An analysis of Bohmian dispositionalism Shan Gao Research Center for Philosophy of Science and Technology, Shanxi University, Taiyuan 030006, P. R. China E-mail: gaoshan2017@sxu.edu.cn. July 4, 2019 Abstract It is argued that Bohmian dispositionalism fails to explain the result of a protective measurement and thus it is plagued by an explanatory deficiency problem. Although adding nonlocal influences between systems being in a product state may avoid the problem, the origin of these influences can hardly be explained, and the revised theory also has unnatural features. By comparison, a natural way to explain the result of a protective measurement is to admit that the wave function is ontic, representing a concrete physical entity, although it remains to see what the physical entity really is. In Bohmian mechanics, there are only particles’ positions and a law of motion (Dürr, Goldstein and Zanghı̀, 1997; Allori et al, 2008; Goldstein, 2017). According to Esfeld et al (2014), there are two main philosophical options to ground a law of motion, namely Humeanism and dispositionalism, that can be applied to Bohmian mechanics.1 On Humeanism, the laws of motion do not have any explanatory function, while on dispositionalism, laws are anchored in the essence of the properties of the objects that there are in the physical world, and they are suitable to figure in explanations answering why-questions. In this paper, I will argue that Bohmian dispositionalism is plagued by an explanatory deficiency problem. In particular, it fails to explain the result of a protective measurement. Let us first see a familiar example in classical mechanics. Suppose in an isolated lab there are a particle with charge Q trapped in an uncharged box and a test electron. The test electron is shot along a straight line near the box, and then detected on a screen after passing by the box. According to 1 There is also a third view, the primitivism about laws (Maudlin, 2007), which regards the law such as the universal wave function in Bohmian mechanics as part of the fundamental ontology. 1 Newton’s laws of motion and Coulomb’s law, the deviation of the trajectory of the test electron is determined by the charge of the measured particle, as well as the distance between the electron and the particle. If there were no charged particle in the box, the trajectory of the electron would be a straight line as denoted by position “0” in Figure 1. Now, the trajectory of the electron will be deviated by a definite amount as denoted by position “1” in Figure 1. Figure 1: Scheme of a measurement of the charge of a classical particle The question is: what makes the test particle deviate from its free trajectory? According to dispositionalism about laws, the cause is that the measured particle has a charge Q in its position in the box as a dispositional property, which by definition has the efficacy to deviate the test electron from its free trajectory. If there is no charge in the box, then the deviation of the test electron as a physical effect will have no cause. In this case, the theory will be plagued by an explanatory deficiency problem. Now consider a similar example in quantum mechanics. For the sake of simplicity, suppose in a universe there are only a measured system with charge Q, trapped in a two-box protective potential, a test electron and a detecting screen (see Figure 2). The wave function of the measured system at the initial instant is ψ(x) = aψ1 (x) + bψ2 (x), where ψ1 (x) and ψ2 (x) are two normalized wave functions respectively localized in their ground states in two small identical boxes 1 and 2, and |a|2 + |b|2 = 1. A test electron, whose initial state is a Gaussian wavepacket narrow in both position and momentum, is shot along a straight line near box 1 and perpendicular to the line of separation between the two boxes. The electron is detected on a screen after passing by box 1. Suppose the separation between the two boxes is large enough so that a charge Q in box 2 has no observable influence on the electron. Then if the system is in box 2, namely |a|2 = 0, the trajectory of the electron wavepacket will be a straight line as denoted by position “0” 2 on the screen. If the system is in box 1, namely |a|2 = 1, the trajectory of the electron wavepacket will be deviated by a maximum amount as denoted by position “1” on the screen. Figure 2: Scheme of a protective measurement of the charge of a quantum system in Bohmian mechanics Let us make a protective measurement of the charge of the system in box 1. An adiabatic-type protective measurement can be realized as follows (Aharonov and Vaidman, 1993; Aharonov, Anandan and Vaidman, 1993; Gao, 2015, 2017). Since the state ψ(x) is degenerate with its orthogonal state ψ ⊥ (x) = b∗ ψ1 (x) − a∗ ψ2 (x), we first need a protection procedure to remove the degeneracy, e.g. joining the two boxes with a long tube whose diameter is small compared to the size of the box. By this protection ψ(x) will be a nondegenerate energy eigenstate. Then we need to realize the adiabatic condition and the weakly interacting condition, which are required for an adiabatic-type protective measurement. These conditions can be satisfied when assuming that (1) the measuring time of the electron is long compared to ~/∆E, where ∆E is the smallest of the energy differences between ψ(x) and other energy eigenstates, and (2) at all times the potential energy of interaction between the electron and the system is small compared to ∆E. Then the measurement by means of the electron trajectory is a realistic protective measurement, and when the conditions approach ideal conditions, the measurement will be an (ideal) protective measurement with certainty. For an (ideal) protective measurement, the deviation of the trajectory of the electron wavepacket is determined by the expectation value of the charge of the system in box 1, namely |a|2 Q, as well as the distance between the electron wavepacket and the box, and thus the electron wavepacket will reach the definite position “|a|2 ” between “0” and “1” on the screen as denoted in Figure 2. Moreover, the wave function of the measured system, 3 ψ(x), does not change during the measurement. Then, what makes the electron wavepacket deviate from its free trajectory? According to dispositionalism about laws, when the wave function is real (Pusey, Barrett and Rudolph, 2012), it seems natural to assume that the cause is that the measured system has a partial charge |a|2 Q in box 1 as a dispositional property, which by definition has an efficacy to deviate the test electron from its free trajectory. This is similar to the classical situation. Also, if the deviation as a definite effect has no cause, then the theory will be plagued by an explanatory deficiency problem. Let us see Bohmian dispositionalism (Esfeld et al, 2014). According to this view, there are only particles that have both positions and dispositions, and the universal wave function represents a holistic disposition of all particles in the universe which determines their motion. In the above example, since the universal wave function is a product state, the holistic disposition is separable, and the measured particle and the test particle have their respective dispositions represented by their wave functions. In particular, the wave function of the measured system represents the disposition of the measured particle that determines only its own motion, letting it be at rest in a position in box 2, and the disposition is realized only in this position and not in all other positons including box 1. Moreover, the measured particle has no properties other than position and this disposition. Thus, according to Bohmian dispositionalism, the measured particle has no influences on the test particle, either local or nonlocal, and the deviation of the trajectory of the test particle has no cause.2 In other words, this view does not explain why the test particle is deviated from its free trajectory during a protective measurement. There is a deeper reason why Bohmian dispositionalism fails to explain the deviation of the trajectory of the test particle during a protective measurement. It is that this view does not consider the usual interactions between quantum systems. Entanglement is indeed important to explain some strange quantum phenomena, but it certainly cannot explain all interactions between quantum systems. The interaction between the measured system and the measuring system during a protective measurement is irrelevant to entanglement, since the wave function of the composite system has been a product state during the measurement. Moreover, it is obvious that the interactions between objects in the macroscopic world are not relevant to entanglement in general. Such interactions include EM and gravitational interactions, and they are represented by the potential terms in the Schrödinger equation. In the above example of protective measurement, the interaction 2 Note that the initial wave functions of the test electron are the same for the situations of |a|2 = 0 and |a|2 6= 0 (i.e. for the situations of free trajectory and deviated trajectory of the test particle). Moreover, the initial positions of the test particle may be also the same for the two situations. Thus, the deviation of the trajectory of the test particle cannot be caused by itself. 4 between the measured system and the measuring system is part of the EM interactions, the electrostatic interaction. The motion of the test particle also depends on the value of Q of the measured system besides its wave function, e.g. when Q = 0 the motion of the test particle is not deviated from its free trajectory. Even if we consider entanglement such as the universal wave function being an entangled state, the EM and gravitational interactions are also important since they determine the evolution of the entangled state over time. In fact, if there were no such interactions, the entangled state could not even be formed in the first place. It seems that Bohmian dispositionalism only emphasizes the entangled nature of the universal wave function being a holistic disposition of all particles in the universe, but ignores the usual interactions that form the entangled universal wave function and further determine its evolution over time. It is the ignorance of usual interactions that makes this view fail to explain the deviation of the trajectory of the test particle in the above example. One way to solve the explanatory deficiency problem is to assume that the measured particle has an additional physical property which enables it excert a nonlocal influence on the test electron in the above example. The nonlocality of the influence is more obvious when the measured particle is in box 2, being far away from the test particle. According to the Bohmian laws of motion, the motion of the test particle is determined by both the wave function of the measured system in box 1 and the charge of the measured system. Concretely speaking, it is determined by the term |a|2 Q. Thus, the cause of the deviation of the trajectory of the test particle, if it exists, should be a physical property described by this mathematical term (when the wave function is real). Since the ontology of Bohmian dispositionalism consists only in particles, this property must be a property of the measured particle being at rest in a position in box 2 (see Figure 2). This means that the measured particle must have a physical property described by |a|2 Q, maybe called charge, which enables it exert a nonlocal influence on the test particle. However, endowing the measured particle in box 2 (not another physical entity in box 1 with such a charge property seems very unnatural, and the origin of the assumed nonlocal influences can hardly be explained either. First, the charge endowed to the measured particle cannot be shielded. When using a Faraday shield for box 2, the influence on the test particle still exists and does not change. But when using a Faraday shield for box 1, the influence on the test particle no longer exists. Next, since there is only the measured particle which influences the test particle, if the degree of the influence depends on a distance, then it seems that the distance must be the distance between the measured particle and the test particle. There is only a distance relation between them after all; there are no other particles interacting with the test particle, and in particular, there is no particle or 5 another physical entity existing in box 1. But the degree of the influence is determined not by the distance between the test particle and the measured particle, but by the distance between the test particle and box 1 where the modulus squared of the wave function of the measured system is |a|2 . Third, the influence is always a repulsion relative to box 1 and its direction is always along the line extending from box 1. But it may be a repulsion or an attraction relative to the measured particle in box 2, depending on the initial position of the test particle; if the initial position of the test particle is between the two boxes, then the influence will be an attraction relative to the measured particle. Morover, the direction of the influence is independent of the measured particle in box 2. Fourth, although the assumed influence exerted by the measured particle is nonlocal, it has a time delay determined by the distance between the test particle and box 1 in the relativistic domain, where the EM interactions are mediated by fields propagating with the speed of light. In fact, all features of the influence relate to box 1 and not to the measured particle in box 2. No matter where box 2 and the measured particle are, how they move, whether the measured particle annihilates with another anti-particle, and what form the wave function of the measured system in box 2 is, and so on, if only box 1 keeps unchanged and the modulus squared of the wave function of the measured system in the box is still |a|2 , the influence will keep unchanged. On the other hand, even if there is no any change in the region of box 2 and the measured particle, if only there is a change in box 1, such as the size of the box being enlarged very slowly, which may influence the modulus squared of the wave function of the measured system in box 1, then the influence will change. Finally, all these strange features can be explained in a natural way when assuming that the physical property described by |a|2 Q is not a charge property of the measured particle, but a charge property of a physical entity which exists in box 1 where the modulus squared of the wave function of the measured system is |a|2 . Similarly, there is also a physical entity existing in box 2 where the modulus squared of the wave function of the measured system is |b|2 . In other words, the wave function of the measured system ψ(x) represents a physical entity extending in space, including both boxs 1 and 2, where the wave function is nonzero. Certainly, what the physical entity really is poses another deep issue (see Gao, 2017; Hubert and Romano, 2018 for a recent analysis). I will analyze this issue in future work. To sum up, I have argued that Bohmian dispositionalism fails to explain the result of a protective measurement and thus it is plagued by an explanatory deficiency problem. Although adding nonlocal influences between systems being in a product state may avoid the problem, the origin of these influences can hardly be explained, and the revised theory also has unnatural features. By comparison, a natural way to explain the result of a protective measurement is to admit that the wave function is ontic, rep6 resenting a concrete physical entity, although it remains to see what the physical entity really is. Acknowledgments I wish to thank Dustin Lazarovici for helpful discussion. This work is supported by the National Social Science Foundation of China (Grant No. 16BZX021). 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