Preprint (2023)
Philip Goyal
Preprint (2023)
Philip Goyal
Abstract: The quantum symmetrization procedure that is used to handle systems of identical quantum particles brings into question whether the elementary constituents of matter, such as electrons, have the fundamental characteristics of persistence and reidentifiability that are attributed to classical particles. However, we presently lack a coherent conception of matter composed of entities that do not possess one or both of these fundamental characteristics. We also lack a clear a priori understanding of why systems of identical particles (as opposed to non-identical particles) require special mathematical treatment, and this only in the quantum mechanical (as opposed to classical mechanical) setting.
Here, on the basis of a conceptual analysis of a recent mathematical reconstruction of the quantum symmetrization procedure, we argue that the need for the symmetrization procedure originates in the confluence of identicality and the active nature of the quantum measurement process. We propose a conception in which detection-events are ontologically primary, while the notion of individually persistent object is relegated to merely one way of bringing order to these events. On this basis, we outline a new interpretation of the symmetrization procedure which gives a new physical interpretation to the indices in symmetrized states and to non-symmetric measurement operators.
In Phenomenology and QBism: New Approaches to Quantum Mechanics, ed. Berghofer, Wiltsche
Philip Goyal
In Phenomenology and QBism: New Approaches to Quantum Mechanics, ed. Berghofer, Wiltsche Philip Goyal
Abstract: The quantum reconstruction program seeks to uncover the deeper physical meaning of the quantum formalism by distilling its content into a set of physically-graspable postulates. Once reconstructed, one can further elucidate the theory by reflecting philosophically upon the postulates rather than the inscrutable formalism in which the theory is couched. Quantum reconstruction thereby opens up a two-step strategy for developing a quantum conceptual framework which reflects the full physical content of quantum theory and can be used to establish precisely what aspects of the classical conceptual framework are retained, modified, or abandoned.
The paper is organized as follows. Sec. II describes the tripartite structure of classical physics, whose coherency I argue underlies its intelligibility. Sec. III discusses the lack of intelligibility of quantum physics, analyses the limitations of the main traditional elucidative approaches, and describes how reconstruction can remove the key interpretative bottleneck. Sec. IV discusses the methodology of reconstruction per se, showing that it is part of the natural life-cycle of physical theories, and summarizes its use in the elucidation of the theories of classical physics. Finally, circling back to quantum theory, Sec. V summarizes the reconstruction of quantum theory, and Sec. VI describes some of the interpretative insights thus far obtained through interpretation of reconstructions of the abstract quantum formalism and the quantum symmetrization postulate.
Continental Philosophy Review (2020)
Philipp Berghofer, Philip Goyal, and Harald Wiltsche
Continental Philosophy Review (2020) Philipp Berghofer, Philip Goyal, and Harald Wiltsche
Abstract: As is well known, the late Husserl warned against the dangers of reifying and objectifying the mathematical models that operate at the heart of our physical theories. Although Husserl’s worries were mainly directed at Galilean physics, the first aim of our paper is to show that many of his critical arguments are no less relevant today. By addressing the formalism and current interpretations of quantum theory, we illustrate how topics surrounding the mathematization of nature come to the fore naturally. Our second aim is to consider the program of reconstructing quantum theory, a program that cur- rently enjoys popularity in the field of quantum foundations. We will conclude by arguing that, seen from this vantage point, certain insights delivered by phenomenology and quantum theory regarding perspectivity are remarkably concordant. Our overall hope with this paper is to show that there is much room for mutual learning between phenomenology and modern physics.
Foundations of Physics 50 1426—1479 (2020)
Philip Goyal
Foundations of Physics 50 1426—1479 (2020)
Philip Goyal
Abstract: The notions of conservation and relativity lie at the heart of classical mechanics, and were critical to its early development. However, in Newton’s theory of mechanics, these symmetry principles were eclipsed with domain-specific laws. In view of the importance of symmetry principles in elucidating the structure of physical theories, it is natural to ask to what extent conservation and relativity determine the structure of mechanics. In this paper, we address this question by deriving classical mechanics—both nonrelativistic and relativistic—using relativity and conservation as the primary guiding principles. The derivation proceeds in three distinct steps. First, conservation and relativity are used to derive the asymptotically conserved quantities of motion. Second, in order that energy and momentum be continuously conserved, the mechanical system is embedded in a larger energetic framework containing a massless component that is capable of bearing energy (as well as momentumin the relativistic case). Imposition of conservation and relativity then results, in the nonrelativistic case, in the conservation of mass and in the frame-invariance of massless energy; and, in the relativistic case, in the rules for transforming massless energy and momentum between frames. Third, a force framework for handling continuously interacting particles is established, wherein Newton’s second law is derived on the basis of relativity and a staccato model of motion-change. Finally, in light of the derivation, we elucidate the structure of mechanics by classifying the principles and assumptions that have been employed according to their explanatory role, distinguishing between symmetry principles and other types of principles (such as compositional principles) that are needed to build up the theoretical edifice.
New Journal of Physics 21 (2019) 063031
Philip Goyal
New Journal of Physics 21 (2019) 063031
Philip Goyal
Abstract: Classical mechanics is based on the notion that matter consists of persistent particles that can be reidentified (or tracked) across time. However, the mathematical symmetrization procedures (due to Dirac [1] and Heisenberg [2], and to Feynman [3]) used to describe identical particles within the quantum formalism are widely interpreted as implying that identical quantum particles are not persistent (so that the concept of ‘the same particle’ is not meaningful) or are persistent but not reidentifiable. However, it has not proved possible to rigorously reconcile these interpretations with the fact that identical particles are routinely assumed to be reidentifiable in particular circumstances—for example, a track in a bubble chamber is interpreted as a sequence of bubbles generated by one and the same particle [4–7]. Moreover, these interpretations do not account for the mathematical form of the symmetrization procedures, leaving open theoretical possibilities other than bosonic and fermionic behavior, such as paraparticles [8], which however do not appear to be realized in nature. Here we propose that the quantum mechanical behaviour of identical particles is a manifestation of a novel kind of complementarity, a complementarity of persistence and nonpersistence. Accordingly, identical ‘particles’ are neither persistent nor nonpersistent; rather, these terms are to be understood as descriptors of different models of the same experimental data. We prove the viability of this viewpoint by showing how Feynman’s and Dirac’s symmetrization procedures arise through a synthesis of a quantum treatment of persistence and nonpersistence models of identical particle-like events, and by showing how reidentifiability emerges in a context-dependent manner. Finally, by drawing on a reconstruction of Feynman’s formulation of quantum theory [9], we construct a precise parallel between the proposed persistence–nonpersistence complementary and Bohr’s wave–particle complementarity for individual particles, and detail their conceptual similarities and dissimilarities.
New Journal of Physics 17 (2015) 013043
Philip Goyal
New Journal of Physics 17 (2015) 013043
Philip Goyal
Abstract: A remarkable feature of quantum theory is that particles with identical intrinsic properties must be treated as indistinguishable if the theory is to give valid predictions in all cases. In the quantum formalism, indistinguishability is expressed via the symmetrization postulate (Dirac P 1926 Proc. R. Soc. A 112 661, Heisenberg W 1926 Z. Phys. 38 411), which restricts a system of identical particles to the set of symmetric states (‘bosons’) or the set of antisymmetric states (‘fermions’). However, the physical basis and range of validity of the symmetrization postulate has not been established. A well-known topological derivation of the postulate implies that its validity depends on the dimensionality of the space in which the particles move (Laidlaw M and DeWitt C 1971 Phys. Rev. D 3 1375–8, Leinaas J M and Myrheim J 1977 Il Nuovo Cimento B 37 1–23). Here we show that the symmetrization postulate can be derived by strictly adhering to the informational requirement that particles which cannot be experimentally distinguished from one another are not labelled. Our key novel postulate is the operational indistinguishability postulate, which posits that the amplitude of a process involving several indistinguishable particles is determined by the amplitudes of all possible transitions of these particles when treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the possibility that identical particles generically exhibit anyonic behavior in two dimensions is excluded.
In "Are we there yet? The search for a theory of everything", ed. M. Emam (Bentham Press, 2011)
Philip Goyal
Phys. Rev. A 89, 032120 (2014)
Philip Goyal
Abstract: Feynman's formulation of quantum theory is remarkable in its combination of formal simplicity and computational power. However, as a formulation of the abstract structure of quantum theory, it is incomplete as it does not account for most of the fundamental mathematical structure of the standard von Neumann–Dirac formalism such as the unitary evolution of quantum states. In this paper, we show how to reconstruct the entirety of the finite-dimensional quantum formalism starting from Feynman's rules with the aid of a single new physical postulate, the no-disturbance postulate. This postulate states that a particular class of measurements have no effect on the outcome probabilities of subsequent measurements performed. We also show how it is possible to derive both the amplitude rule for composite systems of distinguishable subsystems and Dirac's amplitude–action rule, each from a single elementary and natural assumption, by making use of the fact that these assumptions must be consistent with Feynman's rules.
In "Are we there yet? The search for a theory of everything", ed. M. Emam (Bentham Press, 2011)
Philip Goyal
Information 3, 567–594 (2012)
Philip Goyal
Abstract: The concept of information plays a fundamental role in our everyday experience, but is conspicuously absent in framework of classical physics. Over the last century, quantum theory and a series of other developments in physics and related subjects have brought the concept of information and the interface between an agent and the physical world into increasing prominence. As a result, over the last few decades, there has arisen a growing belief amongst many physicists that the concept of information may have a critical role to play in our understanding of the workings of the physical world, both in more deeply understanding existing physical theories and in formulating of new theories. In this paper, I describe the origin of the informational view of physics, illustrate some of the work inspired by this view, and give some indication of its implications for the development of a new conception of physical reality.
In "Are we there yet? The search for a theory of everything", ed. M. Emam (Bentham Press, 2011)
Philip Goyal
Symmetry 3(2), 171-206 (2011)
Philip Goyal, K. H. Knuth
Abstract: Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid.
In "Are we there yet? The search for a theory of everything", ed. M. Emam (Bentham Press, 2011)
Philip Goyal
In "Are we there yet? The search for a theory of everything", ed. M. Emam (Bentham Press, 2011)
Philip Goyal
Abstract: Quantum theory poses deep challenges to the mechanical conception of reality that underlies classical physics. Yet today, over eighty years after its creation, its implications for our picture of reality remain enshrouded in uncertainty. In view of the current search for a more comprehensive theory of physics, it is vital that these implications be clearly elucidated. In this article, I describe the nature of the challenge posed by quantum theory, and outline efforts that have been made to better understand its non-classical features, such as non-locality. In particular, I discuss the informational perspective, which, through the study of quantum information processing, has provided deep insights into the nature of quantum reality, and has also revitalized the long-standing quest to reconstruct the content of the rather mysterious mathematical formalism of quantum theory from a set of crisp physical principles. Finally, I indicate some implications of recent reconstructive work for the search for a theory of quantum gravity, and, more broadly, for our picture of physical reality.
In "Proceedings of the First Interdisciplinary CHESS conference", ed. C. Rangacharyulu & E. Haven (World Scientific, 2010)
Philip Goyal
In "Proceedings of the First Interdisciplinary CHESS conference", ed. C. Rangacharyulu & E. Haven (World Scientific, 2010)
Philip Goyal
Abstract: Quantum theory is an extraordinarily successful physical theory. But what does it mean? What implications does it have for the mechanical conception of Nature that underlies classical physics? Remarkably, some eighty years after the creation of quantum theory, we still lack clear answers to these questions. In this paper, we discuss the nature of the obstacles that stand in our way, and describe recent work to overcome them by attempting to reconstruct the mathematics of quantum theory from a small number of simple physical ideas.
In "Are we there yet? The search for a theory of everything", ed. M. Emam (Bentham Press, 2011)
Philip Goyal
Phys. Rev. A 81, 022109 (2010)
Philip Goyal, K. H. Knuth, J. Skilling
Abstract: Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. In this paper, we show that it is possible to derive the complex nature of the quantum formalism directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, and that the probability of this sequence is a real-valued function of this number pair. By making use of elementary symmetry and consistency conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.
New J. Phys. 12 (2010) 023012
Philip Goyal
New J. Phys. 12 (2010) 023012
Philip Goyal
Abstract: In this paper, we show how information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism. The derivation rests upon three elementary features of quantum phenomena, namely complementarity, measurement simulability, and global gauge invariance. When these features are appropriately formalized within an information geometric framework, and combined with a novel information-theoretic principle, the central features of the finite-dimensional quantum formalism can be reconstructed.
arXiv (2009)
Philip Goyal
arXiv (2009)
Philip Goyal
Abstract: To apply the abstract quantum formalism to a particular physical system, one must specify the precise form of the relevant measurement and symmetry transformation operators. These operators are determined by a set of rules, the correspondence rules of quantum theory. The physical origin of these rules is obscure, and their physical interpretation and their degree of generality is presently unclear. In this paper, we show that all of the commonly-used correspondence rules can be systematically derived from a new physical principle, the Average-Value Correspondence Principle. This principle shows that the correspondence rules result from the systematic translation of relations between measurement results known to hold in a classical model of a system, providing these rules with a clear physical interpretation, and clearly demarcating their domain of applicability.
Phys. Rev. A 78, 052120 (2008)
Philip Goyal
Phys. Rev. A 78, 052120 (2008)
Philip Goyal
Abstract: In this paper, we show how the framework of information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism. The derivation rests upon a few elementary features of quantum phenomena, such as complementarity and global gauge invariance. When appropriately formulated within an information-geometric framework, and combined with a novel information-theoretic principle, these features lead to the abstract quantum formalism for finite-dimensional quantum systems, and the result of Wigner's theorem. By means of a correspondence principle, several correspondence rules of quantum theory, such as the canonical commutation relationships, are also systematically derived. The derivation suggests that information geometry is directly or indirectly responsible for many of the central structural features of the quantum formalism, such as the importance of square roots of probability and the occurrence of sinusoidal functions of phases in a pure quantum state. Global gauge invariance is shown to play a crucial role in the emergence of the formalism in its complex form.
arXiv:quant-ph/0702124 (2007)
Philip Goyal
arXiv:quant-ph/0702124 (2007)
Philip Goyal
Abstract: In this paper and a companion paper, we attempt to systematically investigate the possibility that the concept of information may enable a derivation of the quantum formalism from a set of physically comprehensible postulates. To do so, we formulate an abstract experimental set-up and a set of assumptions based on generalizations of experimental facts that can be reasonably taken to be representative of quantum phenomena, and on theoretical ideas and principles, and show that it is possible to deduce the quantum formalism. In particular, we show that it is possible to derive the abstract quantum formalism for finite-dimensional quantum systems and the formal relations, such as the canonical commutation relationships and Dirac’s Poisson Bracket rule, that are needed to apply the abstract formalism to particular systems of interest. The concept of information, via an information-theoretic invariance principle, plays a key role in the derivation, and gives rise to some of the central structural features of the quantum formalism.
Bayesian Inference and Maximum Entropy Methods (2005)
Philip Goyal
Bayesian Inference and Maximum Entropy Methods (2005)
Philip Goyal
Abstract: General theoretical principles that enable the derivation of prior probabilities are of interest both in practical data analysis and, more broadly, in the foundations of probability theory. In this paper, it is shown that the general rule for the assignment of priors proposed by Jeffreys can be obtained from, and is logically equivalent to, an intuitively reasonable information-theoretical invariance principle. Some of the implications for the priors proposed by Hartigan, Jaynes, and Skilling, are also discussed.