Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.
This doctoral thesis in mathematics consists of three articles that explore the probabilistic structure of quantum mechanics and quantization from a novel perspective. The thesis adopts a probabilistic interpretation of quantum mechanics, which views the archetypical quantum experiments of Bell- and double-slit- type as violating the principle of non-contextuality, i.e., the assertion that all events and observables are always representable on one single Kolmogorovian probability space, rather than the principles of realism or locality. This probabilistic interpretation posits that quantum mechanics constitutes a probability theory that adheres to the principle of contextuality, and that quantum events explicitly occur at the level of measurement, rather than the level of that which is measured, as these are traditionally interpreted. The thesis establishes a natural connection between the probabilistic structure of quantum mechanics, specifically Born’s rule, and the frequentist interpretation of probability. The major conceptual step in establishing this connection is to re-identify quantum observables instead as unitary representations of groups, whose irreducible sub-representations correspond to the observable’s different possible outcomes, rather than primarily as self- adjoint operators. Furthermore, the thesis reformulates classical statistical mechanics in the formalism of quantum mechanics, known as the Koopman-von Neumann formulation, to demonstrate that classical statistical mechanics also adheres to the principle of contextuality. This finding is significant because it raises questions about the existence of a hidden-variable model of classical statistical mechanics of the kind as examined in Bell’s theorem, where this presumed hidden-variable model traditionally has been seen as that which distinguishes "classical" from "quantum" probability.A novel reformulation of quantization is proposed considering it rather in terms of the representation theory of Hamiltonian flows and their associated inherent symmetry group of symplectomorphisms. Contrary to the traditional view of quantization, this formulation can be regarded as compatible with the probabilistic interpretation of quantum mechanics and offers a new perspective on the quantization of gravity.
We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.
As part of a probabilistic reconstruction of quantum theory (QT), we show that spin is not a purely quantum mechanical phenomenon, as has long been assumed. Rather, this phenomenon occurs before the transition to QT takes place, namely in the area of the quasi-classical (here better quasi-quantum) theory. This borderland between classical physics and QT can be reached within the framework of our reconstruction by the replacement
p→M(q,t), where
p is the momentum variable of the particle and
M(q,t) is the momentum field in configuration space. The occurrence of spin, and its special value
1/2 , is a consequence of the fact that
M(q,t) must have exactly three independent components
Mk(q,t) for a single particle because of the three-dimensionality of space. In the Schrödinger equation for a "particle with spin zero", the momentum field is usually represented as a gradient of a single function
S. This implies dependencies between the components
Mk(q,t) for which no explanation exists. In reality,
M(q,t) needs to be represented by three functions, two of which are rotational degrees of freedom. The latter are responsible for the existence of spin. All massive structureless particles in nature must therefore be spin-one-half particles, simply because they have to be described by
4 real fields, one of which has the physical meaning of a probability density, while the other three are required to represent the momentum field in three-dimensional space. We derive the Pauli-Schrödinger equation, the correct value
g=2 of the gyromagnetic ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some other open questions in the borderland between classical physics and QT.