The Miraculous Consilience of Quantum Mechanics! Or Is a Realist Explanation Possible?

Transcription

1 The Miraculous Consilience of Quantum Mechanics! Or Is a Realist Eplanation Possible? Malcolm R. Forster Department of Philosophy, University of Wisconsin, Madison March 8, 00. Introduction Philosophy of science aims to identify patterns of inference, patterns of eplanation, and patterns of support and confirmation that are common within a wide variety of sciences (see Lambert and Brittan 987 for a concise introduction). Quantum mechanics presents a challenge to the philosophy of science because it does not look like any other theory from the outside. I intend to argue that () despite appearances, quantum mechanics is an ordinary probabilistic theory, and () that the pattern by which quantum mechanics accommodates and predicts eperimental phenomena is similar to what occurs in other quantitative sciences. Well before the discovery of quantum mechanics, one philosopher of science (William Whewell 858) claimed that scientific induction proceeds in three steps: () the Selection of the Idea, () the Construction of the Conception, and (3) the Determination of the Magnitudes. In curve-fitting, for eample, Whewell (see Butts 989, pp. 3-37) makes is very clear that selection of the idea is the selection of the independent variable () from which we hope to the predict the dependent variable (y), the construction of the conception is determined by the choice of the formula (family of curves), and the determination of magnitudes is what statisticians now refer to as the estimation of the adjustable parameters. Whewell then makes an insightful claim about curve-fitting: If we thus take the whole mass of the facts, and remove the errours of actual observation, by making the curve which epresses the supposed observation regular and smooth, we have the separate facts corrected by their general tendency. We are put in possession, as we have said, of something more true than any fact by itself is. (Whewell, quoted from Butts 989, p. 7.) In other words, the purpose of the curve, or the formula, is to capture the general tendency of the data, and this is something greater than the sum of the observed facts. In another place, he again

2 emphasizes that the introduction of the conception (the formula and the parameters) is the key to scientific innovation: The particular facts are not merely brought together, but there is a New Element added to the combination by the very act of thought by which they are combined. There is a Conception of mind introduced in the general proposition, which did not eist in any of the observed facts.... The pearls are there, but they will not hang together until some one provides the string. (Whewell, quoted from Butts 989, pp. 40, 4.) In the case of curve-fitting, the curve is the string, but Whewell also insists that the same metaphor applies to all instances of scientific induction. Whewell is also clear that science aims at more than the mere colligation of facts (which is his special name for induction ). The further step is what he calls the consilience of inductions. Note that this is inductions in the plural. According to Whewell, the mark of a good theory lies not in the relationship between the theory and its data in a single narrow contet, but in the way it succeeds tying together separate inductions. A good theory is like a tree that puts out runners that grow into new trees, until there is a huge forest of mutually sustaining trees. The tying together can be achieved in either of two ways: (a) The theory accommodates one set of data, and then predicts data of a different kind. (b) The theory accommodates two kinds of data separately, and then finds that the magnitudes in the separate inductions agree, or that the laws that hold in each case jump together. Whewell was a renowned historian of science who documented instances of both these kinds, and in light of their logical similarity, describes both kinds of cases as leading to a consilience of inductions. There is no curve-fitting in the quantum mechanical eamples that I describe, but there is an equally clear distinction between accommodation and prediction: First, inferences are made about the properties of quantum mechanical observables required for the model to accommodate known eperimental facts, or an inference is made about the quantum mechanical state of systems prepared in the same way. Then the theory applies these theoretically described facts to new eperimental contets in order to make testable predictions. My purpose is not to argue for some historical or eegetical thesis about how Whewell s notion of consilience should be understood, but to use (and adapt) Whewell s idea for the purpose of eplaining how probabilistic theories work in general, and how quantum mechanics works in particular.

3 While Whewell was an innovator in applying statistical methods in the study of tidal motions (see Reidy and Forster, forthcoming), and there were important developments in Whewell s time (Porter 986), the primary eamples from the history of science at the time did not conceive of nature in a fundamentally probabilistic way. So, one of the preliminary tasks of this paper is to argue that quantum mechanics is an ordinary probabilistic theory, and therefore, by implication, to describe an eample of how the consilience of inductions works in a probabilistic theory. In particular, I argue that the failure of quantum mechanics to specify the joint distributions of non-commuting observables does not imply that its probabilities are non-classical in any relevant sense (section 3). This conclusion is etended in section 5 to the contet of the GHZ eample (Greenberger, Horne and Zeilinger 989), which is a version of the famous Bell eperiment (Einstein, Podolsky and Rosen 935; Bell 964, 97). I ease into this eample by showing that local hidden variable theories are also probabilistic in the same sense (section 4). Each theory fits the formal definition introduced by Forster and Kieseppä (forthcoming) for the purpose of defining what it means for a probabilistic theory to maimize predictive accuracy (Forster and Sober 994). Of course, there is a radical difference between hidden variable theories and quantum mechanics. However, this difference, I contend, arises from the way that the theories cover the phenomena, or equivalently, from the way in which the theories generate their predictive probabilities from an initial stage of accommodation. Or, in other words, the difference arises from the way in which they colligate the facts and attempt to achieve a successful consilience of inductions. Having described the success of quantum mechanics in this way (section 5), I turn to the question of how this success might be eplained in terms of some kind of reality behind the phenomena (section 6). It is so common to hear people talk of realism and hidden variables in the same breath that one gets the impression that hidden variable theories provide the only hope for a realist interpretation of quantum mechanics. One also hears it said that quantum mechanics implies the eistence of superluminal action-at-a-distance. So, perhaps all that is missing is an eplanation of the action-at-a-distance in terms of non-local hidden variables? In section 6, I ask how it is that some (commonly held) interpretations of quantum mechanics are led to this conclusion, while at the same time, they strangely deny that this action-at-adistance can be detected or used for superluminal signaling. Why does the theory postulate a 3

4 distinction that makes no difference? It reminds me of Newton s postulation of absolute space and Leibniz s argument that this was a distinction without a difference. In response, I argue for a local collapse interpretation of quantum mechanics, which says that a particle can only change its state by interacting with a local measuring device. At the same time, there is another non-local feature of quantum mechanics that is a universal amongst all these interpretations (hidden variable theories being the notable eception). This is the feature of quantum mechanics that Teller (986) calls relational holism, which says, roughly, that there are quantum properties of whole systems that fail to supervene on (that is, are not determined by) the non-relational properties of the parts. Relational holism, in my view, is here to stay. While there is no completed realist eplanation of the success of quantum mechanics, it is the function of philosophy to ask the right questions.. Sequential Spin Measurements on an Electron In the quantum mechanical model of electron spin, there is a spin observable corresponding to spin in every direction in 3-dimensional space. This makes no sense for spin in the classical sense, which goes to show that we should not try to read too much into the word spin in quantum mechanics. There are really only two facts about quantum mechanical spin that you need to understand for the purposes of this section. The first is that if an electron passes through a device known as a Stern-Gerlach magnet (which creates a non-linear magnetic field), then the electron will eit the device along one of two possible paths, one called the up path and the other called the down path. If the electron is traveling in the z direction and the Stern-Gerlach magnet is aligned, for eample, in the direction, and the electron is detected in the up path, then in common parlance, one says that the electron has been subjected to a spin measurement in the direction, and the outcome of the measurement is spin-up in the direction. However, the reader should not assume that the electron already had the property of being spin-up in the direction prior to the measurement. That is eactly the assumption that is brought into question by Bell s famous proofs against local hidden variables, for to assume that To avoid a possible confusion about the directions, assume that we straighten the paths of the electron after it passes through the first measurement device so it is always traveling in the z direction. 4

5 the outcome of the measurement was spin-up in the direction because the electron had a corresponding property prior to the measurement is eactly what it means to adopt a hidden variable theory (where the theory is local if we further assume that the property that can only be changed by local interactions). On this illicit, or at least controversial, interpretation of measurement, it is always the function of measurements to reveal the values of hidden variables. It s not that the electron cannot have properties prior to the measurement in the quantum theory. It s just that one should not jump to hasty conclusions about what those properties are. Having given the warning, I will drop the scare quotes from now on. Let s return to the quantum theory. First, spin in direction is called an observable in quantum mechanics and there are two possible outcomes for the measurement of a spin observable, up or down. If an electron eits the first magnet in the up path, then we say that the state of the system (the electron) is prepared with spin up in the direction, where is the direction in which the Stern-Gerlach magnet is aligned. One of the basic postulates of quantum mechanics is that the probabilistic predictions for any subsequent measurement can be inferred from this quantum mechanical state, where the state is represented as a vector in a comple valued vector space, also known as a Hilbert space. Since there are two possible outcomes of any spin measurement, we represent the spin state of an electron as a vector in a -dimensional Hilbert space. Following the Dirac notation, we write this vector as +, where the reminds us that the first magnet was aligned in the direction, and the + reminds us that the electron eited from the magnet in the up path. Since the coordinates of the Hilbert space can be chosen arbitrarily, we may chose to represent this vector as a linear combination of the basis vectors, 0 and 0. In fact, any vector in the Hilbert space can be written as a comple combination (a superposition) of these two vectors. Therefore, 0 c + = c c 0 + = c, where c and c are comple numbers. Corresponding to this Hilbert space of column vectors is the so-called dual space of row vectors, where the basis vectors in the dual space are 5

6 [ 0 ] and [ 0 ]. The column vector + can also be represented in this dual space by a row vector, written in the Dirac notation as +. Contrary to what one might epect, + is not defined as c [ 0] c [ 0 ] +, but by [ 0] [ 0 ] + = c + c, * * where * c is the comple conjugate of the comple number c. That is, if c= a+ ib, where a and b are real, then * c = a ib. Given this convention, there is still a well defined one-to-one mapping between vectors and dual vectors. An important property of comple numbers is that the product of a comple number with its comple conjugate is equal to the squared magnitude (modulus) of the comple number, ( )( ) * * cc= a ib a+ ib = a + b = c = c, which is always a non-negative real number. In fact, this is eactly the reason behind the strange definition of the dual vector, for one can now obtain the squared magnitude of a vector in a Hilbert space by multiplying the vector with its dual: 0 ( c [ 0] c [ 0 ] ) c c cc cc c c * * * * = + + = + + = + =. Dirac calls + + a bra-ket (bracket) and therefore calls + a bra vector and + a ket vector. In an ordinary vector space, the scalar product of two vectors, c c and d d, is defined as the product of the dual of one of the vectors times the other vector, as follows: d c c = cd + c d [ ] d This definition applies to comple vector spaces provided that the dual vector is defined as above. For eample, for two arbitrary ket vectors, ψ c = c d and φ = d, 6.

7 d ψφ = c c = c d + c d. * * * * d In general, this will be a comple number. But note that if we reverse the order in the bra-ket, then we get its comple conjugate: * φψ = ψφ. In the special case in which φ = ψ, ψψ * = ψψ, which confirms that ψψ is always a real number. The fundamental postulate of quantum mechanics is that observables are represented by linear operators that map vectors in the Hilbert space to other vectors in the Hilbert space. An arbitrary linear operator, denoted by Â, in a -dimensional Hilbert space is represented by a matri of the form ˆ a b A = c d, where a, b, c, and d are comple numbers. A second fundamental postulate is that the mean value of an observable  when the system is in a state ψ is defined by Note that from the associativity of matri multiplication, ψ (  ψ ) or as ( ψ Â) E Aˆ Aˆ ψ ψ ψ. () ψ  ψ can be read either as ψ. A third fundamental postulate is that the mean value of an observable in any quantum state must be a real number. This restricts the class of operators that are observables in the following way. First note that, in general, ˆ a b c ac + bc ψ A ψ = c c = c c = c ac + bc + c cc + dc + The third postulate therefore requires that the right hand side is real. But, ( ) ( ) * * * * * * c d c cc dc ( ) ( ) ( ) ( ) * * * * c ac + bc + c cc + dc = a c + b c c + c cc + d c. Since c and c are arbitrary comple numbers, the right hand side can only be real in every case if a and d are real. Furthermore, we require that bcc ( * ) ( * ccc ) + is real. With a little algebraic manipulation, it is possible to see that this is true for arbitrary comple numbers c and b only if c * = b. Therefore, every observable in a -dimensional Hilbert space is represented by a matri of the form. 7

8 ˆ a b A = * b d, where a and d are real. Any matri that has this form is said to be Hermitian or self-adjoint (both of these terms mean the same thing). In sum, the fundamental postulates require that observables are Hermitian operators, for otherwise their mean values would be comple in some quantum states. Not only does a linear Hermitian operator have a well defined mean value in every quantum state, but it also has a well defined dispersion, or variance, in every state (in fact, it has a well defined probability distribution in every state, but this detail does not concern us at the moment). The variance of an observable  is defined analogously to the variance of a random variable in standard statistics namely, as the epected value (mean value) of the operator minus its mean value all squared. So, let α = E Aˆ ψ Aˆ ψ. Then ψ ( Aˆ ) Var Aˆ ψ ψ α ψ, () for any state ψ. Note that it is possible to prove that the operator ( )  α is Hermitian if  is Hermitian. So, the variance of  is a (non-negative) real number in every quantum state. In sum, we have shown that the mean and the variance of every quantum observable is well defined in every quantum state. Now, let us eamine the special case in which an observable  is dispersion-free in a state ψ, where dispersion-free is a physicist s way of saying that the variance is 0. Given an observable Â, how do we characterize the states ψ in which Var ˆ ψ A = 0. In other words, which states ψ satisfy the equation Here is the answer: Var ˆ ψ A = 0? Theorem: Var ˆ ψ A = 0 if and only if  ψ = α ψ for some numberα. Proof: See Khinchin (960), pp , or Appendi A. 8

9 Definition: An equation of the form  ψ = λ ψ is called the eigenvalue equation for the operator Â, and any ψ satisfying this equation is called an eigenvector of Â, and the λ is called the eigenvalue corresponding to that eigenvector. Note that λ is the a mean value of the quantum observable in an eigenstate, so we already know that the eigenvalues of Hermitian operators are always real. The eigenvalue equation is most commonly presented as defining the possible outcomes of the measurement of quantum observable, and it does serves this function. But it is also commonly presented as defining the possible values of the observable itself, as if the observable were are variable in the ordinary mathematical sense. But here is must be remembered that an observable only has a well defined mean value in every state. So, we should be careful to maintain a clear distinction between the observed value and mean value of an observable. Let us return the eample of sequential spin measurements on an electron. The electron has passed through the magnetic field of a Stern-Gerlach magnet oriented in the direction. We now place two more Stern-Gerlach oriented in the direction one in the up path eiting the first magnet, and the other in the down path. Finally, we place particle detectors (Geiger counters, or whatever devices are used to detect electrons) in the up and down paths following second magnets (4 Geiger counters in all). What we find, eperimentally, is that no electrons follow the up path and then the down path, or the down path and then the up path. That is to say, repeated spin measurements always produce the same outcomes. Now suppose that we change the eperimental setup. We remove the second set of magnets and place a particle detector in the down path eiting the first magnet. If we know that a particle is passing through the apparatus, and the particle detector does not respond, then we know that the electron is traveling in the up path. Now we place a second magnet in that path that is oriented in some direction d, and place two particle detectors on the other side to capture the electron in either the up path or the down path. Given that the earlier detector was not triggered, how do quantum mechanics determine the probabilities for two possible responses of the second set of detectors? Suppose we begin by assuming that the electron is in some unknown state ψ. Then we can determine the probabilities from the mean value of the spin observable σ d, where d is the alignment of the second magnet. By the fundamental postulates of quantum mechanics, mean value of σ d is ψσd ψ. If we choose the eigenvalues of any spin 9

10 operator to be + and, then we already know that for d =, ψσ ψ =. But this implies that ψ is a dispersion-free state for the observableσ, and so by the Theorem, ψ is an eigenstate of σ, with eigenvalue +, which implies that ψ + =. So, in order to accommodate the first eperimental fact, we need to assume that the state of the electron in the up path eiting the first magnet is +. So, returning to the second setup, if we assume that the state vector of the particle entering the d-aligned magnet is the same, then we can predict the probabilities for this measurement from the value of σ + + d. This is an eample of how prediction and accommodation works in quantum mechanics. Here is a second consequence of the eigenvalue equation. The identity operator, given by ˆ 0 I = 0 always satisfies the eigenvalue equation with α =. Thus, Î is a dispersion-free observable in every state, with a mean value of ψ Î ψ = ψ ψ. But also note that I ˆ ψ =+ ψ. So, if we want the mean value of the identity observable to be equal to its eigenvalue in a dispersion-free state, then we need to assume that all state vectors are normalized. That is, we need to assume that ψψ = ψ =. The net question is how do we evaluate a quantity like ψσ ψ for an arbitrary state vector ψ? What we need to know (besides the state vector) is the matri representation of the operator σ. We already know that it is represented by some Hermitian operator with the form a b σ = * b d, where a and b are real numbers. In order to take the analysis further, we need to make some conventional choices. We know that we can choose any coordinate system we like for the Hilbert space, so we can choose to represent the eigenvector of the observable σ associated with the eigenvalue that represents spin up as: + = 0. 0

11 Let us also denote the epected value of spin up for σ in state + by α +. Because the outcome is dispersion-free in that state, α + also denotes the observed value of the observable when the electron goes up. The eigenvalue equation now requires that which requires that a = α + and b * a b α + * b d = 0 0, = 0 = b. So, the observable is represented by the matri + α 0 σ =, 0 d for some as yet undetermined real value d. What are the other solutions of the equation α 0 c c? + α 0 d c = c We know one solution, and we can also see that ψ = 0 is a trivial solution of every eigenvalue equation. So, what about other non-trivial solutions? By multiplying the matrices, it is easy to see that the only other solution is given by c = 0, c 0, and d = α, for a second eigenvalue α ( α + + ) We make the last requirement because if it were the case that α = α, then σ would be proportional to the identity operator, and this would imply that σ is dispersion-free in every quantum state, contrary to eperimental fact. Thus, we have shown that the observable is represented by the matri σ + α 0 =, 0 α where α α are two eigenvalues, and the corresponding eigenvectors are + + = 0 and 0 = c. Notice that these two eigenvectors are mutually orthogonal in the precise sense that + = 0 = +. We also want to be a unit vector, but this still does not fi the value of c uniquely. The eact choice of c is harmless, so we set c =. The choice of eigenvalues is also a matter of convention, so long as they are not equal, so we set their values to α + =+ and α =. With these choices, we finally arrive at

12 + σ + 0 = 0, = 0 and 0 =. Given these choices, we can check that + σ + =+ and σ =. We may now consider a second eperimental fact namely that if a spin measurement in the direction is followed by a spin measurement in the y direction, then the results are random (50% probability for each outcome). Given that we have chosen to represent spin outcomes by the numbers ±, the second eperimental fact amounts to saying that if the electron is prepared in a state + or, then the epected value of the observableσ y is 0. That is, we require σ + + y = 0 and σ = 0. y In matri form, these equations are written as a b [ 0] 0 * = b d 0 and they imply that a = 0 = d. Therefore, a b 0 0 = 0 b d, and [ ] * 0 b σ y = * b 0. One cannot epect the constraint to result in a unique specification of σ y, since there are many directions orthogonal to, and each is associated with a different spin observable. On the other hand, it cannot be that b = 0, for this would imply that σ y is the zero matri. So, the simplest choice is to set b = i, in which case we arrive at 0 i σ y = i 0. It is straightforward to show that σ y has eigenvalues equal to ±, and that the corresponding eigenvectors are orthogonal to each other, and that each is a linear combination (superposition) of the eigenvectors of σ. Modulo an arbitrary comple factor with modulus, the eigenvectors of σ y are

13 + + i + i y = + and y =. Note that the observable σ y is only dispersion-free in its eigenstates (by the Theorem), so given that its eigenstates are different from the eigenstates of σ, it follows that if the system is forced into an eigenstate of σ, then the observable σ y is not dispersion-free. One could verify that directly by proving that σ ( σ ) + + Var y = y 0 0. So, while the observables σ and σ y have well determined means and variances in every state, there is no state in which they are both dispersion-free. This is the Heisenberg uncertainty principle for spin observables. It is interesting to ask about the correlation between two observables, because this is the point at which quantum observables appear to differ from the random variables familiar to us in classical probability theory. In classical statistics, if X and Y are any two random variables, then the correlation between them can be characterized in terms of their covariance, which is defined by (, ) ( [ ])( [ ]) Cov X Y E X E X Y E Y. (The variance of a single random variable is equal to its self covariance: VarX Cov( X, X ) What this suggests is that we should be able to define the covariance σ and σ y as (, y) ( )( y y ) Cov ψ ψ ψ σ σ ψ σ σ σ σ ψ, =. ) where σ denotes the mean value of σ ψ in the state ψ. However, it turns out that the product observable σσ y is not Hermitian (even though σ and σ y are Hermitian), which means that their covariance is sometimes a comple number. There is no known way to interpret the meaning comple correlations in standard probability theory. Hence, quantum mechanics does not define a joint distribution for every pair of observables. The relevant definitions and theorems behind this limitation are as follows. Definition: The Hermitian conjugate, denoted Â, of a (linear) operator  on a Hilbert space is obtained from its matri representation by transposing the rows and columns of the matri and replacing every element by its comple conjugate. An operator  is Hermitian if and only if ˆ ˆ A= A. 3

14 ˆ ˆ Lemma: For any two operators  and ˆB, ( ) AB B A = ˆ ˆ. (Note the reversal of the ordering.) Definition: Operators  and ˆB commute if and only if AB ˆ ˆ = BA ˆˆ. Theorem: The product of two Hermitian operators is Hermitian if and only if they commute. AB ˆ ˆ = Bˆ A ˆ = BA ˆ ˆ. Proof: Consider two Hermitian operators  and ˆB. Then, by the Lemma, ( ) Therefore, ( AB ˆ ˆ) = AB ˆ ˆ if and only if AB ˆ ˆ = BA ˆˆ. End of proof. The Hermitian operators σ and σ y do not commute, as is verified by multiplying the matrices together in each order: 0 0 b 0 b = * * 0 b 0 b 0, whereas 0 b 0 0 b * * b 0 = 0 b 0. In fact, this shows that σ and σ y anti-commute in the precise sense that σσ y = σσ y. This law will prove to be useful in section 5. So, quantum mechanics does not tell us how the observables σ and σ y are correlated because they are non-commuting observables. However, is this a limitation of quantum mechanics? I would argue that it is not. For it is impossible to measure the observables σ and σ y simultaneously for the simple physical reason that it is impossible to orient a Stern-Gerlach magnet in two directions at the same time. The best that can done is to pass an electron through one magnet and then the other. But quantum mechanics does not need to define the correlations of non-commuting observables in order to predict the probabilities of sequential measurements. 3. Quantum Mechanics as an Ordinary Probabilistic Theory I maintain that the probabilities of quantum mechanics are standard probabilities governed by the normal aioms of probability. Certainly, quantum mechanics generates its probabilities in a different way. But does that mean that the probabilities themselves are different? Fundamentally, the probabilities are probabilities of the possible responses of particle detectors, and the response of a particle detector is represented by a ordinary random variable. To make this point precisely, consider a particular set up for a sequential spin measurement. The state is prepared in an eigenstate of a spin- measurement. So far, there are no probabilities mentioned. The electron can eit in the up or down path. We don t know the relative 4

15 probabilities of these two possibilities, so we just say that there is a miture of electrons in the two states. But still, no probabilities are assigned. Now, place a particle detector in the down path, and no detector in the up path. Now suppose that we know that an electron has just passed through the device, and that the particle does not trigger the detector. Then we know that the electron is in the up path and that it is in the state +. This is an eample of what it takes to force a quantum system in a particular state, and it is only after this stage that the formalism of quantum mechanics produces probabilities. Suppose that we consider two possible measurements an -spin measurement or a y-spin measurement each followed by detectors in the up and down paths and we introduce the variables X and Y to encode the detector responses in each case. The probabilities assigned to the possible values of X and Y are those determined by the mean values of the operatorsσ andσ in the state +. Now, the only basis (that I can see) for arguing that these are not ordinary probabilities is from the fact that they have no joint distribution. I accept the premise. But does it follow that the probabilities are non-classical? I shall argue that the conclusion is non sequitur and false. In ordinary probability theory, it is not required that the joint distribution between any two random variables must be well defined. For if these probability functions pertain to different contets, as they do in the quantum mechanical eample, why must X and Y have a joint distribution? Maybe they can be assigned a joint distribution, but that does not mean that they must have a joint distribution. Consider an everyday eample: There are 3 coins. I toss coin and observe heads up or heads down. Depending on the outcome, I either toss coin or I toss coin 3, but not both. Let X be the random variable whose values are or 0 depending on whether coin lands heads up or heads down, and let Y be the random variable whose values are or 0 depending on whether coin 3 lands heads up or heads down. In each trial of this eperiment either X has a value and Y does not, or Y has a value and X does not. There is nothing etraordinary about this fact. So, why assume that X and Y have a joint distribution just because it is mathematically possible to assign one (for eample, on the assumption that the two variables are probabilistically independent)? Well, one might argue as follows: If X and Y each has a probability distribution simultaneously then each variable must have a value. For, the proposition (X = 0 or X = ) says y 5

16 that X has a value, and so if we are assuming X has a probability distribution, then by the aioms of probability alone, it must be the case that Pr(X = 0 or X = ) =. In the language of probability, this says that X has a value. By the same argument, Y must have a value (in the probabilistic sense) because Pr(Y = 0 or Y = ) =. And this contradicts the earlier assertion that X and Y don t have values at the same time. The correct resolution of the problem is to point out that the two probability functions are conditional on the proposition that the coin in question is actually tossed. So, what we actually have is that Pr (X = 0 or X = ) = and Pr 3 (Y = 0 or Y = ) =, where the first probability statement says that X has a value given that coin is tossed, and the second says that Y has a value given that coin 3 is tossed. So, the argument fails, and there is no inconsistency involved in asserting that X and Y fail to have a joint probability distribution. Certainly, in the case of coin tossing, coins and 3 could be tossed together, so we may need to assign a joint distribution to X and Y in order to predict what would happen in that eperiment. But even here, there is no categorical imperative that says that a classical theory must cover this eperimental situation in order to be bone fide probabilistic theory, nor is there any imperative that says that first two probability functions must be derived from a common joint distribution. In fact, the argument can be taken one step further. In a contet in which X and Y don t both have simultaneous values, it is plainly wrong for any probabilistic theory to assign them a joint distribution. For if X and Y were to have a joint distribution relative to distribution Pr, say, then it would imply that Pr (Y = 0 or Y = ) =. This says that when coin is tossed and coin 3 is not, then coin 3 lands either heads up or heads down. This is absurd. So, in the eample of sequential spin measurements on an electron, let X denote the observable outcome of an interaction with a Stern-Gerlach magnet oriented in the direction, and let Y denote the outcome of an interaction with a Stern-Gerlach magnet oriented in the y direction. Every probabilistic theory, whether classical or non-classical should deny that X and Y have a joint distribution. Quantum mechanics gets it eactly right. What we really need to do in order to be clear up some of the misunderstandings concerning quantum mechanical probabilities is to be very clear about the way in which they are conditional on the eperimental contet. In every token instance, the system has a trajectory defined in the following way. The electron faces a sequence of Stern-Gerlach devices that are placed in the up or down paths of one of the devices prior to it in the sequence, such that there are no 6

17 escape routes (that is, the particle will eventually be detected by a Geiger counter). While this is not the most general kind of eperiment that could be done on an electron, it is sufficient to illustrate the point. Let encode the alignment of the first Stern-Gerlach magnet, while encodes the positions and alignments of the two Stern-Gerlach magnets placed in the subsequent up and down paths along which the particle would be detected if Geiger counters were placed there instead. If a particle detector is actually placed on of these two paths, then we encode that eperimental fact by y, followed by z, which records the response of the detector. Suppose, as a concrete illustration, that after the first device, a Geiger counter is placed in the down path, while a second Stern-Gerlach magnet is placed in the up path with two detectors placed after it. The trajectory would be represented by a sequence (,,,,, ) y z y z, where z can be or 0, depending on whether the first detector is triggered or not, and z would be ( 0,0 ), (, 0) or ( 0, ) depending on whether neither of the second set of detectors responds, or just the top one responds, or just the bottom one responds, respectively. Clearly, if the value of z is, then z has to be ( 0,0 ), since we are assuming that there is only one particle, and it can only be detected in one place. Moreover, we are assuming that the particle must be detected by one of the three detectors, so if z is 0, z must be(, 0) or ( 0, ). Without ambiguity, the sequence (, y, z,, y, z ) can be written as ((, y, z,, y), z ), where i z records the detector response of the detector placed at the positions specified by y i, for i =,. What every probabilistic theory should do in this eample is to assign probabilities to all the possible values of z conditional on the proposition that the eperimental setup is described by (,,,, ) The eperiment contet, (,,,, ) y z y. y z y, implies which quantum observable is being measured, and quantum mechanics generates a classical probability distribution from the mean value of this observable. So, quantum mechanics does eactly what is required. It is beyond the scope of any theory in physics to predict which eperiments devices will be placed where that is up to the eperimenter. So, it is natural that the quantum mechanics only assigns probabilities conditional on a description of the eperimental setup encoded by (,,,, ) y z y. 7

18 The purpose of the net two sections is to etend these ideas to the GHZ eperiment, where three electrons are created by a high energy collision, and fly apart towards measuring devices. This is an eample of an entangled quantum state, which give rise to so many of the perpleing features of quantum mechanics. Let me ease into the eample slowly by introducing the local hidden variable theory of the same eperimental phenomena, so that I can also eplain the sense in which it is an ordinary probabilistic theory. 4. The Hidden Variable Theory of the GHZ Phenomena Three electrons fly apart towards 3 widely separated measuring devices, labeled,, and 3, each containing a Stern-Gerlach magnet that can be aligned in one of two directions, and y. The directions and y are orthogonal to each other and to the direction of motion of the incoming particle (which is traveling in the local z direction). Each device contains two particle detectors, one in the up path and one in the down path, such that if the incoming electron is detected in the up path, the light bulb attached to the device flashes red and if it is detected in the down path, the same light bulb flashes green. The particle is always detected by one or other of the particle detectors, and never both. In common parlance, we say that the electron is spin-up if the light is red and spin-down if the light is green in whichever direction the Stern-Gerlach magnet is aligned. First, consider the following set of eperimental facts: When any of the measurement devices are set to y and the third is set to (that is, for settings y-y-, y--y, or -y-y) then there is always an odd number of red lights flashing in every trial of the eperiment that is, either all 3 red lights flash or one light flashes red and the other two flash green. A local hidden variable theory accommodates these eperimental facts in the following way: Suppose that each particle, after separation from the others, carries with it a set of properties that determine which light bulb will flash for every possible settings of the device it enters. Let us represent the property that the particle approaching device would cause the red bulb to flash if the device were set to position by X = +, and the property that the green bulb would flash were device set to position by X =. According to the hidden variable story, the particle has the property X = + or the property X =, but not both. 8

19 Note that while the value of the variable determines the eperimental outcome, the variable is not being defined as representing the observable outcome (which is why the X here is very different from the X in the previous section). The eistence of these values is postulated by the theory to eplain the observed outcome. However, the theory also assumes that the hidden variables have values when they are not measured, for this will then provide the theory with the power to predict. Similarly, let Y = + and Y = represent the two properties that determine the outcome when the measuring device is set at y. So, a particle heading towards device will have eactly one of 4 possible sets of properties, which Mermin (990) refers to as instruction sets : Either {X = +, Y = +}, {X = +, Y = }, {X =, Y = +}, or {X =, Y = }. Note that in a single run of the eperiment, a measurement device cannot be oriented in two directions simultaneously, so we cannot determine all the hidden variable values by direct measurement. For instance, if we see the bulb flash red when device is set to y, then we would only know that the instruction set was either {X = +, Y = +} or {X =, Y = +}. This is why the variables are called hidden variables. The laws or regularities that must hold amongst hidden variables in order for the theory to accommodate the first set of eperimental facts then lead to a prediction. What needs to be accommodated is that fact that the outcome for the third particle must be R if first two outcomes are either R-R or G-G. Otherwise, the outcome for the third particle is G. The theoretical laws that are necessary and sufficient to accommodate this regularity are: X Y Y 3 = +, Y X Y 3 = +, and Y Y X 3 = +. (3) For eample, if the setting is y--y, then the second law tells us that if the outcome for particles and 3 is R-R, then Y = = X, and therefore Y 3 =, and so the outcome for particle 3 will be R. To present this theory as a probabilistic theory, think of an ensemble of repeated instances of this eperiment of the following type: First, assume that the electron triples are prepared in the same way in each case, and denote this description by (here has nothing to do with the hidden variable X, or the directions of any of the Stern-Gerlach magnets). According to the hidden variable theory, this description is incomplete in that it does not, by itself, imply or constrain any of the values of the hidden variables. But this does not prevent the theory from assigning probabilities to the outcomes based on the information that it has. 9

20 The second piece of available information is the alignment of the Stern-Gerlach magnets. We encode this by, where can take on the values -y-y, y--y, y-y-, or some other alignment triple. The value of will be different in different token eperiments, and so each value will define a sub-type within the ensemble. Net, there is the information that electron detectors are placed at all possible eit paths, and this fact is encoded by y. Finally, z encodes the detector responses, which can be any one of eight possible triples ( RRR,, ), ( RRG,, ), (,, ) ( RGG,, ), ( GRR,,, ) ( GRG,, ), ( GGR,, ), or (,, ) encoded by a trajectory (,, ), ( ) values to each of the eight values of RGR, GGG. So, each token of the general type is y z. The function of the theory is to assign probability z conditional on the value of (,, ) y. As we have described the theory so far, the laws in (3) do not uniquely determine what probabilities should be assigned. They only imply that 4 of the 8 possible outcomes have probability 0 (at least when takes on one of the values -y-y, y--y, or y-y-). So, there are many hidden variable theories that satisfy the laws in (3), each of which distributes the probabilities amongst the 4 possible outcomes in a different way. Since the eperimental facts of this eample are such that the 4 cases, ( RRR,, ), ( RGG,, ), ( GRG,, ), (,, ) GGR, and ( GGG,, ), occur with equal frequency (modulo epected sampling errors), then I will assume that the hidden variable accommodates this fact by assigning a probability of ¼ in each case. It should be evident by now that it cannot be required of a probabilistic theory that it assign these probability distributions in advance. It is normal that prediction is preceded by accommodation. However, it is also epected of a good theory that, after a period of accommodation, it will be able to assign probability distributions to eperimental situations that have never been encountered before. Although Bell did not consider this particular eample, it was Bell (964, 97) who first saw that local hidden variable theories of the kind I have described do make predictions. If the laws in (3) are assumed to apply in all eperimental situations (in which the electron triples are prepared in same way, as specified by ), then the following deductive consequence of the laws holds in all situations. First, multiply the 3 equations in (3) together, to obtain (X Y Y 3 )(Y X Y 3 )(Y Y X 3 ) = X X X 3 = +, (4) 0

21 where I have used the fact that Y = Y = Y 3 =. The derived law, X X X 3 = +, now implies that there will also be an odd number of red flashes when all 3 devices are set to --. This prediction is dramatically different from the prediction made by quantum mechanics, which predicts that there must be an even number of red flashes in this contet! The almost universal opinion of physicists is that the hidden variable prediction is wrong! 5. The Consilience of Quantum Mechanics I will now prove that quantum mechanics makes the following prediction: If the number of red flashes in the y-y-, y--y, and -y-y settings is always odd, then with probability, there will be an even number of red flashes in the -- setting. Instead of using variables, quantum mechanics assigns spin observables to each particle in every spin direction. So, for eample, the observable σ replaces the hidden variable X in the hidden variable theory, and the observable σ 3 y replaces the hidden variable Y 3, and so on. The outcome probabilities are then determined from the appropriate observable in the way described in section. For eample, if the system were in a + eigenstate of the observable σ, then we would predict with probability that device will flash red. If the system is in a eigenstate of the same observable, then the green light will flash, and so on. If the system is not in an eigenstate of σ, then the probability that device flashes red still be inferred from the mean value of σ, which is equal to ψσ ψ, where ψ is the quantum state of the system as a whole. All of this is eactly what we should epect from the quantum mechanical treatment of sequential spin measurements (section ). There are 6 spin observables involved in our story: σ, σ y, σ, σ y, σ 3, σ 3 y. The new feature of the GHZ eample is that we can also construct new observables by considering products and sums of the 6 observable. However, not every function of the 6 observables is a Hermitian operator. Just as in section, an operator has to be Hermitian for it to be guaranteed a real mean value in every state, and a product of two Hermitian operators is Hermitian if and only they commute. So, it is important to understand that operators pertaining to different particles always commute. For eample, σ commutes with σ y, and so on. And the product observable (σ y σ ) commutes with σ 3 y, and so on. By using these facts alone, it follows that every product variable, like (σ y σ )σ 3 y, is Hermitian. The product observables that correspond the three random

22 variable products that appear in (3) are (σ σ y σ 3 y ), (σ y σ σ 3 y ), and (σ y σ y σ 3 ). We have just shown that these are quantum observables, which therefore have well defined mean values and variances in every quantum state. Analogously to the hidden variable theory, the mean values of these product operators will determine the correlations amongst the light flashes. The quantum mechanical story begins with the assumption that all the electron triples are prepared in eactly the same quantum state, ψ. The fact that there is always an odd number of red flashes in the settings y-y-, y--y, and -y-y tells us that the three product observables have dispersion-free distributions in the state ψ, which implies that ψ is an eigenstate of all three product observables. So, the first set of facts is accommodated by the supposition that ψ is in a + eigenstate of all of the product observables (σ σ y σ 3 y ), (σ y σ σ 3 y ), and (σ y σ y σ 3 ). Therefore, in quantum mechanics, (3) is replaced by the laws: σ σ y σ 3 y ψ = + ψ, σ y σ 3 σ ψ y = + ψ, and σy σ y σ 3 ψ = + ψ. (6) What predictions can be made from these quantum mechanical laws? By using the anticommutation property of the spin operators, as proved in section, and the fact that any spin operator times itself is equal to the identity operator (as can be verified directly by squaring the matrices derived in section ), and the fact that operators pertaining to different particles commute, tells us that (σ σ y σ 3 y )(σ y σ σ 3 y )(σ y σ y σ 3 ) = (σ σ y σ 3 y )(σ 3 y σ σ y )(σ y σ y σ 3 ) = σ σ y σ σ y σ 3. From here, note that σ (σ y σ )σ y σ 3 =σ ( σ σ y )σ y σ 3 = σ σ (σ y σ y )σ 3 = σ σ σ 3, where the minus sign arises from the anti-commutation law σ y σ = σ σ y. Recall that the anti-commutation law is required in order to accommodate the facts about sequential measurements (section ) But now, from (6), it follows that (σ σ y σ 3 y )(σ y σ σ 3 y )(σ y σ y σ 3 ) ψ = + ψ, and therefore, σ σ σ 3 ψ = ψ. (7)

23 I have just proved what I promised: The only way for the quantum model to accommodate the first set of eperimental facts is to assume that (6) is true, which implies (7), which implies that that there will always be an even number of red flashes when the magnets are set to --. The spin operators of quantum mechanics are projectible in a way that is reminiscent of the idea that Goodman (965) had about projectible predicates, ecept that operators do not function in the same as variables in ordinary logical languages. I will spell out the philosophical consequences in the net section. In the meantime, I want to establish that spin observables are far better projectors than hidden variables in the GHZ eample. At first sight, the quantum account appears to be fleible about the eact probabilities distributions assigned by ψ to the 6 spin observables. However, a deeper analysis of the eample reveals that the ψ is uniquely determined by the quantum laws in (6). This is worth proving in detail because it shows that quantum mechanical accommodation of the first three eperimental results is predictively far stronger than the corresponding accommodation of hidden variables. Because there are 8 possible outcomes, the state vector ψ is a vector in an 8-dimensional Hilbert space,. This space is constructed out of the three -dimensional Hilbert spaces that would be used to represent the states of electrons separately. Let us choose the basis vectors for the first Hilbert space to be + and, etc., where the subscripts keep track of the Hilbert space in question. It is now possible to prove that the eight vector products, ± ± ±, 3 form a natural basis for the 8-dimensional Hilbert space, where the products are formed by generalizing ordinary matri multiplication in a natural way. For instance, = = = = is obtained treating the second column matri in the product as if it where simply a matri with a column vector as its element. By etending the same idea to the product of three - vectors, one may easily confirm that the 8 products ± ± ± 3 are each equal to an eightelement column matri in which eactly one element is equal to and the rest are 0. In each of 3