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The Strong Free Will Theorem John H. Conway and Simon Kochen T he two theories that revolutionized fortunate because these theoretical notions have physics in the twentieth century, rela- led to much confusion. For instance, it is often said tivity and quantum mechanics, are full that the probabilities of events at one location can of predictions that defy common sense. be instantaneously changed by happenings at an- Recently, we used three such para- other space-like separated location, but whether doxical ideas to prove “The Free Will Theorem” that is true or even meaningful is irrelevant to (strengthened here), which is the culmination of our proof, which never refers to the notion of a series of theorems about quantum mechanics probability. that began in the 1960s. It asserts, roughly, that if For readers of the original version [1] of our indeed we humans have free will, then elementary theorem, we note that we have strengthened it particles already have their own small share of by replacing the axiom FIN together with the as- this valuable commodity. More precisely, if the sumption of the experimenters’ free choice and experimenter can freely choose the directions temporal causality by a single weaker axiom MIN. in which to orient his apparatus in a certain The earlier axiom FIN of [1], that there is a ﬁnite measurement, then the particle’s response (to upper bound to the speed with which informa- be pedantic—the universe’s response near the tion can be transmitted, has been objected to by particle) is not determined by the entire previous history of the universe. several authors. Bassi and Ghirardi asked in [3]: Our argument combines the well-known conse- what precisely is “information”, and do the “hits” quence of relativity theory, that the time order of and “ﬂashes” of GRW theories (discussed in the space-like separated events is not absolute, with Appendix) count as information? Why cannot hits the EPR paradox discovered by Einstein, Podolsky, be transmitted instantaneously, but not count as and Rosen in 1935, and the Kochen-Specker Para- signals? These objections miss the point. The only dox of 1967 (See [2].) We follow Bohm in using a information to which we applied FIN is the choice spin version of EPR and Peres in using his set of 33 made by the experimenter and the response of directions, rather than the original conﬁguration the particle, as signaled by the orientation of the used by Kochen and Specker. More contentiously, apparatus and the spot on the screen. The speed the argument also involves the notion of free will, of transmission of any other information is irrel- but we postpone further discussion of this to the evant to our argument. The replacement of FIN last section of the article. by MIN has made this fact explicit. The theorem Note that our proof does not mention “probabil- has been further strengthened by allowing the ities” or the “states” that determine them, which is particles’ responses to depend on past half-spaces rather than just the past light cones of [1]. John H. Conway is professor of mathematics at Princeton University. His email address is jhorcon@yahoo.com. Simon Kochen is professor of mathematics at Prince- The Axioms ton University. His email address is kochen@math. We now present and discuss the three axioms on princeton.edu. which the theorem rests. 226 Notices of the AMS Volume 56, Number 2 quantum mechanics, but only two of its testable the choice an experimenter makes is not a func- consequences, namely this axiom SPIN and the tion of the past. We explicitly use only some very axiom TWIN of the next section. special cases of these assumptions in justifying It is true that these two axioms deal only with our ﬁnal axiom. idealized forms of experimentally veriﬁable pre- The MIN Axiom: Assume that the experiments dictions, since they refer to exact orthogonal and performed by A and B are space-like separated. parallel directions in space. However, as we have Then experimenter B can freely choose any one of shown in [1], the theorem is robust, in that approx- the 33 particular directions w , and a’s response imate forms of these axioms still lead to a similar is independent of this choice. Similarly and inde- conclusion. At the same time, this shows that any pendently, A can freely choose any one of the 40 more accurate modiﬁcations of special relativity triples x, y, z, and b’s response is independent of (such as general relativity) and of quantum theory that choice. will not aﬀect the conclusions of the theorem. It is the experimenters’ free will that allows the free and independent choices of x, y, z, and w . But (ii) The TWIN Axiom and the EPR Paradox in one inertial frame—call it the “A-ﬁrst” frame— One of the most curious facts about quantum B’s experiment will only happen some time later mechanics was pointed out by Einstein, Podolsky, than A’s, and so a’s response cannot, by temporal and Rosen in 1935. This says that even though the causality, be aﬀected by B’s later choice of w . In a results of certain remotely separated observations B-ﬁrst frame, the situation is reversed, justifying cannot be individually predicted ahead of time, the ﬁnal part of MIN. (We shall discuss the mean- they can be correlated. ing of the term “independent” more fully in the In particular, it is possible to produce a pair of Appendix.) “twinned” spin 1 particles (by putting them into the “singleton state” of total spin zero) that will The (Strong) Free Will Theorem give the same answers to the above squared spin Our theorem is a strengthened form of the original measurements in parallel directions. Our “TWIN” version of [1]. Before stating it, we make our terms axiom is part of this assertion. more precise. We use the words “properties”, The TWIN Axiom: For twinned spin 1 particles, “events”, and “information” almost interchange- suppose experimenter A performs a triple exper- ably: whether an event has happened is a property, iment of measuring the squared spin component and whether a property obtains can be coded by of particle a in three orthogonal directions x, y, z, an information-bit. The exact general meaning of while experimenter B measures the twinned par- these terms, which may vary with some theory ticle b in one direction, w . Then if w happens to that may be considered, is not important, since we be in the same direction as one of x, y, z, experi- only use them in the speciﬁc context of our three menter B’s measurement will necessarily yield the axioms. same answer as the corresponding measurement To say that A’s choice of x, y, z is free means by A. more precisely that it is not determined by (i.e., In fact we will restrict w to be one of the 33 di- is not a function of) what has happened at earlier rections in the Peres conﬁguration of the previous times (in any inertial frame). Our theorem is the section, and x, y, z to be one of 40 particular or- surprising consequence that particle a’s response thogonal triples, namely the 16 such triples of that must be free in exactly the same sense, that it is conﬁguration and the 24 further triples obtained not a function of what has happened earlier (with by completing its remaining orthogonal pairs. respect to any inertial frame). (iii) The MIN Axiom, Relativity, and Free Will The Free Will Theorem. The axioms SPIN, TWIN One of the paradoxes introduced by relativity and MIN imply that the response of a spin 1 parti- was the fact that temporal order depends on the cle to a triple experiment is free—that is to say, is choice of inertial frame. If two events are space- not a function of properties of that part of the uni- like separated, then they will appear in one time verse that is earlier than this response with respect order with respect to some inertial frames, but to any given inertial frame. in the reverse order with respect to others. The Proof. We suppose to the contrary—this is the two events we use will be the above twinned spin “functional hypothesis” of [1]—that particle a’s measurements. response (i, j, k) to the triple experiment with It is usual tacitly to assume the temporal causal- directions x, y, z is given by a function of proper- ity principle that the future cannot alter the past. ties α, . . . that are earlier than this response with Its relativistic form is that an event cannot be inﬂu- respect to some inertial frame F. We write this as enced by what happens later in any given inertial F frame. Another customarily tacit assumption is θa (α) = one of (0, 1, 1), (1, 0, 1), (1, 1, 0) that experimenters are free to choose between (in which only a typical one of the properties α is possible experiments. To be precise, we mean that indicated). 228 Notices of the AMS Volume 56, Number 2 Similarly we suppose that b’s response 0 or 1 We now deﬁne for the direction w is given by a function G G θ0 (w ) = θb (x, y, z, w ; β0 ) , G θb (β) = one of 0 or 1 noting that since by MIN the response of b cannot G of properties β, . . . that are earlier with respect to vary with x, y, z, θ0 is a function just of w . a possibly diﬀerent inertial frame G. Similarly there is a value α0 of α′ for which the F (i) If either one of these functions, say θa , is function F F inﬂuenced by some information that is free in the θ1 (x, y, z) = θa (x, y, z, w ; α0 ) above sense (i.e., not a function of A’s choice of is deﬁned for all 40 triples x, y, z, and it is al- directions and events F-earlier than that choice), so independent of w , which argument we have then there must be an an earliest (“inﬁmum”) therefore omitted. F-time t0 after which all such information is avail- But now by TWIN we have the equation able to a. Since the non-free information is also F G G G available at t0 , all these information bits, free and θ1 (x, y, z) = (θ0 (x), θ0 (y), θ0 (z)) . non-free, must have a value 0 or 1 to enter as However, since by SPIN the value of the left-hand F arguments in the function θa . So we regard a’s side is one of (0, 1, 1), (1, 0, 1), (1, 1, 0), this shows G response as having started at t0 . that θ0 is a 101 function, which the Kochen- If indeed, there is any free bit that inﬂuences Specker paradox shows does not exist. This com- a, the universe has by deﬁnition taken a free pletes the proof. decision near a by time t0 , and we remove the pedantry by ascribing this decision to particle a. Locating the Response (This is discussed more fully in the section “Free We now provide a fuller discussion of some delicate Will Versus Determinism”.) points. (ii) From now on we can suppose that no such (i) Since the observed spot on the screen is the new information bits inﬂuence the particles’ re- result of a cascade of slightly earlier events, it sponses, and therefore that α and β are functions is hard to deﬁne just when “the response” really of the respective experimenters’ choices and of starts. We shall now explain why one can regard events earlier than those choices. a’s response (say) as having already started at any Now an α can be expected to vary with x, y, z time after A’s choice when all the free information and may or may not vary with w . However, whether bits that inﬂuence it have become available to a. the function varies with them or not, we can intro- Let N(a) and N(b) be convex regions of space- duce all of x, y, z, w as new arguments and rewrite time that are just big enough to be “neighborhoods F θa as a new function (which for convenience we of the respective experiments”, by which we mean give the same name) that they contain the chosen settings of the ap- F paratus and the appropriate particle’s responses. θa (x, y, z, w ; α′ ) (⋆) Our proof has shown that if the backward half- of x, y, z, w and properties α independent of ′ space t < tF determined by a given F-time tF is x, y, z, w . disjoint from N(a), then the available informa- To see this, replace any α that does depend tion it contains is not enough to determine a’s on x, y, z, w by the constant values α1 , . . . , α1320 it response. On the other hand, if each of the two takes for the 40 × 33 = 1320 particular quadru- such half-spaces contains the respective neigh- ples x, y, z, w we shall use. Alternatively, if each borhood, then of course they already contain the α is some function α(x, y, z, w ) of x, y, z, w , we responses. By varying F and G, this suﬃces to lo- may substitute these functions in (⋆) to obtain cate the free decisions to the two neighborhoods, information bits independent of x, y, z, w . which justiﬁes our ascribing it to the particles G themselves. Similarly, we can rewrite θb as a function G θb (x, y, z, w ; β′ ) (ii) We remark that not all the information in the G-backward half-space (say) need be available to of x, y, z, w and properties β′ independent of G b, because MIN prevents particle b’s function θb x, y, z, w . from using experimenter A’s choice of directions Now for the particular choice of w that B will x, y, z. The underlying reason is of course, that rel- make, there is a value β0 for β′ for which ativity allows us to view the situation from a B-ﬁrst G frame, in which A’s choice is made only later than θb (x, y, z, w ; β0 ) b’s response, so that A is still free to choose an is deﬁned. By the above independence of β′ from arbitrary one of the 40 triples. However, this is our G w , the function θb (x, y, z, w ; β0 ) is deﬁned with only use of relativistic invariance—the argument the same value β0 for all 33 values of w . (The fact actually allows any information that does not re- that MIN allows B to freely vary his choice of w veal A’s choice to be transmitted superluminally, makes this intuitively clear.) or even backwards in time. February 2009 Notices of the AMS 229 (iii) Although we’ve precluded the possibility Historically, this kind of correlation was a great G that θb can vary with A’s choice of directions, it is surprise, which many authors have tried to ex- conceivable that it might nevertheless vary with a’s plain away by saying that one particle inﬂuences G (future!) response. However, θb cannot be aﬀected the other. However, as we argue in detail in [1], by a’s response to an unknown triple chosen by the correlation is relativistically invariant, unlike A, since the same information is conveyed by the any such explanation. Our attitude is diﬀerent: responses (0, 1, 1), to (x, y, z), (1, 0, 1) to (z, x, y), following Newton’s famous dictum “Hypotheses and (1, 1, 0) to (y, z, x). For a similar reason θa F non ﬁngo”, we attempt no explanation, but accept cannot use b’s response, since B’s experiment the correlation as a fact of life. might be to investigate some orthogonal triple Some believe that the alternative to determin- u, v, w and discard the responses corresponding ism is randomness, and go on to say that “allowing to u and v. randomness into the world does not really help in understanding free will.” However, this objec- (iv) It might be objected that free will itself tion does not apply to the free responses of the might in some sense be frame-dependent. Howev- particles that we have described. It may well be er, the only instance used in our proof is the choice true that classically stochastic processes such as of directions, which, since it becomes manifest in tossing a (true) coin do not help in explaining free the orientation of some macroscopic apparatus, will, but, as we show in the Appendix and in §10.1 must be the same as seen from arbitrary frames. of [1], adding randomness also does not explain (v) Finally, we note that the new proof involves the quantum mechanical eﬀects described in our four inertial frames—A-ﬁrst, B-ﬁrst, F, and G. theorem. It is precisely the “semi-free” nature of This number cannot be reduced without weak- twinned particles, and more generally of entan- ening our theorem, since we want it to apply to glement, that shows that something very diﬀerent arbitrary frames F and G, including for example from classical stochasticism is at play here. those in which the two experiments are nearly Although the FWT suggests to us that determin- simultaneous. ism is not a viable option, it nevertheless enables us to agree with Einstein that “God does not play Free Will Versus Determinism dice with the Universe.” In the present state of We conclude with brief comments on some of the knowledge, it is certainly beyond our capabilities to understand the connection between the free more philosophical consequences of the Free Will decisions of particles and humans, but the free Theorem (abbreviated to FWT). will of neither of these is accounted for by mere Some readers may object to our use of the randomness. term “free will” to describe the indeterminism of The tension between human free will and phys- particle responses. Our provocative ascription of ical determinism has a long history. Long ago, free will to elementary particles is deliberate, since Lucretius made his otherwise deterministic parti- our theorem asserts that if experimenters have a cles “swerve” unpredictably to allow for free will. certain freedom, then particles have exactly the It was largely the great success of deterministic same kind of freedom. Indeed, it is natural to classical physics that led to the adoption of deter- suppose that this latter freedom is the ultimate minism by so many philosophers and scientists, explanation of our own. particularly those in ﬁelds remote from current The humans who choose x, y, z, and w may of physics. (This remark also applies to “compati- course be replaced by a computer program con- balism”, a now unnecessary attempt to allow for taining a pseudo-random number generator. If we human free will in a deterministic world.) dismiss as ridiculous the idea that the particles Although, as we show in [1], determinism may might be privy to this program, our proof would formally be shown to be consistent, there is no remain valid. However, as we remark in [1], free longer any evidence that supports it, in view of the will would still be needed to choose the random fact that classical physics has been superseded by number generator, since a determined determinist quantum mechanics, a non-deterministic theory. could maintain that this choice was ﬁxed from the The import of the free will theorem is that it is not dawn of time. only current quantum theory, but the world itself We have supposed that the experimenters’ that is non-deterministic, so that no future theory choices of directions from the Peres conﬁguration can return us to a clockwork universe. are totally free and independent. However, the freedom we have deduced for particles is more Appendix. Can There Be a Mechanism for constrained, since it is restricted by the TWIN Wave Function Collapse? axiom. We introduced the term “semi-free” in [1] Granted our three axioms, the FWT shows that to indicate that it is really the pair of particles nature itself is non-deterministic. It follows that that jointly makes a free decision. there can be no correct relativistic deterministic 230 Notices of the AMS Volume 56, Number 2 theory of nature. In particular, no relativistic ver- choices of directions x, y, z, and w .1 There are sion of a hidden variable theory such as Bohm’s 40 × 33 = 1320 possible ﬁelds in question. For well-known theory [4] can exist. each such choice, we have a distribution X(FA , FB ) Moreover, the FWT has the stronger impli- of ﬂashes, i.e., we have diﬀerent distributions cation that there can be no relativistic theory X1 , X2 , . . . , X1320 . Let us be given “in advance” that provides a mechanism for reduction. There all such random sequences, with their diﬀerent are nonlinear extensions of quantum mechanics, weightings as determined by the diﬀerent ﬁelds. which we shall call collectively GRW theories (after Note that for this to be given, nature does not have Ghirardi, Rimini, and Weber, see [5]) that attempt to know in advance the actual free choices FA (i.e., to give such a mechanism. The original theories x, y, z) and FB (i.e., w ) of the experimenters. Once were not relativistic, but some newer versions the choices are made, nature need only refer to make that claim. We shall focus here on Tumul- the relevant random sequence Xk in order to emit ka’s theory rGRWf (see [6]), but our argument the ﬂashes in accord with rGRWf. If we refer to the proof of the FWT, we can see below applies, mutatis mutandis, to other rela- that we are here simply treating the distributions tivistic GRW theories. We disagree with Tumulka’s X(FA , FB ) [= X(x, y, z, w )] in exactly the same way claim in [7] that the FWT does not apply to rGRWf, we treated any other information-bit α that de- for reasons we now examine. pended on x, y, z, w . There we substituted all the (i) As it is presented in [6], rGRWf is not a de- values α1 , . . . , α1320 for α in the response function terministic theory. It includes stochastic “ﬂashes” θa (x, y, z, w ; α). Thus, the functional hypothesis that determine the particles’ responses. However, does apply to rGRWf, as modiﬁed in this way by in [1] we claim that adding randomness, or a the recipe. stochastic element, to a deterministic theory does Tumulka [7] grants that if that is the case, then not help: rGRWf acquires some nasty properties: In some Λ “To see why, let the stochastic element in a frame Λ, “[the ﬂash] fy will entail inﬂuences to putatively relativistic GRW theory be a sequence the past.” Actually, admitting that the function- of random numbers (not all of which need be al hypothesis applies to rGRWf has more dire used by both particles). Although these might consequences—it leads to a contradiction. For if, only be generated as needed, it will plainly make as we just showed, the functional hypothesis ap- no diﬀerence to let them be given in advance. plies to the ﬂashes, and the ﬁrst ﬂashes determine But then the behavior of the particles in such a the particles’ responses, then it also applies to theory would in fact be a function of the informa- these responses, which by the FWT leads to a tion available to them (including this stochastic contradiction. element).” (ii) Another possible objection is that in our Tumulka writes in [7] that this “recipe” does statement of the MIN axiom, the assertion that a’s not apply to rGRWf: response is independent of B’s choice was insuf- “Since the random element in rGRWf is the set ﬁciently precise. Our view is that the statement of ﬂashes, nature should, according to this recipe, must be true whatever precise deﬁnition is given make at the initial time the decision where-when to the term “independent”, because in no inertial ﬂashes will occur, make this decision ‘available’ frame can the past appearance of a macroscopic to every space-time location, and have the ﬂash- spot on a screen depend on a future free decision. es just carry out the pre-determined plan. The It is possible to give a more precise form of MIN problem is that the distribution of the ﬂashes by replacing the phrase “particle b’s response is depends on the external ﬁelds, and thus on the independent of A’s choice” by “if a’s response is free decision of the experimenters. In particular, determined by B’s choice, then its value does not vary with that choice.” However, we actually need the correlation between the ﬂashes in A and those precision only in the presence of the functional in B depends on both external ﬁelds. Thus, to hypothesis, when it takes the mathematical form let the randomness ‘be given in advance’ would F that a’s putative response function θa cannot in make a big diﬀerence indeed, as it would require fact vary with B’s choice. To accept relativity but nature to know in advance the decision of both deny MIN is therefore to suppose that an exper- experimenters, and would thus require the theory imenter can freely make a choice that will alter either to give up freedom or to allow inﬂuences to the past, by changing the location on a screen of the past.” a spot that has already been observed. Thus, he denies that our “functional hypothe- sis”, and so also the FWT, apply to rGRWf. However, 1 This unfortunately makes rGRWf non-predictive—it can we can easily deal with the dependence of the only ﬁnd the ﬂash distribution that “explains” either distribution of ﬂashes on the external ﬁelds FA particle’s behavior when both experimenters’ ﬁelds are and FB , which arise from the two experimenters’ given. February 2009 Notices of the AMS 231 Tumulka claims in [7] that since in the twinning [3] A. Bassi and G. C. Ghirardi, The Conway-Kochen experiment the question of which one of the ﬁrst argument and relativistic GRW models, Found. Phys. ﬂashes at A and B is earlier is frame-dependent, it 37(2) (2007), 169–185. follows that the determination of which ﬂash inﬂu- [4] D. Bohm, Quantum Theory in terms of “hidden” variables, I, Phys. Rev. 85 (1952), 166–193. ences the other is also frame-dependent. However, [5] G. C. Ghirardi, A. Rimini, and T. Weber, Uniﬁed dy- MIN does not deal with ﬂashes or other occult namics for microscopic and macroscopic systems, events, but only with the particles’ responses as Phys. Rev. D34 (1986), 470–491. indicated by macroscopic spots on a screen, and [6] R. Tumulka, arXiv:0711.oo35v1 [math-ph] Octo- these are surely not frame-dependent. ber 31, 2007. In any case, we may avoid any such questions [7] , Comment on “The Free Will Theorem”, Found. about the term “independent” by modifying MIN Phys. 37 (2) (2007), 186–197. to prove a weaker version of the FWT, which nev- ertheless still yields a contradiction for relativistic GRW theories, as follows. Authors’ Note: We thank Eileen Olszewski for MIN′ : In an A-ﬁrst frame, B can freely choose any typesetting the paper and Frank Swenton for the one of the 33 directions w , and a’s prior response graphics. is independent of B’s choice. Similarly, in a B-ﬁrst frame, A can independently freely choose any one of the 40 triples x, y, z, and b’s prior response is independent of A’s choice. To justify MIN′ note that a’s response, signaled by a spot on the screen, has already happened in an A-ﬁrst frame, and cannot be altered by the later free choice of w by B; a similar remark applies to b’s response. In [7], Tumulka apparently accepts this justiﬁcation for MIN′ in rGRWf: “. . . the ﬁrst Mathematicians rarely make it into ﬂash fA does not depend on the ﬁeld FB in a frame the newspaper, much less into fashion in which the points of B are later than those of A.” spreads. But on September 21, 2008, the This weakening of MIN allows us to prove a fashion section of the New York Times weaker form of the FWT: Sunday magazine carried a picture of An- FWT′ : The axioms SPIN, TWIN, and MIN′ imply nalisa Crannell, a professor of mathemat- that there exists an inertial frame such that the re- ics at Franklin & Marshall College (and a sponse of a spin 1 particle to a triple experiment is book reviewer in this issue of the Notices). not a function of properties of that part of the uni- The magazine contacted Crannell saying verse that is earlier than the response with respect that it was doing a photo portfolio about to this frame. academics and had chosen her because This result follows without change from our she had received the Haimo Award from present proof of the FWT by taking F to be an the Mathematical Association of America. A-ﬁrst frame and G a B-ﬁrst frame, and applying “By phone, I learned that the ‘photo port- F MIN′ in place of MIN to eliminate θa ’s dependence folio’ was really about fashion, which was G pretty funny because part of my Haimo on w and θb ’s dependence on x, y, z. Award speech included the price tags of We can now apply FWT′ to show that rGRWf’s Λ the clothes I usually wear (US$1 or less),” ﬁrst ﬂash function (fy of [4]), which determines she said. a’s response, cannot exist, by choosing Λ to be the A crew of eight spent three hours frame named in FWT′ . dressing Crannell and doing her hair and The Free Will Theorem thus shows that any makeup. One of them told Crannell he such theory, even if it involves a stochastic ele- loves mathematics and asked her some ment, must walk the ﬁne line of predicting that for questions about fractals. The picture of certain interactions the wave function collapses to Crannell, which can be found on the Web, some eigenfunction of the Hamiltonian, without does not show the US$2,500 Gucci boots, being able to specify which eigenfunction this is. which Crannell called “pretty darned un- If such a theory exists, the authors have no idea comfortable”. “I was wearing my blue AMS what form it might take. ‘I love math’ bracelet that I’d picked up at MathFest, and the guy admired it so I gave References it to him,” Crannell recalled. “I didn’t get [1] J. Conway and S. Kochen, The Free Will Theorem, to keep any of the clothes they brought, Found. Phys. 36 (2006), 1441–1473. but he got to keep my bracelet!” [2] S. Kochen and E. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17 (1967), 59–88. 232 Notices of the AMS Volume 56, Number 2