independent of the outcome of a free selection among the possible positions of GA at and around t1. And so the sort of quantum-mechanical non-locality that comes up in the GRW theory can be of no help at all with the project of sending messages outside of the ordinary Hamiltonian-governed dynamical channels. And thats that.
The case of Bohms Theory - which threatens on the face of it to be a much more complicated business, which is much more the sort of theory that Bell seems to have been worrying about in the passage quoted above - is (as a matter of fact) just as easy.
The first thing to do is to identify some Bohmian-Mechanical signifigable of the world which has to do with the positions of golf-balls. And in the case of Bohmian Mechanics - as opposed to that of the GRW theory - there will be more than one at least superficially plausible candidate on offer.4 And the beauty of this technique (remember) is precisely that one such signifigable is all we need, and that any such signifigable, that whatever such signifigable we are otherwise in the mood for, that whatever such signifigable we happen to stumble across first, will do.
Say (then) that the center-of-mass of a golf-ball G is effectively located in the region Z if and only if
I(Z) Tr[[x><x]( EG)]dx . 1, (2)
where EG is the effective reduced density matrix of the center of mass of the golf-ball in question, which is calculated not from the total wave-function of the world at the time in question, but from the Bohmian-Mechanical effective wave-function of the world at the time in question. And note that (as required) the laws of Bohmian Mechanics are indeed going to entail that the effective positions of the centers-of-masses of golf-balls are the sorts of things that we can routinely observe and manipulate.
Now, it is an elementary exercise in quantum mechanics to show that if the wave-function of the world invariably evolves in accord with the linear dynamical equations of motion, then the above-mentioned restrictions on the Hamiltonians of interaction between the various systems in our example will entail that the reduced density matrix of GB at and around t2 will be completely independent of the strength and the direction and the time-dependence and the physical character of whatever external fields GA may happen to encounter at and around t1. And it follows almost immediately (since effective wave-functions, in so far and for so long as they are effective wave-functions - evolve in accord with precisely the same dynamical laws as wave-functions simpliciter do) that those same restrictions on the Hamiltonians of interaction between the various systems in our example will entail that the Bohmian-Mechanical effective reduced density matrix of GB at and around t2 will be completely independent of the strength and the direction and the time-dependence and the physical character of whatever external fields GA may happen to encounter at and around t1. And note that the probabilities of various different effective positions of GB at and around t2 - on Bohms theory - can be read off of the effective reduced density matrix of GB at and around t2. And so it follows that the probabilities of the various possible positions of GB at and around t2 will be completely independent - on Bohms theory - of whatever external fields GA may happen to encounter at and around t1. And the imposition of external fields can (again) be put to work as a means of selecting a position for GA, at and around t1. And so it is a theorem of Bohmian Mechanics, given the above-mentioned restrictions on the Hamiltonians of interaction between the various systems in our example, that the position GB at and around t2 is statistically independent of the outcome of a free selection among the possible positions of GA at and around t1. And so the sort of quantum-mechanical non-locality that comes up in Bohms theory can be of no help at all with the project of sending messages outside of the ordinary Hamiltonian-governed dynamical channels. And that (again) is that.
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But these last two arguments - the one about GRW and the one about Bohm - go by very quickly. And it will be worth taking a minute to slow them down, and to compare them with one another, and to consider precisely what it is that they accomplish.
Both of these arguments involve demonstrations that the probabilities of the various possible positions of GB at and around t2 will be completely independent - given the restrictions on the Hamiltonians of interaction between the various systems in the examples we were considering - of whatever external fields GA may happen to encounter at and around t1. And both of those demonstrations follow precisely the course marked out by Bell in the passage I quoted at the beginning of this paper. And so it is worth emphasizing that the use that this demonstration was put to in Bells argument is altogether different than the use it is being put to here.
Bells intuition was that the entirety of what we can do by way of controlling physical conditions in space-time region 1 will be exhausted by what can be done by means of imposing external fields. And if that intuition is right, and if it can be demonstrated that the values of all observables of space-time region 2 are completely independent of the strength and the direction and the time-dependence and the physical character of whatever external fields may be at work in space-time region 1, then it will follow that we cannot send a message from space-time region 1 to space-time region 2.
The trouble - as Bell himself seems to have understood - is that there is no clear argument for this intuition. And that is precisely the trouble that this paper means to cure.
All thats being supposed about external fields in the arguments we have been considering here is that external fields can, in principle, be put to work moving golf-balls around. If they can be put to that sort of work, and if it can be demonstrated that the probabilities of the various possible positions of GB at and around t2 are completely independent of whatever external fields GA may happen to encounter at and around t1, then it follows that the position GB at and around t2 is statistically independent of the outcome of a free selection among the possible positions of GA at and around t1. And it will follow from that - by means of a straightforward application of the technique of signifigables - that messages cannot be transmitted from region 1 to region 2.
The Bohman-Mechanical argument makes essential use of the language of effective wave-functions, and the relationship between the language of effective wave-functions and the language of the exact fundamental Bohmian-mechanical condition of the world is vague, and yet the impossibility of sending messages that that argument is designed to establish is supposed be exact and principled and rigorous and universal. And it will be worth thinking for a minute about how all that can possibly hang together.
The thing to keep in mind is that the language of effective wave-functions is the only Bohmian-mechanical language in which the discourse of observes and experiments and measuring-instruments can be carried on. The thing to keep in mind (to put it another way) is that the language of effective wave-functions is the only Bohmian-mechanical language in which the question of sending messages can even be brought up. Its the question (then), and not the business of answering it, that requires us to speak an inexact language. And the sort of thing one wants to say is that in precisely so far as the question itself can even be raised, the sort of thing one wants to say is that relative to precisely that standard of accuracy which the question itself imposes on us, it can indeed be given an exact and rigorous and universal answer.5
In both of these demonstrations, what gets shown is that (A+B) can not be fashioned into a contraption whereby the position of GB at t2 can be made to counterfactually depend, in some knowable and particular way, in any knowable and particular way, on a free selection among the possible positions of GA at t1. In each case, what gets shown (more particularly) is that it would amount to a direct violation of the laws of the fundamental theory of the world in question for (A+B) to be fashioned into a contraption whereby the position of GB at t2 can be made to counterfactually depend, in some knowable and particular way, in any knowable and particular way, on the outcome of a free selection among the possible positions of GA at t1.
But it turns out that these two demonstrations differ a great deal from one another - notwithstanding the above-mentioned superficial structural appearances to the contrary - just underneath the surface. In the case of the GRW theory, what gets shown is that there is no physically possible microscopic state of (A+B) on which there is any counterfactual dependence whatsoever, knowable or otherwise, between the position of GB at t2 and the outcome of a free selection among the possible positions of GA at t1. But that manifestly can not be whats going on in the Bohmian case. Its a well-known consequence of Bohmian Mechanics, it is of the very essence of the sort of non-locality that comes up in Bohmian Mechanics, after all, that there are physically possible states of systems like (A+B) on which there is a very definite sort of counterfactual dependence between the position of GB at t2 and the outcome of a free selection among the possible positions of GA at t1.
Suppose (for example) that A and B each contain, among other things, an electron - call them electron 1 and electron 2, respectively. And suppose that the Bohmain-Mechanical effective wave-function of electrons 1 and 2 at t0 is
12 = {(1/o2)([8x>1[9x>2 - [9x>1[8x>2)}Ca(x1, y1, z1)Cb(x2, y2, z2);
where {(1/o2)([8x>1[9x>2 - [9x>1[8x>2)} is the spin-space singlet, and *a(x1, y1, z1)* is uniform throughout region a (in figure 1) and vanishes elsewhere, and *b(x2, y2, z2)* is uniform throughout region b, and vanishes elsewhere.
It turns out (see, for example, my Quantum Mechanics and Experience pages 158-160) that every physically possible fully specified Bohmain-mechanical state of affairs of a pair of electrons which is compatible with 12 will instantiate one or another of four perfectly definite sets of counterfactual dependencies between the orientation of a Stern-Gerlach magnet in region 1 and the position of the electron in region 2. Which one of those four dependencies actually obtains will be determined by the exact position of the first of the above-mentioned two electrons within region a and by the exact position of the second of those two electrons within region b. In particular, there will be some division of region a into (say) sub-regions a+ and a-, and some division of region b into sub-regions b+ and b- - as shown in figure 2 - such that the first of those four counterfactual dependencies will obtain if the effective wave-function of the two-electron system is 12 and the first electron is in region a+ and the second electron is in region b+, and the second of those four counterfactual dependencies will obtain if the effective wave-function of the two-electron system is 12 and the first electron is in region a+ and the second electron is in region b-, and so on. And any one of t5wq s
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h670Jjh670JUhD\hD\5CJ\aJhD\CJH*aJUhD\>*CJaJhD\CJH*aJhD\CJaJBhose dependencies can of course straightforwardly be parlayed into an analogous dependency between the effective position of GA at t1 and the outcome of a free selection among the possible effective positions of GB at t2.
But it is a theorem of Bohmian Mechanics that the frequencies of the four above-mentioned complete Bohmian-mechanical states of affairs over the course of any typical history of the world will be roughly equal to one another; and it is a theorem of Bohmian mechanics that the order in which occurrences of one or another those four complete Bohmian-mechanical states of affairs succeed one another in time, over the course of any typical history of the world, will satisfy every well-motivated criterion of randomness; and it is a theorem of Bohmian Mechanics that there will be no statistical correlations whatever, over the course of any typical history of the world, between occurrences of any particular one of those four complete Bohmian-mechanical states of affairs and the effective positions of any particular set of golf-balls; and so on. And it is one of the difficult and beautiful lessons of the work of Shelly Goldstein and his collaborators on Quantum Equilibrium and the Origin of Absolute Uncertainty that all of this, and the way all of it hangs so elegantly together, is just what it means to say that it is a law of Bohmain mechanics that the probability of this or that particular effective position of GB at and around t2 is completely independent of the effective position of GA at and around t1.6
David Z Albert
Philosophy
Columbia University
FOOTNOTES
PAGE
PAGE 1
1. The argument that follows (then) is not going to apply to those versions of quantum mechanics - versions like Bohms theory - which solve the measurement problem by adding extra variables to the standard wave-functional description of physical systems. Mind you - a conclusion analogous to the one about to be argued for here can be argued for in Bohms theory as well - but that latter argument will need to be a good deal more elaborate than the one I am about to describe.
2. In this sentence, and in the previous one, the rest of the world refers to the compliment of (A+B+GA+GB).
3. Of course, the positions of the golf-balls are not the sorts of things that we expect to supervene on the wave-function directly - the positions of golf-balls are (rather) the sorts of things that we expect to supervene on the wave-function via something more fundamental, something along the lines of the positions of elementary particles. And the rule that we have just now written down for reading the positions of golf-balls off of the wave-function is presumably going to follow - if everything works out right - from the more fundamental rule for reading the positions of elementary particles off of the wave-function, together with the appropriate sort of conceptual analysis of what it is to be a golf-ball.
4. One can imagine candidates for such a significable, for example, which (unlike the significable we are about to settle on here) involve only the positions of the Bohmian particles, and make no reference to wave-functions at all.
5. The same sort of worry is going to arise, by the way, and the same sorts of considerations are going to apply, in connection with the vagueness of the . sign in equations 1 and 2.
6. It is precisely here, by the way, that the difference between the impossibility of exploiting the non-locality of something like Bohmian Mechanics for the purposes of transmitting readable information between space-like sep@AKLMNOPYZ[fghwwh]hgd67&`#$gd677$
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