arXiv subjects

Philip Goyal

Publications and source records attributed to Philip Goyal.

At least 19 recordsLinked to original sources

arXiv2026

Understanding Quantum Theory: An Operational Reconstructive Approach

One hundred years after the creation of quantum theory, there is no consensus on the kind of reality that is described by the theory. Here, I attribute the lack of progress to the prevailing interpretative methodology, which invariably takes the quantum formalism as the starting point for philosophical reflection and analysis. I argue that this methodology is particularly inappropriate, for it invariably marginalizes much of the theory's content, both that implicit in modelling heuristics and experimental practices, and that encapsulated in the mathematical structures of its formalism. In addition, the prevailing methodology offers little protection against undue influence by metaphysically-laden language which invariably accompanies the formalism. Here, I summarize an alternative methodology whose goal is to ensure that an interpretational project take into account all forms of theoretic content. The methodology harnesses the recent results of the quantum reconstruction program. These results distil the mathematical content of the quantum formalism into physical principles and assumptions, which are more readily philosophically digestible than the formalism itself, and bracket much of its metaphysically-laden language. As a case study of reconstruction-based interpretation, I describe the reconstruction of the identical particle formalism, and its step-by-step interpretation, highlighting the key questions that drive the interpretation forwards and the techniques and stances that are employed in each step. The interpretation yields a novel metaphysical profile for systems of identical particles as potential parts of a whole, which can be traced step-by-step to elementary experimental data and the reconstruction's physical postulates and assumptions. I also describe some of the pitfalls that one faces in any attempt to directly interpret the identical particle formalism.

quant-ph↗

arXiv2026

From Complementarity to Quantum Properties: An Operational Reconstructive Approach

Quantum theory brings into question the compatibility of the twin desiderata of exact knowability of the present state of the physical world and perfect predictability of its future states. Bohr's coordination-causality complementarity principle transforms this tension into one between properties (as ordinarily understood in classical physics) and deterministic causality. Here, we develop an explicit model of quantum properties which accommodates this essential tension. Our approach integrates operational, reconstructive, and metaphysical standpoints. In particular, we make use of an operational framework employed in a recent operational reconstruction of Feynman's formulation of quantum theory; base our property model on an analysis of property types; and use the notions of actuality and potentiality to frame the model. We show that this quantum property model provides a natural resolution of Zeno's paradox of motion, and provides reliable intuitions about phenomena such as electron diffraction and the non-local behaviour of entangled states of non-identical particles.

quant-ph↗

arXiv2026

Identical Quantum Particles as Potential Parts

The mathematical rules used to handle systems of identical quantum particles bring into question whether the elementary constituents of matter, such as electrons, have the fundamental characteristics of persistence and reidentifiability that are usually attributed to classical particles. However, despite considerable philosophical debate, the metaphysical profile of these entities remains elusive. Previous debates have taken the mathematical rules, and the language in which these are usually couched, as a starting point. Here, we argue that this methodology is inherently limited, and develop a new conception of identical particles based on a recent mathematical reconstruction of these rules. Using this reconstruction, we demonstrate that the special behaviour of identical particles originates in the confluence of identicality and the active nature of the quantum measurements. We propose that identical particles are appropriately viewed as potential parts of a whole, and show how this leads to striking consequences such as restricted transtemporal identity.

quant-ph↗

arXiv2025

Systematizing the Interpretation of Quantum Theory via Reconstruction

For a century, quantum theory has posed a fundamental challenge to philosophical thinking. On its face, it repudiates many of the key features of the mechanical conception of physical reality. However, the challenge of developing a precise, coherent alternative to that conception has yet to be met. Here, I argue that a major hindrance to the project of quantum interpretation is its existing interpretative methodologies, which suffer from a lack of systematicity in their judgements about what aspects of the theory are interpretational relevant. In particular, I argue that current interpretations tend to marginalize the informal part of the theory in favour of its formal part, and place inappropriate emphasis on the natural language component of the formalism over its detailed mathematical structure. To counterbalance these biases, I propose that an interpretation-free zone be constructed around the theory, wherein an interpreter initially adopt a descriptive stance which considers all parts of the theory, and that the results of this deliberation~(and the judgements about what facts are interpretationally relevant) are reported as part of their interpretation. I argue that the interpretation of quantum theory poses special challenges and difficulties which necessitate this interpretation-free zone, and that existing interpretative methodologies are insufficient to address them. Further, I argue that a reconstructive interpretative methodology, which harnesses the recent results of the quantum reconstruction program, provides a powerful means to identify almost all facts that could be interpretationally relevant, and naturally meets these challenges and difficulties. Moreover, I argue that the quantum reconstruction program offers a powerful way to discover new physical principles, and offers a systematic pathway to build a rich, coherent conception of quantum reality.

quant-ph↗

arXiv2025

Operational reconstruction of Feynman rules for quantum amplitudes via composition algebras

This article explores an operational model for transition amplitudes between measurements proposed by Goyal et al. within the quantum reconstruction program. To classify suitable amplitude algebras, we distinguish mathematical axioms, physical choices, and their consequences. This leads to several improvements on the published work: Our coordinate-independent approach requires no two-dimensional amplitudes a priori. All scalar field and vector space axioms are traced from model axioms and observer choices, including additive and multiplicative units and inverses. Existing mathematical characterizations identify allowable amplitude algebras as the real associative composition algebras, namely the complex numbers and the quaternions, as well as their split forms. Observed probabilities are quadratic in amplitudes, akin to the Born rule. We examine selected implications of the proposed axioms, reformulate observer questions, and highlight the broad applicability of our framework to subsequent discovery.

quant-ph↗

arXiv2024

The Structure of Quantum Questions

In classical physics, a single measurement can in principle reveal the state of a system. However, quantum theory permits numerous non-equivalent measurements on a physical system, each providing only limited information about the state. This set of various measurements on a quantum system indicates a rich internal structure. We illuminate this structure for both individual and composite systems by conceptualizing measurements as questions with a finite number of outcomes. We create a mathematical question structure to explore the underlying properties, employing the concept of information as a key tool representing our knowledge gained from asking these questions. We subsequently propose informational assumptions based on properties observed from measurements on qubits, generalizing these to higher dimensional systems. Our informational assumptions shape the correlations between subsystems, which are symbolized as classical logical gates. Interestingly, systems with prime number dimensions exhibit unique property: the logical gate can be expressed simply as a linear equation under modular arithmetic. We also identify structures in quantum theory that correspond to those in the structure of quantum questions. For instance, the questions determining the system correspond to generalized Pauli matrices, and the logical gate connecting questions in subsystems is directly related to the tensor product combining operators. Based on these correspondences, we present two equivalent scenarios regarding the evolution of systems and the change of information within both quantum questions and quantum mechanics.

quant-ph↗

arXiv2023

Principle of Information Increase: An Operational Perspective of Information Gain in the Foundations of Quantum Theory

A measurement performed on a quantum system is an act of gaining information about its state, a view that is widespread in practical and foundational work in quantum theory. However, the concept of information in quantum theory reconstructions is multiply-defined, and its conceptual foundations remain surprisingly under-explored. In this paper, we investigate the gain of information in quantum measurements from an operational viewpoint. We show that the continuous extension of the Shannon entropy naturally admits two distinct measures of information gain, differential information gain and relative information gain, and that these have radically different characteristics. In particular, while differential information gain can increase or decrease as additional data is acquired, relative information gain consistently grows, and moreover exhibits asymptotic indifference to the data or choice of Bayesian prior. In order to make a principled choice between these measures, we articulate a Principle of Information Increase, which incorporates Summhammer's proposal that more data from measurements leads to more knowledge about the system, and also takes into consideration black swan events. This principle favors differential information gain as the more relevant metric in two-outcome quantum systems, and guides the selection of priors for these information measures. Finally, we show that, of the beta distribution priors, the Jeffreys' binomial prior is the prior ensures maximal robustness of information gain to the particular data sequence obtained in a run of experiments.

quant-ph↗

arXiv2023

The Born Rule -- Axiom or Result?

The Born rule is part of the collapse axiom in the standard version of quantum theory, as presented by standard textbooks on the subject. We show here that its signature quadratic dependence follows from a single additional physical assumption beyond the other axioms - namely, that the probability of a particular measurement outcome (the state $\phi_k$, say) is independent of the choice of observable to be measured, so long as one of its eigenstates corresponds to that outcome. We call this assumption ``observable independence.'' As a consequence, the Born rule cannot be completely eliminated from the list of axioms, but it can, in principle, be reduced to a more physical statement. Our presentation is suitable for advanced undergraduates or graduate students who have taken a standard course in quantum theory. It does not depend on any particular interpretation of the theory.

quant-ph↗

arXiv2018

Persistence and Nonpersistence as Complementary Models of Identical Quantum Particles

According to our understanding of the everyday physical world, observable phenomena are underpinned by persistent objects that can be reidentified (or tracked) across time by observation of their distinctive properties. This understanding is reflected in classical mechanics, which posits that matter consists of persistent, reidentifiable particles. However, the mathematical symmetrization procedures used to describe identical particles within the quantum formalism have led to the widespread belief that identical quantum particles lack either persistence or reidentifiability. However, it has proved difficult to reconcile these assertions with the fact that identical particles are routinely assumed to be reidentifiable in particular circumstances. Moreover, neither of these assertions accounts for the mathematical form of the symmetrization procedures used to describe identical particles within the quantum framework, leaving open theoretical possibilities other than bosonic and fermionic behavior, such as paraparticles, which do not appear to be realized in nature. Here we propose the novel idea that both persistence and nonpersistence models must be employed in order to fully account for the behaviour of identical particles. Thus, identical particles are neither persistent nor nonpersistent. 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 these models, and by showing how reidentifiability emerges in a context-dependent manner. We further show that the persistence and nonpersistence models satisfy the key characteristics of Bohr's concept of complementarity, and thereby propose that the behavior of identical particles is a manifestation of a persistence-nonpersistence complementarity, analogous to Bohr's wave-particle complementarity for individual particles.

quant-ph↗

arXiv2015

Fundamental groupoids in quantum mechanics: a new approach to quantization in multiply-connected spaces

Quantization of multiply-connected spaces requires tools which take these spaces' global properties into account. Applying these tools exposes additional degrees of freedom. This was first realized in the Aharonov-Bohm effect, where this additional degree of freedom was a magnetic flux confined to a solenoid, which an electron cannot enter. Previous work using Feynman path integrals has either only dealt with specific cases, or was limited to spaces with finite fundamental groups, and therefore, was in fact inapplicable to the Aharonov-Bohm effect, as well as to interesting systems such as anyons. In this paper we start from the fundamental groupoid. This less familiar algebraic-topological object is oriented towards general paths. This makes it a more natural choice for the path integral approach than the more commonly known fundamental group, which is restricted to loops. Using this object, we provide a method that works for spaces with infinite fundamental groups. We adapt a tool used in the previous, restricted result, in order to clearly delineate physically significant degrees of freedom from gauge freedom. This allows us to explicitly build phases which take account of symmetries of the space, directly from topological considerations; we end by providing a pertinent example relating to anyons.

math-ph↗

arXiv2014

Anyons in the Operational Formalism

The operational formalism to quantum mechanics seeks to base the theory on a firm foundation of physically well-motivated axioms [1]. It has succeeded in deriving the Feynman rules [2] for general quantum systems. Additional elaborations have applied the same logic to the question of identical particles, confirming the so-called Symmetrization Postulate [3]: that the only two options available are fermions and bosons [4,5]. However, this seems to run counter to results in two-dimensional systems, which allow for anyons, particles with statistics which interpolate between Fermi-Dirac and Bose-Einstein (see [6] for a review). In this talk we will show that the results in two dimensions can be made compatible with the operational results. That is, we will show that anyonic behavior is a result of the topology of the space in two dimensions [7], and does not depend on the particles being identical; but that nevertheless, if the particles are identical, the resulting system is still anyonic.

quant-ph↗

arXiv2014

On the Origin of the Quantum Rules for Identical Particles

We present a proof of the Symmetrization Postulate for the special case of noninteracting, identical particles. The proof is given in the context of the Feynman formalism of Quantum Mechanics, and builds upon the work of Goyal, Knuth and Skilling (Phys. Rev. A 81, 022109 (2010)), which shows how to derive Feynman's rules from operational assumptions concerning experiments. Our proof is inspired by an attempt to derive this result due to Tikochinsky (Phys. Rev. A 37, 3553 (1988)), but substantially improves upon his argument, by clarifying the nature of the subject matter, by improving notation, and by avoiding strong, abstract assumptions such as analyticity.

quant-ph↗

arXiv2014

Derivation of Quantum Theory from Feynman's Rules

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.

quant-ph↗

arXiv2013

Informational Approach to Identical Particles in Quantum Theory

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 the quantum formalism, indistinguishability is expressed via the symmetrization postulate, 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. However, this derivation relies on the labeling of indistinguishable particles, a notion that cannot be justified on an informational basis. 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 labeled. 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.

quant-ph↗

arXiv2010

Origin of the Correspondence Rules of Quantum Theory

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.

quant-ph↗

arXiv2010

From Information Geometry to Quantum Theory

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.

quant-ph↗

arXiv2010

Origin of Complex Quantum Amplitudes and Feynman's Rules

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 the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry 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.

quant-ph↗

arXiv2008

An Information-Geometric Reconstruction of Quantum Theory, II: The Correspondence Rules of Quantum Theory

In a companion paper (hereafter referred to as Paper I), we have presented an attempt to derive the finite-dimensional abstract quantum formalism within the framework of information geometry. In this paper, we formulate a correspondence principle, the Average-Value Correspondence Principle, that allows relations between measurement results which are known to hold in a classical model of a system to be systematically taken over into the quantum model of the system. Using this principle, we derive the explicit form of the temporal evolution operator (thereby completing the derivation of the abstract quantum formalism begun in Paper I), and derive many of the correspondence rules (such as operator rules, commutation relations, and Dirac's Poisson bracket rule) that are needed to apply the abstract quantum formalism to model particular physical systems.

quant-ph↗