Philip Goyal

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 characteristic feature. In this article, 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 Feynmans sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.

Information-geometric reconstruction of quantum theory

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 unitarilyand antiunitarily 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.

Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

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.

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.

The Origin of Complex Quantum Amplitudes

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.

Informational approach to the quantum symmetrization postulate

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.

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.

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 [1], which shows how to derive Feynmans rules from operational assumptions concerning experiments. Our proof is inspired by an attempt to derive this result due to Tikochinsky [2], 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.

WHAT IS QUANTUM THEORY TELLING US ABOUT HOW NATURE WORKS

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. Quantum theory is perhaps the most empirically successful theory in the history of physics. In the eighty or so years since its creation, it has pro- vided us with precise mathematical models that account for a vast range of physical phenomena ranging from the principles of chemical bonding, the nuclear reactions that fuel the stars, exotic phenomena like supercon- ductivity and superuidity, and many others besides. In short, quantum theory has been the faithful companion of physicists for eighty years, and has consistently met the challenge of providing mathematical models of new physical phenomena. Quantum theory also underlies much of the modern technology that fuels our lives. The transistor, the basis for the modern computer that underlies all information processing technology, requires quantum theory for its design and modelling. The same holds true for the laser and the light- emitting diode, which jointly provide the basis for optical communication networks (the backbone of the telecommunications industry) and optical data storage (in the form of CDs and DVDs, for example), and many other

The Average‐Value Correspondence Principle

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 Diracs Poisson bracket rule) that are needed to apply the abstract quantum formalism to model particular physical systems.