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Quantum Mechanics as Classical Physics Charles Sebens University of Michigan July 31, 2013 Dirk-André Deckert Michael Hall Howard Wiseman UC Davis Griffith University Griffith University 2 A Strange Interpretation of QM Deterministic, no collapse No wave function, no Schrödinger equation Many worlds Worlds do not branch The number of worlds is finite Particles follow Bohmian trajectories Equation of motion of the form 𝐹 = 𝑚𝑎 This is a development of the hydrodynamic interpretation, originally proposed by Madelung (1927) and developed by Takabayasi (e.g., 1952) among others. 3 Outline I. A Series of Solutions to the Measurement Problem I. The Many-worlds Interpretation II. Bohmian Mechanics III. Prodigal QM II. Newtonian Quantum Mechanics III. A Strength: Probability IV. A Weakness: Non-quantum States 4 Refresher: The Double-slit Experiment 5 Solution 1: Everettian QM What there is (Ontology): The Wave Function What it does (Laws): Schrödinger Equation 6 Solution 2: Bohmian QM Ontology: The Wave Function Particles Laws: Schrödinger Equation Guidance Equation 7 Solution 3: Prodigal QM Ontology: The Wave Function Particles (in many worlds) Laws: Schrödinger Equation Guidance Equation This is a significantly altered variant of the proposal in Dorr (2009). 8 Two Important Characters: 𝜌 & 𝑣𝑘 is the density of worlds in configuration space, scaled so that . By hypothesis, . gives the velocity of the 𝑘-th particle in the world where particles are arranged . The evolution of 𝜌 is determined by the velocities of the particles in the various worlds by a continuity equation: 9 The Evolution of 𝑣𝑘 From the following facts one can derive the double boxed equation below. Worlds are distributed in accordance with psi-squared: The Schrödinger equation: The guidance equation: The quantum potential Q. This equation gives a way of calculating the evolution of 𝑣𝑘 , and hence 𝜌, which never references the wave function Ψ. 10 Solution 4: Newtonian QM Ontology: Particles (in many worlds) Law: Newtonian Force Law 11 Review of the Alternatives 12 The Wave Function in Newtonian QM The wave function is not part of the fundamental ontology in Newtonian QM. Still, one can introduce an object Ψ which conforms to the following conditions: Worlds are distributed in accordance with psi-squared The guidance equation is obeyed 13 The Quantitative Probability Problem for Everettian QM Alice makes 10 measurements of the z-spin of the following state: In a highly idealized scenario, 210 = 1,024 branches… In 252 branches (25%) she will see half up, half down. In 10 branches (1%) she will see nine up, one down (which should be most likely, happening about 40% of the time). In general, most branches don’t exhibit Born Rule statistics. 14 No Similar Problem for Newtonian QM Since the wave function is not fundamental but introduced as a means of summarizing information about 𝜌 & 𝑣𝑘 , it is impossible for the density of worlds in a region of configuration space to deviate from psi-squared. Most worlds are in high-amplitude regions. In Newtonian QM the uncertainty present is self-locating uncertainty. One needs to determine which world they are in since many are consistent with any particular set of experiences. Since, in general, many worlds will be consistent with one’s experiences, each should be assigned equal credence (given the state of the universe; see Elga 2004). 15 Versus Bohmian Mechanics In Newtonian QM, each world follows a Bohmian trajectory through configuration space. So, if Bohmian mechanics can reproduce the predictions of textbook QM, Newtonian QM should be able to also. Some Bohmian trajectories don’t predict quantum statistics. The correct long-run quantum statistics will be observed if the universe satisfies the quantum equilibrium hypothesis. Quantum Equilibrium Hypothesis (Teufel 2013, rough version): “The initial wave function Ψ(0) and configuration 𝑄(0) of the "universe" are such that the empirical distributions of subsystem configurations 𝑋𝑖 (𝑡𝑖 ) of (different) subsystems at (different) times 𝑡𝑖 with the same conditional wave function 𝜑𝑐𝑜𝑛𝑑 are close to the |𝜑𝑐𝑜𝑛𝑑 |2 -distribution.” 16 Avoiding Anomalous Statistics In Newtonian QM, it is also possible that one’s own world does not exhibit quantum statistics. However, it is necessarily true that the majority of worlds in any universe satisfy the quantum equilibrium hypothesis since (by definition of Ψ). Thus, one should always expect to be in a world that is in quantum equilibrium. So, one should expect to see Born Rule statistics in long-run frequencies of measurements (see Durr et al. 1992). 17 How Many Worlds? We’ve been treating the number of worlds as continuously infinite, described by a density, 𝜌, and a collection of velocity fields, . Meanwhile, I’ve claimed that the number of worlds is actually finite. Why finitely many worlds? 𝜌 is easier to interpret. How many? Enough that they can be well-modeled by a continuum. This is similar to the hydrodynamic limit in fluid mechanics. If there isn’t really a continuum, is the force law given earlier really a fundamental law? No. 18 The Quantization Condition Not all collections of 𝜌 & 𝑣𝑘 can be described by a wave function. The velocity of a particle is proportional to the gradient of the wave function’s complex phase. For there to exist a wave function which describes* 𝜌 & 𝑣𝑘 , the velocity field must satisfy the following requirement (see Wallstrom 1994; Takabayasi 1952). Quantization Condition: Integrating particle momenta along any closed loop in configuration space gives a multiple of Planck’s constant. *That is, a wave function for which by the particle velocities. and the guidance equation is obeyed 19 Limitations of Newtonian QM Yet to be extended to relativistic quantum physics Yet to be extended to multiple particles with spin We don’t yet have the fundamental law(s) The state space is too large in two ways: States that violate the Quantization Condition States with too few worlds to use the hydrodynamic limit 20 Neat Features of Newtonian QM Wave function is a mere summary of the properties of particles No superpositions No entanglement No collapse No mention of “measurement” in the laws All dynamics arise from Newtonian forces The theory is deterministic Worlds are fundamental, not emergent (so avoids the need to explain how people and planets arise as structures in the WF) Worlds do not branch (so avoids concerns about personal identity) No qualitative probability problem No quantitative probability problem Immune to Everett-in-denial objection,* not in denial 21 * See Deutsch (1996), Brown & Wallace (2005). Neat Features of Newtonian QM Wave function is a mere summary of the properties of particles No superpositions No entanglement No collapse No mention of “measurement” in the laws All dynamics arise from Newtonian forces The theory is deterministic Worlds are fundamental, not emergent (so avoids the need to explain how people and planets arise as structures in the WF) Worlds do not branch (so avoids concerns about personal identity) No qualitative probability problem No quantitative probability problem Immune to Everett-in-denial objection, not in denial 22 Neat Features of Newtonian QM Wave function is a mere summary of the properties of particles No superpositions No entanglement No collapse No mention of “measurement” in the laws All dynamics arise from Newtonian forces The theory is deterministic Worlds are fundamental, not emergent (so avoids the need to explain how people and planets arise as structures in the WF) Worlds do not branch (so avoids concerns about personal identity) No qualitative probability problem No quantitative probability problem Immune to Everett-in-denial objection, not in denial 23 References 24 End 25 Ontological Options Option 1: World-particles in Configuration Space Option 2: World-particles in Configuration Space and 3D Worlds Option 3: Distinct 3D Worlds Option 4: Overlapping 3D Worlds Shown below for two particles in one dimensional space… 26 An Unnatural Constraint To get a feel for the Quantization Condition, let’s look at the following case: Consider the 𝑛 = 2, 𝑙 = 1, 𝑚 = 1 energy eigenstate of an electron in the Hydrogen atom (treating the force from the nucleus as an external potential). If the electrons were circling the 𝑧-axis a little faster or a little slower, they would not be representable by a wave function. 27 The Orbital 28