Derivation of Quantum Theory from Feynman's Rules (2014)

Philip Goyal

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.

Phys. Rev. A 89, 032120 (2014)




[arxiv:1403.3527]

Informational Approach to Identical Particles in Quantum Theory (2013)

Philip Goyal

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. For example, our understanding of the structure of the periodic table hinges on treating the electrons in multi-electron atoms as indistinguishable. 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 precise connection between particle indistinguishability and the symmetrization postulate has not been established. There exist a number of variants of the postulate that appear to be compatible with particle indistinguishability, and a well-known derivation of the postulate implies that its validity depends on the dimensionality of space. These variants leave open the possibility that there exist elementary particles, such as anyons, which violate the symmetrization postulate. Here we show that the symmetrization postulate can be derived on the basis of the indistinguishability postulate. This postulate establishes a functional relationship between the amplitude of a process involving indistinguishable particles and the amplitudes of all possible transitions when the particles are treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The key to the derivation is a strictly informational treatment of indistinguishability which prohibits the labelling of particles that cannot be experimentally distinguished from one another. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the existence of anyons as elementary particles is excluded.

arXiv version, 2nd Sept 2013

[arxiv:1309.0478]

Information Physics—Towards a New Conception of Physical Reality (2012)

Philip Goyal

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.

Information 2012, 3, 567–594

Quantum Theory and Probability Theory: their Relationship and Origin in Symmetry (2011)

Philip Goyal, Kevin H. Knuth

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.

Symmetry 2011, 3(2), 171-206

Origin of Complex Quantum Amplitudes and Feynman's Rules (2009)

Philip Goyal, Kevin H. Knuth, John Skilling

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.

Phys. Rev. A 81, 022109 (2010)




[arxiv:0907.0909]

Origin of the Correspondence Rules of Quantum Theory (2009)

Philip Goyal

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.


arXiv:0910.2444

Origin of Complex Quantum Amplitudes (2009)

Philip Goyal, Kevin H. Knuth, John Skilling

Physics is real. Measurement produces real numbers. Yet quantum mechanics uses complex arithmetic, in which √ −1 is necessary but mysteriously relates to nothing else. By applying the same sort of symmetry arguments that Cox [1, 2] used to justify probability calculus, we are now able to explain this puzzle. The dual device/object nature of observation requires us to describe the world in terms of pairs of real numbers about which we never have full knowledge. These pairs combine according to complex arithmetic, using Feynman’s rules.

Presented at the 29th International Workshop on Bayesian Inference and Maximum Entropy Methods, Oxford, MS (Jully 2009)

Published in "Proceedings of the 29th International Workshop on Bayesian Inference and Maximum Entropy Methods", ed. P.Goggans, 2009

From Information Geometry to Quantum Theory (2008)

Philip Goyal

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.


New J. Phys. 12 (2010) 023012



[
arXiv:0805.2770]

Information-Geometric Reconstruction of Quantum Theory (2008)

Philip Goyal

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.


Phys. Rev. A 78, 052120 (2008)


An Information-Geometric Reconstruction of Quantum Theory, I: Abstract Quantum Formalism (2008)

Philip Goyal

In this paper and a companion paper, we show how the framework of information geometry, a geometry of discrete probability distributions, can form the basis of a derivation of the quantum formalism. The derivation rests upon a few elementary features of quantum phenomena, such as the statistical nature of measurements, complementarity, and global gauge invariance. It is shown that these features can be traced to experimental observations characteristic of quantum phenomena and to general theoretical principles, and thus can reasonably be taken as a starting point of the derivation. When appropriately formulated within an information geometric framework, these features lead to (i) the abstract quantum formalism for finite-dimensional quantum systems, (ii) the result of Wigner's theorem, and (iii) the fundamental correspondence rules of quantum theory, such as the canonical commutation relationships. The formalism also comes naturally equipped with a metric (and associated measure) over the space of pure states which is unitarily- and anti-unitarily invariant. The derivation suggests that the information geometric framework 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 seen to play a crucial role in the emergence of the formalism in its complex form.


arxiv:0805.2761

An Information-Geometric Reconstruction of Quantum Theory, II: Correspondence Rules (2008)

Philip Goyal

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.

arxiv:0805.2765

The Role of Information in the Probabilistic Reconstruction of Quantum Theory (2007)

Philip Goyal

In this paper, we explore the possibility that the concept of information may enable a derivation of the quantum formalism from a set of physically comprehensible postulates. Taking the probabilistic nature of measurements as a given, we introduce the concept of information via a novel invariance principle, the Principle of Information Gain. Using this principle, we then show that it is possible to deduce the abstract quantum formalism for finite-dimensional quantum systems from a set of postulates, of which one is a novel physical assumption, and the remainder are based on experimental facts characteristic of quantum phenomena or are drawn from classical physics. The concept of information 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, ed. K. Knuth et. al., 2007

The Average-Value Correspondence Principle (2007)

Philip Goyal

In previous work [1], we have presented an attempt to derive the finite-dimensional abstract quantum formalism
from a set of physically comprehensible assumptions. In this paper, we continue the derivation of the quantum formalism
by formulating a correspondence principle, the Average-Value Correspondence Principle, that allows relations between
measurement outcomes which are known to hold in a classical model of a system to be systematically taken over into the
quantum model of the system, and by using this principle to 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.


Quantum Theory: Reconsideration of Foundations – 4, ed. Adenier et. al. (2007)

An information-Theoretic Approach to Quantum Theory, I: The Abstract Quantum Formalism (2007)

Philip Goyal

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.


arXiv:quant-ph/0702124

An Information-Theoretic Approach to Quantum Theory, II: The Formal Rules of Quantum Theory (2007)

Philip Goyal

In a companion paper (hereafter referred to as Paper I), we have presented an attempt to derive the finite-dimensional abstract quantum formalism from a set of physically comprehensible assumptions. In this paper, we formulate a correspondence principle, the Average-Value Correspondence Principle, that allows relations between measurement outcomes 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 formal 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.


arxiv:quant-ph/0702149

Prior Probabilities: An Information-Theoretic Approach (2005)

Philip Goyal

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.


Bayesian Inference and Maximum Entropy Methods, ed. A. Abbas, K. Knuth, 2005