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Book Review The Beginning of Infinity: Explanations That Transform the World Reviewed by William G. Faris

elements in the interior of an exploding supernova. The Beginning of Infinity: Explanations That Gold, among other elements, is created and then Transform the World distributed throughout the universe. A small David Deutsch fraction winds up on planets like Earth, where it can Viking Penguin, New York, 2011 be mined from rock. The other source is intelligent US$30.00, 496 pages beings who have an explanation of atomic and ISBN: 978-0-670-02275-5 nuclear matter and are able to transmute other metals into gold in a particle accelerator. So gold is created either in extraordinary violent stellar The Cosmic Scheme of Things explosions or through the application of scientific A number of recent popular scientific books treat insight. a wide range of scientific and philosophical topics, In the claim there is a potential ambiguity in the from quantum physics to evolution to the nature use of the word “people”. Deutsch characterizes of explanation. A previous book by the them (p. 416) as “creative, universal explainers”. David Deutsch [2] touched on all these themes. His This might well include residents of other planets, new work, The Beginning of Infinity, is even more which leads to a related point. Most physical effects ambitious. The goal is to tie these topics together diminish with distance. However, Deutsch argues to form a unified world view. The central idea is (p. 275): the revolutionary impact of our ability to make There is only one known phenom- good scientific explanations. enon which, if it ever occurred, In books written by one expects to find would have effects that did not fall awe of the universe and delight that we have come off with distance, and that is the to understand something about it. There might creation of a certain type of knowl- also be an implicit assumption that people are not edge, namely a beginning of infinity. particularly significant in the cosmic scheme of Indeed, knowledge can aim itself things. Deutsch makes the contrary claim (p. 45): at a target, travel vast distances People are significant in the cosmic having scarcely any effect, and then scheme of things. utterly transform the destination. He presents the following example. The atomic Deutsch is a physicist who believes that matter composition of the universe consists mainly of is strictly governed by the laws of physics. However, the light elements hydrogen and helium. Consider he is no reductionist. He believes that atoms are a heavy element such as gold. Where can it be real, but also that abstractions are real. One level found, and where did it come from? There are two of abstraction is knowledge, which he defines as very different sources. One is the transmutation of “information which, when it is physically embodied in a suitable environment, tends to cause itself to William G. Faris is professor of mathematics at the remain so” (p. 130). This includes both knowledge University of Arizona. His email address is faris@math. embedded in DNA and ideas in human brains. arizona.edu. Biological knowledge is nonexplanatory, but hu- DOI: http://dx.doi.org/10.1090/noti821 man knowledge includes explanations that solve

544 Notices of the AMS Volume 59, Number 4 unexpected problems. The gold example shows they are used in his sys- that an explanation can have physical effects just tem. Thus “” as real as those created by an exploding star. is defined as the “miscon- Deutsch’s book is also a manifesto in praise of ception that science cannot the notion of progress. His claim is that a new describe reality, only predict kind of progress began with the Enlightenment. outcomes of observations” In the best case the growth of knowledge may (p. 31). As is clear from this proceed indefinitely in the future without bound. example, definitions often This is the “beginning of infinity” of the title come with a judgment. and of the quotation above. The argument has Deutsch’s main work as a several threads. It begins with a definite position physicist has been on quan- on philosophy of science, centered on an argument tum computing, and it is that a notion of “good explanation” of reality is natural that he would test Reproduced with the permissionLibrary of of the the London School of Economics. the key to progress. It includes a theory of cultural his ideas in the framework : philosopher of evolution, which explains obstacles to progress of quantum theory. This “theory-laden” and the lucky circumstance that swept them away. leads to a problem. Quan- science. There is extensive discussion of the anomalous tum theory is the accepted status of quantum theory, which on some accounts framework for understand- is incompatible with a realistic world view. The ing properties of matter, viewed as consisting of argument connects views on topics as varied as constituents on the molecular, atomic, and sub- mathematical reality, voting systems, aesthetics, atomic scale. It explains such properties of matter and sustainability. The author rejects sustainability as density and strength and conductivity and color. as an aspiration or as a constraint on planning; It also underlies the deeper understanding of the he favors an open-ended journey of creation and chemical bond. It is so pervasive in fundamental exploration. physics that it is difficult to see how to make the smallest modification to it without destroying the Good Explanations whole edifice. The problem is that the formulation of quantum Deutsch attributes recent progress to the discov- theory is abstract and mathematical, and, as usually ery of how to create good explanations. A “good presented, it also has a peculiar dual character. explanation” is defined as an explanation “that There is a law of time evolution given by the is hard to vary while still accounting for what it Schrödinger equation: For every time interval there purports to account for” (p. 30). This philosophy is a transformation U obtained by solving this is inspired by the writings of Karl Popper. In his equation. It maps the initial state of a system to the influential works, Popper argued against empiri- final state of a system. (The notation U indicates a cism, the notion that we derive all our knowledge unitary linear transformation.) from sensory experience. There is no such thing There is another law of time evolution, a random as raw experience; scientific observation is always transformation R that determines the result of theory-laden. Furthermore, there is no principle the measurement. Various possibilities for this of inductive reasoning that says that patterns of transformation may occur, according to certain the past will be repeated in the future. In Popper’s calculated probabilities. The state change under view the path to scientific progress is to make this transformation is called reduction or collapse. conjectures and then test them. Deutsch insists The origin of the transformation R is that that even explanations that are testable are not it describes an intervention from the outside. enough; they should be good explanations in the The system is in a certain state. A physicist sense that he defines. sets up a measuring apparatus and performs the His version of philosophy of science has two experiment. The observed outcome occurs. The complementary strands. One is realism, the notion new state after the experiment is determined by the that there is a physical world about which it particular R that corresponds to the experimental is possible to obtain knowledge. The other is outcome. If the experiment is repeated under fallibilism, the doctrine that there is no sure path identical conditions, then the outcomes vary, but to justify the knowledge that has been obtained at the frequencies are predicted by the calculated any stage. These together support the metaphor probabilities. of a knowledge base of good explanations that is Such instrumentalist accounts have a long and increasing but bounded above by a reality that is complicated history. An early version is sometimes never fully knowable. referred to as the “Copenhagen interpretation”, One useful feature of the book is the explicit since it stems from ideas of Niels Bohr and his definitions of philosophical terms, explaining how school. (See [8] for a critical account.) Deutsch

April 2012 Notices of the AMS 545 will have none of this. For him instrumentalism Deutsch contrasts two ways that a may is defeat; the only acceptable account is realistic. successfully replicate itself. A meme may survive Here is his explicit definition (or dismissal) of as a meme of conformity, because it is never ques- Bohr’s approach (p. 324): tioned. It relies on disabling the recipients’ critical Copenhagen interpretation facilities. When such dominate, the result is Niels Bohr’s combination of instru- a static society in which dangerous dysfunctional mentalism, anthropocentrism and memes are suppressed. The other possibility is studied ambiguity, used to avoid a dynamic society. In such a society memes are understanding quantum theory as replicated in a rapidly changing environment. This being about reality. can take place in a culture of criticism, because If the instrumentalist account is not acceptable, the memes embody new truths. then what are the alternatives? One is to develop The transition from a static society to a dynamic a theory that includes the deterministic U time society depends on an almost accidental shift in evolution but also has additional structure. This how meme transmission is employed. A meme structure should introduce randomness in some cannot be simply copied from behavior. There has way; perhaps the reduction transformation R will to be a mental capacity to infer the idea from the emerge as a consequence. The other way is to behavior. This creative ability developed in service develop a theory with only the deterministic time to the task of replicating memes of conformity. It evolution U. Such a theory must explain why the was hijacked to the task of creating new knowledge. outcomes of experiments on quantum systems Deutsch considers the transition to a society appear random. where there is deliberate creation of new knowledge The final section of this review gives a more to begin with the Enlightenment. There is no precise detailed discussion of these alternatives. It will date, but the founding of the Royal Society in 1660 come as no surprise that Deutsch prefers a realistic is a landmark. There may have been previous version of quantum theory with only the dynamical transitions that did not survive. Deutsch calls such time evolution given by U. This seems the happiest a period a “mini-enlightenment”. He speculates that outcome, but we shall see the price he must pay. two such mini-enlightenments may have occurred. One was at the time of the Athenian advances in Cultural Evolution political freedom and openness to new ideas in Many of the ideas in this book are related to philosophy and science. Another was when Florence evolutionary theory. In biological became a center of in art, accompanied by evolution genes are replicated, advances in science, philosophy, and technology. In and they change through variation both cases the initial spark was extinguished. This and selection. In cultural evolution has major enduring consequences (pp. 220–221): the analog of a gene is a “meme”. The inhabitants of Florence in 1494 (The term was coined by Richard or Athens in 404 BCE could be for- Dawkins in 1976.) Memes are given for concluding that optimism replicated, and they also change just isn’t factually true. For they through variation and selection. know nothing of such things as The copying mechanism is dif- the reach of explanations or the ferent for genes and memes. A power of science or even laws of gene exists in a physical form nature as we understand them, let as DNA that may be copied in- alone the moral and technological tact through several generations progress that was to follow when without ever being expressed as the Enlightenment got under way. behavior. It acts much like a com- At the moment of defeat, it must

Copyright of Max Alexander, STARMUS. puter program. A meme can be have seemed at least plausible to : copied only if it is enacted. In fact, the formerly optimistic Athenians evolutionary a meme has two forms. It may be that the Spartans might be right, theorist. an idea in a brain. This idea may and to the formerly optimistic Flo- provoke a behavioral embodiment rentines that Savonarola might be. of the meme, such as action or body language or Like every other destruction of speech. The idea in the next recipient brain has optimism, whether in a whole civ- to be guessed from the observed behavior. The ilization or in a single individual, successful meme variant is the one that changes these must have been unspeakable the behavior of its holders in such a way as to catastrophes for those who had make itself best at displacing other memes from dared to expect progress. But we the population. should feel more than sympathy

546 Notices of the AMS Volume 59, Number 4 for these people. We should take Quantum Theory it personally. For if any of those From here on this review concentrates on quantum earlier experiments in optimism theory and is more technical. The formulation had succeeded, our species would of quantum theory centers on the wave function, be exploring the stars by now, and a quantity that is difficult to interpret in terms you and I would be immortal. of a realistic world view. In compensation the theory has an elegant mathematical structure. As we have seen, Deutsch’s book combines What follows is a brief review of this structure, various threads to create a vision of continuing followed by a discussion of three possibilities progress in enlightenment and human flourishing. for interpretation. (Deutsch would not agree that It is a unified portrait that explains scientific interpretation is an issue, since he sees only one advance and puts it in a moral framework. The reasonable way to think of the theory.) vision is supported by a great number of assertions, The state of a quantum system at a given some of them extravagant. Take, for instance, his moment in time is described by a complex valued claim that “everything that is not forbidden by laws wave function ψ(x). This depends on the positions of nature is achievable, given the right knowledge” of all the particles in the system. For instance, say (p. 76). Here is a supporting argument (p. 56): that there are N particles (no spin or statistics). The position of particle i at fixed time is described by a ...every putative physical transfor- coordinate xi in three-dimensional space. The wave mation, to be performed in a given function depends on x = (x1,..., xN ), which ranges time with given resources or under over configuration space of dimension 3N. This is any other conditions, is either: already an incredible picture of nature: everything — impossible because it is for- is related to everything else through the wave bidden by laws of nature; function. In particular, the joint probability density or for the system of N particles is proportional to — achievable, given the right (1) ρ(x) = |ψ(x)|2. knowledge. That momentous dichotomy exists It is possible for the different particles to be highly because if there were transforma- correlated, even when they are widely separated in space. tions that technology could never The change in the state over a given time achieve regardless of what knowl- interval is ordinarily given by a transformation edge was brought to bear, then U. This transformation is deterministic, and it is this fact would itself be a testable computed by solving the Schrödinger equation. regularity in nature. But all regulari- This is a complicated linear partial differential ties in nature have explanations, so equation, and much of consists the explanation of that regularity of attempts to solve it for some particular system.1 would itself be a law of nature, or a Since the Schrödinger equation is linear, the consequence of one. And so, again, corresponding transformations U are also linear. everything that is not forbidden by This naturally leads to a more abstract point of laws of nature is achievable, given view. The characteristic feature of vectors is that it the right knowledge. is possible to take linear combinations of vectors to produce new vectors. In particular, if ψ and A passage like this is tough to decipher. 1 ψ are vectors, then the sum ψ + ψ is also a The question of whether to believe every detail 2 1 2 vector. Also, every scalar multiple of a vector is a may be beside the point. A manifesto is not a vector. Since wave functions share these properties, scientific or philosophical treatise; it is an outline there is a convention of calling a wave function a of a world view that one can try on for comfort. vector. This in turn leads to the use of geometrical Some readers will find satisfaction in a systematic view of the world that is compatible with notions 1In some accounts the complete description of the physi- of science and progress. Others may find it too cal state of a quantum system involves more than its wave simple or too optimistic. In the latter case they are function; it brings in additional structure, namely the actual positions of the particles. This leads to the question of how violating yet another of Deutsch’s maxims (p. 212): to specify the time evolution of particle positions. This issue is not treated in orthodox , but Bohmian The Principle of Optimism mechanics and Féynes-Nelson stochastic mechanics each All evils are caused by insufficient knowledge. specify a possible continuous time evolution for particle posi- tions. The reviewer thanks Sheldon Goldstein for comments So much the worse for them. on this topic.

April 2012 Notices of the AMS 547 ψ1 + ψ2 = ψ ψ2(x) BB B  ψ1(x) B B ψ B BM B 1 ψ2 B   B  ψ1 B    B  B  2 2 2  Bkψ1k +kψ2k =kψk =1 B B B B B BB

Figure 1. Example of orthogonal functions. Figure 2. Probability = Pythagoras.

2 analogies. (The fact that the scalars are complex If kψk = 1, then the theorem of Pythagoras numbers is only a minor problem.) has the special form Each pair of vectors ψ1 and ψ2 has a scalar X 2 (6) 1 = kψj k . product hψ1, ψ2i. This too has an analog for wave j functions; the scalar product is defined by the definite integral These give positive numbers that sum to one, just what is needed for probability. In fact, this Z is the standard framework for the interpretation (2) hψ1, ψ2i = ψ1(x)ψ2(x) dx. of quantum mechanics. For simplicity consider The space of all wave functions is regarded as an observable quantity with discrete values, such a vector space of functions, in fact, as a Hilbert as the energy of a bound system. The possible space. Each vector ψ in the Hilbert space has a values are indexed by j. The observable quantity norm (length) kψk which is determined by the determines a decomposition of the Hilbert space 2 usual formula kψk2 = hψ, ψi for the square of the into orthogonal subspaces. The state vector is a norm. In the case of wave functions the explicit vector ψ of unit length. It is expressed as a sum expression is (4) of the projected vectors. Different observable quantities define different decompositions. For Z (3) kψk2 = |ψ(x)|2 dx. each observable quantity there are probabilities given by the terms in (6).

Vectors ψ1 and ψ2 are orthogonal (perpendicu- This geometrical picture of quantum mechanics, lar) if their scalar product has the value zero, that abstract and beautiful, is immensely appealing. is, hψ1, ψ2i = 0. (Figure 1 shows an example of The reviewer is tempted to summarize it in a orthogonal wave functions. In this example ψ1(x) slogan: is even and ψ2(x) is odd, so their product has Probability = Pythagoras integral zero.) With these definitions various no- To see such a surprising and satisfying connection tions of geometry have attractive generalizations. (Figure 2) is to be seduced by perfection. However, Suppose, for instance, that there is a sequence of as we shall see, there are reasons to resist its allure. orthogonal vectors ψj with sum ψ, so

X 2The mathematical structure is an orthogonal direct sum (4) ψ = ψj . H = L H j decomposition j j of the Hilbert space. There are corresponding real numbers λj that are possible values for In this situation the theorem of Pythagoras takes the observable quantity. These data determine a self-adjoint the form operator A whose action on each Hj is multiplication by λj . If ψ is the state vector and ψj is the orthogonal projection 2 X 2 H (5) kψk = kψj k . of ψ onto j , then the expected value of this quantity has P 2 j the elegant expression j λj kψj k = hψ, Aψi.

548 Notices of the AMS Volume 59, Number 4 The trouble begins with a counterexample. of experiments, saying nothing Consider a quantum system. Each possible decom- (else) about reality. This move is position (4) defines probabilities. It is tempting to still popular today, and is known think of these probabilities as describing outcomes to its critics (and even to some associated with this system. This is not a consistent of its proponents) as the ‘shut- view. An early argument to this effect was given up-and-calculate interpretation of by . A more recent example of Lucian quantum theory’. Hardy [5] makes the same point in an even more In an instrumentalist account the relevant convincing way. Something else is needed to select decomposition into orthogonal subspaces is de- a particular decomposition relevant to a given termined by the measurement one performs on situation. See the appendix for a brief account of the system. This leads to the nontrivial problem: Hardy’s argument. how does one characterize measurement? There What is it that determines which decomposition is no universally accepted solution. The notion is relevant? If in a given situation probabilities are of “measurement” is defined with varying degrees to predict frequencies, then which probabilities of precision. In some versions a measurement are to be used, and which are to be discarded? is said to necessarily involve interactions on a Any serious account of quantum mechanics must macroscopic scale. This is in spite of the fact that consider this problem. There are various ways to the macroscopic world is supposed to be made of address it, but in the reviewer’s view they fall into atoms. three general classes: The mathematical formulation ignores most • Instrumentalism of this detail. The particular measurement is • Quantum theory with additional structure specified by an orthogonal decomposition, which • Pure quantum theory then determines a decomposition of the state The instrumentalist account of quantum the- vector ψ as a sum of components ψj as in (4). For ory emphasizes its ability to make experimental each j there is a reduction operator Rj . The effect predictions. This solves the problem, because the of reduction on the state vector ψ is another vector outcomes emerge as a result of the particular ex- χj = Rj ψ. In some accounts it is required that perimental setup. Deutsch summarizes the usual χj = ψj , but this is unnecessarily restrictive. All 2 2 rule for such predictions (p. 307): that is needed is that kχj k = kψj k . The particular j that is used is random; its probability is given With hindsight, we can state the by the squared norm kψ k2. Strictly speaking, the rule of thumb like this: whenever j vector χ is not a state vector, since it does not a measurement is made, all the j have norm one. However, it may be multiplied by histories but one cease to exist. a scalar to give a normalized state vector, which The surviving one is chosen at could be taken as a new state of the system. random, with the probability of In general, a probability describes the statistics each possible outcome being equal of what happens when an experiment is repeated to the total measure of all the many times. But what happens in a particular histories in which that outcome experiment? As always in probability, when the occurs. experiment is performed, a particular outcome He then describes the adoption of the instrumen- value j0 occurs. In this instrumentalist version talist interpretation (p. 307): of quantum mechanics there is an additional At that point, disaster struck. In- postulate: the new reduced wave function is stead of trying to improve and χj0 = Rj0 ψ. No mechanism is given. integrate these two powerful but Another direction is quantum theory with addi- slightly flawed explanatory theo- tional structure. One ingredient might be to try to ries [of Schrödinger and Heisen- model the experimental apparatus along with the berg], and to explain why the system of interest. There can also be an explicit rule of thumb worked, most of mechanism for introducing randomness. Some of the theoretical-physics community the ideas go back to von Neumann; others have retreated rapidly and with remark- been elaborated by various authors. There is no able docility into instrumentalism. agreement on details. Here is a brief account of If the predictions work, they rea- a possible view, taken from [4]. It is not univer- soned, then why worry about the sally accepted; in particular, Deutsch presumably explanation? So they tried to regard would be appalled at the ad hoc introduction of quantum theory as being nothing randomness. but a set of rules of thumb for The complete description involves the system predicting the observed outcomes of interest together with an environment, forming

April 2012 Notices of the AMS 549 what one might call the total system. The system y ψ(x) = ψ (x) + ψ (x) of interest might be a particle, an atom, or a 1 2 molecule. To make the description concrete, it will be called an atomic system. The environment could be an experimental apparatus or, more initial final generally, a larger system of some complexity. ψ(x)φ (y) For brevity call it the apparatus. The combined 0 2 system is described by a wave function (x, y), § ¤ Ψ χ1(x)φ1(y) + χ2(x)φ2(y) where x describes the positions of particles in the § ¤ ¦ ¥ ¦ ¥ 1 atomic system, and y describes the positions of § ¤ particles in the apparatus. Typically there is no ¦ ¥ natural way for the wave function for the combined x system to determine a wave function for the atomic

subsystem. One situation where it is meaningful to have Figure 3. Support of system-apparatus wave a wave function ψ(x) for the atomic subsystem function. is when the total system wave function is of the product form Suppose that there is such (7) (x, y) = ψ(x)φ (y). Ψ0 0 an interaction. Since the transformation is linear, Suppose that this is an initial state that has no it maps Ψ0 to a state Ψ in which the atomic system immediate interaction between the atomic system is coupled to the states of the apparatus. The new and apparatus. This means that the wave function wave function is is nonzero only if the x are so far away from the X i (9) (x, y) = χ (x)φ (y). y that the interaction is negligible. The atoms Ψ j j j j are headed toward the apparatus, but they are not there yet. As long as the combined wave One can think of χj (x) as the result of applying function has this form, the dynamics of the atomic a reduction operator Rj to the state ψ(x). The subsystem is described by the Schrödinger time probability associated with such an atomic wave 2 2 evolution U for the atomic system alone. function is kχj k = kψj k . Now the system is to interact with the apparatus. The apparatus wave functions φj (y) are The starting point is a decomposition of the wave functions of the apparatus configuration function ψ(x) of the atomic system as a sum y = (y1, y2,..., yM ), where M is the number X of particles in the apparatus. When M is very large (8) ψ(x) = ψj (x). it is plausible that the apparatus wave functions j φj (y) with various indices j are macroscopically The measurement itself is accomplished by the different. Such effects have been studied under the unitary dynamics of the combined system. This name quantum decoherence. (See [7] for a recent deterministic transformation must be appropriate survey.) In the present account the condition to the decomposition (8) in the following sense. that the apparatus states are macroscopically It should map each wave func- different is interpreted to mean that the wave tion ψj (x)φ0(y) to a new wave functions φj (y) with different j are supported function χj (x)φj (y) for which on subsets with negligible overlap in apparatus the new environment wave func- configuration space (Figure 3). The corresponding tion φj (y) factor also depends atomic wave functions χj (x) are the result of on j. These wave functions dynamical interaction. Each of them is a candidate φj (y) should be normalized for the result of the reduction process. and form an orthogonal family. In order to have an actual result, additional This implies that the atomic structure is required. This is obtained by introduc- wave function normalization is ing a new dynamics with random outcome. One of preserved, in the sense that the indices is randomly selected. Say that this is 2 2 0 kχj k = kψj k . For such a the j index. Then the corresponding reduced wave transformation to exist, there function of the atomic system is the χj0 (x) given by Courtesy of Mark Everett. must be a physical interac- applying R 0 to ψ(x). This reduction process does Hugh Everett: originator j tion of a suitable type—for not contradict the unitary dynamics for the atomic of many-worlds instance, electric or magnetic— subsystem. The wave function for the atomic sub- quantum theory. between the atomic system and system is not even defined while the atomic system apparatus. and apparatus are interacting; it is only defined

550 Notices of the AMS Volume 59, Number 4 before the interaction and after the interaction is fundamental constants of nature over. Suppose that for a subsequent time interval such as the speed of light or the a decomposition (9) with nonoverlapping φj (y) charge on an electron are different. persists and there is no immediate interaction As with much of what Deutsch writes, the question between the atomic system and apparatus. Then is not so much whether to believe it as whether to over this interval of time the wave function of the march under his banner. atomic subsystem continues to be defined, and the deterministic dynamics U describes its time References evolution. [1] Valia Allori, Sheldon Goldstein, Roderich The reader will notice that the above account is Tumulka, and Nino Zanghì, Many-worlds and complicated and artificial. However, it is at least Schrödinger’s first quantum theory, British Journal consistent. This is because it is a consequence of a for the Philosophy of Science 62 (2011), 1–27. variant of quantum theory [3] that is itself known [2] David Deutsch, : The Science of Parallel Universes—and Its Implications, Allen Lane to be consistent. (Viking Penguin), New York, 1997. The final possibility is pure quantum theory. [3] Detlef Dürr, Sheldon Goldstein, and Nino Zanghì, Almost everyone agrees on the deterministic Quantum equilibrium and the role of operators as ob- dynamics given by the Schrödinger equation. servables in quantum theory, J. Statistical Physics 116 Perhaps this is all that is needed. This idea is the (2004), 959–1055. genesis of the many-worlds theory (see [6] for a [4] William G. Faris, Outline of quantum mechanics, pp. 1–52 in Entropy and the Quantum, edited by Robert recent survey). This theory began with hints by Sims and Daniel Ueltschi, Contemporary Mathematics, Schrödinger himself as early as 1926 (see [1] for 529, American Mathematical Society, Providence, RI, a modern account) and was developed in more 2010. detail by Everett in 1957. The rough idea is that [5] Lucian Hardy, Nonlocality for two particles without when there are macroscopically separated wave inequalities for almost all entangled states, Phys. Rev. Letters 71 (1993), 1665–1668. functions φj (y), each of these describes a world with its own history. We live in and experience only [6] , Jonathan Barrett, Adrian Kent, and David Wallace (editors), Many Worlds? Everett, one such world, but they are all equally real. There Quantum Theory, and Reality, Oxford University Press, is no miraculous reduction of the wave function New York, 2010. and no randomness. The apparent randomness [7] Maximilian Schlosshauer, Decoherence and the is perhaps due to the effect that we experience a Quantum-to-Classical Transition, Springer, Berlin, 2010. history that is typical and hence appears random. [8] David Wick, The Infamous Boundary: Seven Decades Or there may be some other mechanism. of Heresy in Quantum Physics, Copernicus (Springer- Deutsch is an enthusiastic proponent of such Verlag), New York, 1996. a theory. It does everything he wants, avoiding instrumentalism and ad hoc introduction of ran- Appendix: The Hardy Example domness. He refers to the resulting picture of In quantum mechanics each observable defines a physics as the “,” and he is willing to decomposition of the Hilbert space into orthogonal draw the consequences (p. 294): subspaces. Each state vector ψ (length one) is then written as a sum of projections ψ onto ...there exist histories in which any j these subspaces. The probability corresponding given person, alive in our history to j is kψ k2. The Hardy example shows that at any time, is killed soon after- j these probabilities cannot simultaneously predict wards by cancer. There exist other outcomes for all the observables together. histories in which the course of a First consider a single particle. Suppose that for battle, or a war, is changed by such such a particle there are two distinct observable an event, or by a lightning bolt at quantities D and U. Each can have either of two exactly the right place and time, or values, 1 or 0. The event that D = 1 is denoted d, by any of countless other unlikely, and the event that D = 0 is denoted d¯. Similarly, ‘random’ events. the event that U = 1 is denoted u, and the event It is not the case that everything is permitted. that U = 0 is denoted u¯. A great deal of fiction is therefore Now consider two particles, perhaps widely close to a fact somewhere in the separated in space. For the first particle one multiverse. But not all fiction. For can observe either D or U, and for the second instance, there are no histories in particle one can observe either D or U. This gives which my stories of the transporter four possibilities for observation. There are four malfunction are true, because they corresponding decompositions of the state vector require different laws of physics. describing the two particles: Nor are there histories in which the

April 2012 Notices of the AMS 551 MATHEMATICS AT THE N ATIONAL S ECURITY A GENCY

(1) ψ = ψdd + ψdd¯ + ψdd¯ + ψd¯d¯,

Make a calculated difference (2) ψ = ψdu + ψdu¯ + ψdu¯ + ψd¯u¯,

with what you know. (3) ψ = ψud + ψud¯ + ψud¯ + ψu¯d¯,

(4) ψ = ψuu + ψuu¯ + ψuu¯ + ψu¯u¯. Furthermore, Hardy constructs the quantities D and U and the state ψ in such a way that

(5) ψdd ≠ 0,

(6) ψdu¯ = 0,

(7) ψud¯ = 0,

(8) ψuu = 0. For fixed ψ he chooses the parameters that define D and U to satisfy the last three of these equations. He then optimizes the parameters that specify ψ to 2 maximize the probability kψddk . The maximum 1 √ value works out to be 2 (5 5 − 11), which is about 9 percent. Suppose that in a given physical situation the observables all have values. According to the first equation above, the probability of dd is greater than zero. So it is possible that the outcome is d Tackle the coolest problems ever. for the first particle and d for the second particle. You already know that mathematicians like complex challenges. Suppose that this is the case. The second equation But here’s something you may not know. says that d for the first particle implies u for the The National Security Agency is the nation’s largest employer of second particle, and the third equation says that mathematicians. In the beautiful, complex world of mathematics, d for the second particle implies u for the first we identify structure within the chaotic and patterns among particle. It follows that the outcome is u for the first the arbitrary. particle and u for the second particle. However, the fourth equation says that this is impossible. Work with the finest minds, on the most challenging problems, The conclusion is that in a given physical situa- using the world’s most advanced technology. tion the observables cannot all have values. The usual explanation for this is that measurement in KNOWINGMATTERS quantum mechanics does not reveal preexisting values but in effect creates the values. The four Excellent Career Opportunities for Experts in the Following: decompositions correspond to four different ex- periments, and each decomposition provides the n Number Theory n Combinatorics correct probabilities for the result of the corre- n n Probability Theory Linear Algebra sponding experiment. For a given experiment, only n Group Theory >> Plus other opportunities one of the four decompositions is relevant to de- n Finite Field Theory termining what actually happens. The other three decompositions give mathematical probabilities that are not relevant to this context.

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552 Notices of the AMS Volume 59, Number 4

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