The Intrinsic Structure of Quantum Mechanics

The wave function in quantum mechanics presents an interesting challenge to our understanding of the physical world. In this paper, I show that the wave function can be understood as four intrinsic relations on physical space. My account has three desirable features that the standard account lacks: (1) it does not refer to any abstract mathematical objects, (2) it is free from the usual arbitrary conventions, and (3) it explains why the wave function has its gauge degrees of freedom, something that are usually put into the theory by hand. Hence, this account has implications for debates in philosophy of mathematics and philosophy of science. First, by removing references to mathematical objects, it provides a framework for nominalizing quantum mechanics. Second, by excising superfluous structure such as overall phase, it reveals the intrinsic structure postulated by quantum mechanics. Moreover, it also removes a major obstacle to "wave function realism.

An Intrinsic Theory of Quantum Mechanics: Progress in Field's Nominalistic Program, Part I

In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (1980), responds to David Malament’s long-standing impossibility conjecture (1982), and establishes an important first step towards a genuinely intrinsic and nominalistic account of quantum mechanics. I will also compare the present account to Mark Balaguer’s (1996) nominalization of quantum mechanics and discuss how it might bear on the debate about “wave function realism.” In closing, I will suggest some possible ways to extend this account to accommodate spinorial degrees of freedom and a variable number of particles (e.g. for particle creation and annihilation). Along the way, I axiomatize the quantum phase structure as what I shall call a “periodic difference structure” and prove a representation theorem as well as a uniqueness theorem. These formal results could prove fruitful for further investigation into the metaphysics of phase and theoretical structure. (For a more recent version of this paper, please see "The Intrinsic Structure of Quantum Mechanics" available on PhilPapers.

Surreal Decisions

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives

Bell’s Theorem, Quantum Probabilities, and Superdeterminism

In this short survey article, I discuss Bell’s theorem and some strategies that attempt to avoid the conclusion of non-locality. I focus on two that intersect with the philosophy of probability: (1) quantum probabilities and (2) superdeterminism. The issues they raised not only apply to a wide class of no-go theorems about quantum mechanics but are also of general philosophical interest

From Time Asymmetry to Quantum Entanglement: The Humean Unification

Two of the most difficult problems in the philosophical foundations of physics are (1) what gives rise to the arrow of time and (2) what the ontology of quantum mechanics is. The first problem is puzzling since the fundamental dynamical laws of physics do not include an arrow of time. The second problem is puzzling since the quantum-mechanical wave function describes a non-separable reality that is remarkably different from the objects in our ordinary experiences. In this paper, we propose a unified ``Humean'' solution to the two problems. Humeanism allows us to incorporate the Past Hypothesis and the Statistical Postulate into the best system, which we then use to simplify the quantum state of the universe. This allows us to confer the nomological status to the quantum state in a way that adds no significant complexity to the best system and solves the ``supervenient-kind problem'' facing the original version of the Past Hypothesis. We call this strategy the "Humean unification." It brings together the origins of time asymmetry and quantum entanglement. On this theory, what gives rise to the arrow of time is also responsible for the non-separable phenomena in nature. The result is a more unified theory, with a separable mosaic, a best system that is simple and non-vague, less tension between quantum mechanics and special relativity, and more theoretical and dynamical unity. We then compare our proposals to those in the literature that focus on only one of the two problems. Our analysis further suggests that, in order to obtain a deeper understanding about the problems in philosophy of science, it can be tremendously illuminating to explore the full resources of Humeanism, even if one is not a Humean

Time's Arrow in a Quantum Universe I: On the Simplicity and Uniqueness of the Initial Quantum State

In a quantum universe with a strong arrow of time, we postulate a low-entropy boundary condition (the Past Hypothesis) to account for the temporal asymmetry. In this paper, I show that the Past Hypothesis also contains enough information to significantly simplify the quantum ontology and clearly define a unique initial condition in such a world. First, I introduce Density Matrix Realism, the thesis that the quantum universe is described by a fundamental density matrix (a mixed state) that corresponds to some physical degrees of freedom in the world. This stands in sharp contrast to Wave Function Realism, the thesis that the quantum universe is described by a wave function (a pure state) that represents something physical. Second, I suggest that the Past Hypothesis is sufficient to determine a unique and simple density matrix. This is achieved by what I call the Initial Projection Hypothesis: the initial density matrix of the universe is the projection onto the special low-dimensional Hilbert space. Third, because the initial quantum state is unique and simple, we have a strong case for the Nomological Thesis: the initial quantum state of the universe is completely specified by a law of nature. This new package of ideas has several interesting implications, including on the dynamic unity of the universe and the subsystems, the theoretical unity of statistical mechanics and quantum mechanics, and the alleged conflict between Humean supervenience and quantum entanglement

Nomic Vagueness

If there are fundamental laws of nature, can they fail to be exact? In this paper, I consider the possibility that some fundamental laws are vague. I call this phenomenon nomic vagueness. I propose to characterize nomic vagueness as the existence of borderline lawful worlds. The existence of nomic vagueness raises interesting questions about the mathematical expressibility and metaphysical status of fundamental laws. For a case study, we turn to the Past Hypothesis, a postulate that (partially) explains the direction of time in our world. We have reasons to take it seriously as a candidate fundamental law of nature. Yet it is vague: it admits borderline (nomologically) possible worlds. An exact version would lead to an untraceable arbitrariness absent in any other fundamental laws. However, the dilemma between nomic vagueness and untraceable arbitrariness is dissolved in a new quantum theory of time’s arrow

The Wentaculus: Density Matrix Realism Meets the Arrow of Time

Two of the most difficult problems in the foundations of physics are (1) what gives rise to the arrow of time and (2) what the ontology of quantum mechanics is. They are difficult because the fundamental dynamical laws of physics do not privilege an arrow of time, and the quantum-mechanical wave function describes a high-dimensional reality that is radically different from our ordinary experiences. In this paper, I characterize and elaborate on the ''Wentaculus'' theory, a new approach to time's arrow in a quantum universe that offers a unified solution to both problems. Central to the Wentaculus are (i) Density Matrix Realism, the idea that the quantum state of the universe is objective but can be impure, and (ii) the Initial Projection Hypothesis, a new law of nature that selects a unique initial quantum state. On the Wentaculus, the quantum state of the universe is sufficiently simple to be a law, and the arrow of time can be traced back to an exact boundary condition. It removes the intrinsic vagueness of the Past Hypothesis, eliminates the Statistical Postulate, provides a higher degree of theoretical unity, and contains a natural realization of ''strong determinism.'' I end by responding to four recent objections. In a companion paper, I elaborate on Density Matrix Realism.Comment: 14 pages. Written for the book Bassi, A., Goldstein, S., Tumulka, R., Zangh\`i, N. (eds.) Physics and the Nature of Reality: Essays in Memory of Detlef D\"ur

An Intrinsic Theory of Quantum Mechanics: Progress in Field's Nominalistic Program, Part I

In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (1980), responds to David Malament’s long-standing impossibility conjecture (1982), and establishes an important first step towards a genuinely intrinsic and nominalistic account of quantum mechanics. I will also compare the present account to Mark Balaguer’s (1996) nominalization of quantum mechanics and discuss how it might bear on the debate about “wave function realism.” In closing, I will suggest some possible ways to extend this account to accommodate spinorial degrees of freedom and a variable number of particles (e.g. for particle creation and annihilation). Along the way, I axiomatize the quantum phase structure as what I shall call a “periodic difference structure” and prove a representation theorem as well as a uniqueness theorem. These formal results could prove fruitful for further investigation into the metaphysics of phase and theoretical structure

From Time Asymmetry to Quantum Entanglement: The Humean Unification

Two of the most difficult problems in the foundations of physics are (1) what gives rise to the arrow of time and (2) what the ontology of quantum mechanics is. I propose a unified 'Humean' solution to the two problems. Humeanism allows us to incorporate the Past Hypothesis and the Statistical Postulate into the best system, which we then use to simplify the quantum state of the universe. This enables us to confer the nomological status to the quantum state in a way that adds no significant complexity to the best system and solves the ''supervenient-kind problem'' facing the original version of the Past Hypothesis. We call the resultant theory the Humean unification. It provides a unified explanation of time asymmetry and quantum entanglement. On this theory, what gives rise to time's arrow is also responsible for quantum phenomena. The new theory has a separable mosaic, a best system that is simple and non-vague, less tension between quantum mechanics and special relativity, and a higher degree of theoretical and dynamical unity. The Humean unification leads to new insights that can be useful to Humeans and non-Humeans alike