Why the Quantum World is Deterministic

Why the Quantum World is Deterministic Kunihisa Morita Graduate School of Human Sciences, Osaka University, JAPAN Abstract In this study, I argue that the quantum world is deterministic if quantum mechanics is complete. At first glance, quantum world seems to be deterninistic, because it cannot always predict measurement value with certainty. However, many interpretations regard quantum mechanics as deterministic. These interpretations only suggest that the quantum mechanical world can be deterministic. I argue that, although quantum mechanics cannot predict the future with certainty, the quantum mechanical world must be deterministic, and the value observed by the observer is determined. I examine the following two cases: (1) the wave-function completely describes the physical state and (2) the wave-function does not describe the physical state. Then, I argue that the quantum world must be deterministic in either case when quantum mechanics is complete. 1. Introduction Roughly speaking, an open future means that the events of the future are not, at present, determined.1 Whether or not the future is open is an important issue, in particular, for the philosophy of time. If the future is really open, then the B-theory, or the static view of time, is false because this view does not accept the existence of the “absolute present.” The absolute present is objective and metaphysically privileged time, and is 1 I will discuss this concept fully below. 1 not relative to the speaker (e.g., Merricks 2006, p. 103; Zimmerman 2008, p. 212; Dowe 2009, p. 642; Olson 2009, p. 3). Suppose that the future is open. The state of the world, Σ, at time t is undetermined when t is in the future, but Σ will become definite when t is the present. This means that the present is a privileged time because the state of the world changes depending on whether t is the future or the present. Therefore, if the future is open, the static view of time is false, because there is an absolute present. On the other hand, the growing block universe theory and presentism, both of which are models of the Atheory (or, the dynamic view of time) which accept the concept of the absolute present, insist that future things are not real. Therefore, while the open future reinforces these models, if the future is not open, their persuasiveness is diminished, although they are not falsified. The moving spotlight theory is another model of the A-theory. However, since this model accepts that future things are real, the open future thesis undermines the persuasiveness of this model. Nevertheless, the open future thesis does not necessarily falsify the moving spotlight theory because the future can change over time (though this is, to some extent, a strange model). Therefore, considering whether or not the future is open is a significant issue. In this paper, I argue that if quantum mechanics (QM) is a complete theory, then the future is not open, contrary to appearances. When a theory, T, is complete, any information other than that which the theory T requires is unnecessary to explain the experimental data; in other words, there are no so-called “hidden variables.” Therefore, since, in principle, QM does not predict the measurement values of physical quantities with certainty except in certain specific cases, the world is essentially probabilistic if 2 QM is complete.2 Accordingly, at first sight, QM strongly supports the open future thesis. However, there is a possibility that a physical quantity (observable), Q, has one definite value before the measurement, even though it is impossible, in principle, to predict which value of Q the observer will find through measurement. Actually, there are many interpretations of QM insisting that physical quantities that are not yet measured have definite values, such as modal interpretations. However, as discussed below (§3), those interpretations do not necessarily exclude the open future thesis at first glance (and I consider that their proponents also do not intend to reject the open future thesis; instead, they would like to support realism). Therefore, I argue in this paper that, if QM is complete and if there is no non-physical process, QM cannot be interpreted as a theory that supports the open future thesis. In the rest of this section, I clarify the concept of the open future. F(x), O(x, y), □ are respectively: F(x): “in x time units it will be the case that …” O(y, z): “an observer observes one definite value, y, by the measurement of a physical quantity, z” □: “it is now necessarily the case that …” Therefore, the open future thesis is defined as: ¬□F(x)O(y, z) ∧ ¬□F(x) O(¬y, z). 2 [OP] Bohmian mechanics (Bohm 1952) and GRW theory (Ghirardi et al. 1986) do not assume that QM is complete, and thus I do not include them in this paper’s argument. 3 That is, the open future thesis means both that it is not necessarily the case now that the observer will observe one definite value y by the measurement of the observable z in time x in the future, and that it is not necessarily the case now that the observer will not observe one definite value y by the measurement of z in time x in the future. That Q possesses one definite value, a, means:3 □O(a, Q). [D] That is, it is now necessarily the case that the observer observes the definite value, a, by the measurement of Q. I also assume that the observer always obtains one definite value of Q by the measurement: ∃x, O(x, y). [AD] For example, interpretations such as the many-worlds interpretation and the manyminds interpretation assume that the observer cannot arrive at one definite value of Q by way of measurement; the observer believes that they are observing one definite value, but that is an illusion (thus, condition [AD] is false). However, such cases are not open 3 There are controversies regarding how to interpret the state of quantum mechanical indeterminacy, or the superpositional state (Darby 2010; Skow 2010; Torza 2017; Calosi and Wilson 2019). There arises the question of which is the adequate account: the supervaluationism account, or the determinable-based account. However, definition [D] does not support either side of this debate, because it does not refer to state itself (rather, it only refers to measurement value). 4 future situations, even if [OP] is satisfied, because according to the open future thesis, the undetermined future will become a determined present and past. Therefore, not only [OP] but also [AD] is a necessary condition for the open future thesis. This paper consists of two parts. First, I argue that if the wave-function completely describes the physical state of the system, then we have to accept that non-physical processes causally influence physical ones (thus, the interpretation that the wavefunction completely describes the physical state is unacceptable), although such an interpretation can support the open future thesis (§2). Second, I argue that those interpretations of QM in which the wave-function does not describe the physical state cannot support the open future thesis (§3). 2. The wave-function completely describes the physical state In this section, I assume that the wave-function completely describes the physical state of a physical system. The implications of this are as follows. Suppose that φ(t) is a wave-function of a system, S, at a certain point in time, t. Suppose also that t1 and t2 are points in time, and that t2 is further in the future than t1. Possible values (eigenvalues) of physical quantity Q are qi (i = 1, 2, …, N), and their corresponding eigenstates are |qi>. Now, suppose that φ(t2) = a1(t2)|q1> + … +aN (t2)|qN>, (2-1) where ai are normalized constants. For φ(t2) to completely describe the physical state of S means that Q does not possess any definite value at t2. Thus, when t1 is the present ¬□F(t2 − t1)O(q1, Q) ∧ … ∧ ¬□F(t2 − t1)O(qk, Q) ∧ … ∧ ¬□F(t2 − t1)O(qN, Q) 5 (2-2) However, when t2 becomes the present and Q is measured, the observer observes one definite value: for example, q1. Thus, Q possesses the definite value q1 at t2: □O(q1, Q) (2-3) Since the wave-function completely describes the physical state of S, the wave-function changes into φ(t2) = |q1>. (2-4) On the other hand, if (2-4) is true then (2-3) is true, and if (2-2) is true then (2-1) is true. These relations are referred to as the “eigenstate-eigenvalue link.” In other words, for the wave-function to completely describe the physical state means that the eigenstateeigenvalue link works. The interpretation I have discussed above is called the orthodox interpretation or the standard interpretation. The features of this interpretation are as follows. (a) QM is complete (b) The Schrödinger equation, which is the fundamental equation of QM, completely describes the temporal behavior of the wave-function, except in the measurement process (c) The observer obtains one definite value by measurement (d) A physical quantity, Q, possesses a definite value if and only if the wavefunction is an eigenfunction of Q (eigenstate-eigenvalue link) 6 (e) The square of probability to observe the measurement value, qi, is ai*ai (the Born Rule) For (2-2) and (2-3), the orthodox interpretation satisfies the conditions for the open future thesis, [OP] and [AD]. However, the transition from (2-1) to (2-4) is mysterious. This instantaneous and discontinuous change from (2-1) into (2-4) is called the “collapse of the wave-function.” The process of the collapse of the wave-function cannot be described by the Schrödinger equation because this process is discontinuous, and the Schrödinger equation is a continuous differential equation. Thus, Wigner (1961) argued that the orthodox interpretation of QM, which is characterized by (a)–(e), implies mind-body dualism.4 If there is no non-physical process, the measurement process is also a physical one. If the measurement process is a physical process, it must be described by the Schrödinger equation, because there is no reason to distinguish between measurement processes and other physical ones. Nevertheless, since the Schrödinger equation cannot describe the measurement process as discussed above, the measurement process cannot be considered a physical one. Therefore, the orthodox interpretation implies that nonphysical processes causally influence physical ones. However, one might object that the reason the Schrödinger equation cannot describe the measurement process is that the Schrödinger equation is applied only to a closed system, whereas the system in question, S, is not a closed system, due to its 4 Recently, Barrett (2006) and Morita (2020) also argue that the orthodox interpretation implies mind-body dualism. 7 interaction with the measurement apparatus and the observer. Thus, this interaction might cause indefinite values to become definite. Actually, certain physicists and philosophers of physics try to explain the collapse of the wave-function by arguing that the interaction between the system and the measuring apparatus, including the observer, brings about the collapse of the wave-function (Myrvold 2016, §2.3.2). However, supposing that the Schrödinger equation could completely describe the behavior of the wave-function of the whole system S and its environment system, including the measurement apparatus and observers, and that the Schrödinger equation could deterministically predict the state of this whole system, this would also contradict the assumption that the future is open ([OP] is not satisfied). Therefore, if QM is complete and the future is open, the measurement process that changes the indefinite value of Q into a definite value cannot be a physical process. Of course, there is a possibility that there are some non-physical processes in the world. However, if such non-physical processes can causally influence physical processes, it means that physics is not closed. For physics not to be closed implies, in turn, that physics cannot explain all physical phenomena, a hypothesis that is difficult to accept in general. 3. The wave-function does not completely describe the physical state As discussed in §2, if the wave-function completely describes the physical state and if the future is open, then one has to accept that non-physical processes influence physical processes. However, one does not need to accept the assumption that the wave-function completely describes the physical state, even if one accepts the completeness of QM. Van Fraassen (1991) suggests that there are two states of systems, the dynamic state and the value state. The dynamic state determines which physical properties 8 (values) the system may possess, and which properties the system may have at later times. Thus, the wave-function represents the dynamic state. On the other hand, the value state represents what actually is the case: that is, all the system’s physical properties that are sharply defined at the instant in question (Lombardi and Dieks 2017). This idea is common to so-called “modal interpretations.” An essential feature of this interpretation is that a physical quantity, Q, can possess a definite value even if the dynamical state is not an eigenstate of Q (thus, the wave-function is not an eigenfunction of Q). Therefore, the eigenstate-eigenvalue link is violated. This seems to mean that [OP] is not satisfied even if the wave-function at t (in the future) is not an eigenstate of Q that will be measured at t. However, there seems to be a loophole to save the open future thesis, contrary to appearances. Suppose that t1 is the present point in time, and t2 and t3 are points in time in the future (t1 < t2 < t3). Also suppose that an electron e1 interacts with another electron e2 at t2, that they spatially separate soon after t2, and that no external forces act on e1 and e2 between t2 and t3. After the interaction, the spin state (the dynamic state) of the system consisting of e1 and e2 is |+1/2>I |−1/2>II − |−1/2>I |+1/2>II (where | >I and | >II respectively represent the state of e1 and e2, normalized constants are ignored, and the unit is ħ); namely, e1 and e1 are entangled. However, according to the modal interpretation, the value state of e1 must be either |+1/2>σ or |−1/2>σ (σ = x, y, z); namely, the e1 system and e2 system are separated. Therefore, when t2 is the present point in time, and the observer measures the xspin of e1 at t3, □F(t3− t2)O(+1/2, x-spin) ∨ □F(t3− t2)O(−1/2, x-spin) 9 (3-1) Thus, at t2, the future is not open with regard to the x-spin value of e1. However, there is no guarantee that the following (3-2) is also true when t1 is present. □F(t3− t1)O(+1/2, x-spin) ∨ □F(t3− t1)O(−1/2, x-spin), (3-2) which implies that what will happen (which value of the x-spin the observer will observe) at t3 might not be determined at t1 (before the interaction of e1 and e2). Accordingly, there still seems a possibility that the future is open at t1. Nevertheless, I argue that, according to the modal interpretation, the future cannot be open. In general, the Kochen-Specker (K-S) theorem states that all physical quantities cannot have definite values at the same time if QM is complete (Kochen and Specker 1967). Thus, (3-1) and the following propositions, (3-3) and (3-4), cannot simultaneously be true when t2 is the present. □F(t3− t2)O(+1/2, y-spin) ∨ □F(t3− t2)O(−1/2, y-spin). (3-3) □F(t3− t2)O(+1/2, z-spin) ∨ □F(t3− t2)O(−1/2, z-spin). (3-4) This means that the physical quantities that can possess definite values are restricted by a certain rule. Although different versions of modal interpretation have different rules, the physical quantity that will be measured must possess a definite value; otherwise, the same problem as discussed in §2 arises. Therefore, if the observer wishes, at t2, to measure the x-spin at t3, (3-1) is true while (3-3), (3-4), or both (3-3) and (3-4) are not true; according to the K-S theorem, these three observables cannot possess definite values at the same time. In contrast, if 10 the observer wishes, at t2, to measure the y-spin at t3, (3-3) is true while (3-1), (3-4), or both (3-1) and (3-4) are not true. However, suppose that, although the observer wished at t2 to measure the x-spin at t3, the observer changes his/her mind at t4 (t2 < t4 <t3) and decides instead to measure the y-spin at t3, and that the observer again changes his/her mind at t5 (t4 < t5 <t3) to measure the z-spin at t3, Then, the truth value of at least one of (3-1), (3-3), and (3-4) changes between t2 and t3. Since we assume that there is no external force on e1 (and e2) between t2 and t3, this is unacceptable; in addition, it seems to violate the concept of “necessary truth.” To emphasize this absurdity, consider the socalled “EPR situation,” which was originally suggested by Einstein et al. (1945) and revised by Bohm (1989, pp. 611ff.). Consider also the state of e2. Suppose that the total spin of e1 and e2 is 0 at t2. t4 (and thus t3) is long enough after t2 that e2 is a few light years away from e1 and the measuring apparatus on the earth. The observer set the measurement apparatus for measuring the x-spin of e1 at t1, and thus both the x-spins of e1 and e2 possess definite values at t2 according to the modal interpretation, while either the y-spin of e1, the y-spin of e2 (or both) does not possess a definite value at t2 (for the convenience of the discussion, we assume that the x-spin and y-spin cannot simultaneously possess definite values because of the K-S theorem). Thus far, this is not strange at all. However, the observer changes the set of the measurement apparatus so that it is appropriate for measuring the y-spin at t4. Then, the y-spin of e2 begins to possess a definite value at t4, if it does not already possess one, despite the fact that e2 is far from both e1 and the measuring apparatus (of course, a case where the observable that does not possess one definite value is the z-spin, and the same line of argument is applied). Note that there is no quantum non-local correlation in this situation because the value states of the x(y)-spins of e1 and e2 are either |+1/2>x(y) or |−1/2>x(y); that is, they 11 are not the quantum entangled states: |+1/2>I |−1/2>II − |−1/2>I|+1/2>II. Therefore, there must be some superluminal mechanical interaction between e1 and e2 (or, between the measuring apparatus and e2) because these states are physical states. However, this is absurd according to the relativity theory.5 In order to avoid such an absurd conclusion, exactly which physical quantity the observer will measure at t3 must be determined before t2; even though the observer can change his/her mind between t1 and t3, the final state of his/her mind at t3 is determined before t2. In other words, in the above discussion leading to an absurd conclusion, we have presupposed the open future thesis. Namely, ¬□F(x)M(y) ∧ ¬□F(x)¬M(y), (3-5) where M(y) designates the propositional function that the observer measures the physical quantity, y, meaning that which observable the observer will measure in future is, thus, not determined now. Now, let us return to our current problem. The problem is this: the x-spin of e1 might not possess one definite value before t2. Namely, (3-2) is possibly false, and thus, the open future thesis might be correct even under the modal interpretations. We have seen that the physical quantity that the observer will measure is determined before t2. Still, (3-2) does not seem necessarily to be true. However, what is the significance of the fact that which physical quantity the observer will measure at t3 is determined before t2? This means that the state of the observer’s brain (mind) at t3 has been determined 5 Although this interaction cannot transmit any information with superluminal speed, this clearly changes the physical state (the value state) of e2 by a distant cause (the measuring apparatus). 12 before t2.6 The brain consists of many quantum particles, and these particles must interact with each other many times before t3. Notwithstanding this, as discussed, the physical quantity which the observer will measure is determined. It follows that the definite values possessed by the physical quantities (of elements of the observer’s brain) are also determined before the interactions of these particles in the observer’s brain. Therefore, the definite value which will be possessed by the x-spin of e1 must be also determined before t2, because there is no fundamental difference between the electron system, e1, and the observer’s brain system. Thus, (3-2) is true at t1. In conclusion, under the modal interpretation, the open future thesis is false. However, while (3-2) is true, either (3-3) or (3-4) is false. This seems to satisfy the condition for the open future [OP]. Nevertheless, remember that [AD] should be satisfied for the open future thesis. As discussed above, the y-spin and z-spin of e1 will not be measured at t3, and they will not satisfy [AD] at t3. Of course, they can be measured after t3, and the observer would observe a definite value of the y-spin or z-spin at t3. However, in this case, one of the following is true. 6 □F(t6− t2)O(+1/2, y-spin) ∨ □F(t6− t2)O(−1/2, y-spin), (3-6) □F(t6− t2)O(+1/2, z-spin) ∨ □F(t6− t2)O(−1/2, z-spin) , (3-7) One might object that the physical quantity which the observer will measure is determined at t2 when e1 and e2 intact. However, the observer can be far from e1, e2, and the measuring apparatus before and at t2; thus, if the interaction of e1 and e2 influences the observer’s brain, this can follow from the absurdity as discussed. 13 where t6 > t3. And, as discussed above, whether (3-6) or (3-7) is true is determined before t2. Finally, there is another interpretation assuming that the wave-function does not completely describe the physical state, although accepting the completeness of QM. This interpretation is referred to as QBism (Fuchs 2010). QBism claims that the wavefunction describes the state of our knowledge, and it regards the actual physical state as a kind of black box so that one can avoid committing to the unobserved state. QBism considers QM as a tool to explain experimental data; it suggests that QM does not need to offer any description of the unobserved state. Therefore, QBism can yield no metaphysical claims about future quantum states. Accordingly, I do not consider QBism in this paper. 4. Summary Since, according to quantum mechanics, measurement values are principally unpredictable, if quantum mechanics is complete then the future must be open, or at least it might be open, at first glance. However, as I have argued in this paper, the future cannot be open if quantum mechanics is complete. An open future means both that it is not necessarily the case now that the observer will obtain a particular definite measurement value, a, and that it is also not necessarily the case now that the observer will not obtain the value, a, when the observer measures the physical quantity, Q, in the future. First, I examined a case where the wave-function completely describes the physical state. Since, in most cases, the wave-function of Q at t in the future is not an eigenfunction, it is considered that Q possesses no definite value at t and that the future is therefore open. However, when t becomes the present point in time and the observer 14 measures Q, the observer obtains one definite value of Q. Therefore, this measurement process is discontinuous, and the Schrödinger equation cannot describe this process. If this measurement process is a physical one, it must be described by the Schrödinger equation, because there is no fundamental difference between the measurement process and other physical processes that can be described by the Schrödinger equation. Accordingly, if the future is open, non-physical processes can influence physical ones, but this is unacceptable. However, there is a possibility that the measurement process can be described by the Schrödinger equation when considering the observer, the measuring apparatus, and the system in question. Nevertheless, if this is the case, then the world is deterministic, and thus the future is not open. In conclusion, if quantum mechanics is complete, the wave-function completely describes the physical state, and non-physical processes cannot influence the physical ones, the future cannot be open. Second, I examined a case where the wave-function does not completely describe the physical state. Since this case does not assume the eigenstate-eigenvalue link, Q can possess one definite value even when the wave-function is not an eigenfunction of Q. However, because of the Kochen-Specker theorem, not all physical quantities can possess definite values at the same time. Nevertheless, at least physical quantities that will be measured possess definite values, to avoid the discontinuous collapse of the wave-function; this kind of interpretation is called the modal interpretation. Although the modal interpretation seems to exclude the open future thesis, it also seems possible that precisely which definite value the physical quantity, Q, of the system, S, will possess is not determined before the system S finally interacts with the external system. However, I argue that the physical quantity which the observer will measure must be determined before the final interaction of S with the external system. 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