Several recent studies have suggested that incompatible variables, which play an essential role in quantum mechanics (QM), are, somewhat surprisingly, not necessarily unique to QM. To investigate this possibility and obtain a better understanding of two central postulates of QM, namely the commutator postulate and the Born postulate, we introduce a classical probabilistic theoretical framework which is more general than QM and contains QM as a special case. We call this framework the General Incompatible Variables (GIV) theory, and we show that not only QM systems but also any probabilistic systems (classical or quantal) that possess incompatible variables will exhibit the quantal properties of uncertainty and interference, and we illustrate this with a roulette-like classical system (which we call the Arrow) that shows precisely these properties. We show that QM emerges naturally from the GIV framework when the fundamental variables are taken to be symmetries, so that the incompatibility of the QM variables is actually just the incompatibility of the corresponding symmetries (or their generators). Specifically, when the variables are taken to be the elements of the Poincare group (the symmetry of spacetime) we find that the commutator postulate and the Born postulate follow automatically, and are therefore no longer postulates. QM thus emerges as a special case of GIV theories for which the variables are symmetries.
Several recent studies have suggested that incompatible variables, which play an essential role in quantum mechanics (QM), are, somewhat surprisingly, not necessarily unique to QM. To investigate this possibility and obtain a better understanding of two central postulates of QM, namely the commutator postulate and the Born postulate, we introduce a classical probabilistic theoretical framework which is more general than QM and contains QM as a special case. We call this framework the General Incompatible Variables (GIV) theory, and we show that not only QM systems but also any probabilistic systems (classical or quantal) that possess incompatible variables will exhibit the quantal properties of uncertainty and interference, and we illustrate this with a roulette-like classical system (which we call the Arrow) that shows precisely these properties. We show that QM emerges naturally from the GIV framework when the fundamental variables are taken to be symmetries, so that the incompatibility of the QM variables is actually just the incompatibility of the corresponding symmetries (or their generators). Specifically, when the variables are taken to be the elements of the Poincare group (the symmetry of spacetime) we find that the commutator postulate and the Born postulate follow automatically, and are therefore no longer postulates. QM thus emerges as a special case of GIV theories for which the variables are symmetries.
We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set A of incompatible experiments, and a transformation group G on the cartesian product Π of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of Π, an orbit or a set of orbits of G. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space H. A state is equivalent to a question together with an answer: the choice of an experiment a∈A plus a value for the corresponding parameter. Finally, probabilities are introduced through Born’s formula, which is derived from a recent version of Gleason’s theorem. This then leads to the usual formalism of elementary quantum mechanics in important special cases. The theory is illustrated by the example of a quantum particle with spin.
Quantal physics is established as a manifestation of symmetry more far-reaching than hitherto appears to have been recognized. In this primary manifestation, the coordinate transformations of spacetime invariance are themselves the elementary variables, which define their own properties without appeal to an assumed quantal formalism. In irreducible representations, the symmetry variables are inherently indeterminate, and the probabilistic laws invoked in the interpretation of traditional quantum physics are found to originate in geometric relations between these variables. Completeness is, therefore, not an issue, and the quantum of action is not part of the theory of symmetry variables. Quantal physics thus emerges as but an implication of relativistic invariance, liberated from a substance to be quantized and a formalism to be interpreted. A symmetry variable appears in a measurement with one of its eigenvalues, but does not have a value (cannot be represented by a number) in an irreducible representation, which combines sets of eigenvalues. It is this generalized significance of a measurement that allows for correlations that cannot arise for classical variables. The observation of symmetry variables is illustrated by an interferometer experiment measuring reflection symmetry and by the equivalent coincidence experiment registering the polarization of two quanta. The measurement process becomes a matter of following the state of affairs of the symmetry variables in their unitary evolution. For the resolution of the dilemmas that quantal phenomena have been felt to pose, it appears crucial to recognize that indeterminacy, as an inherent property of a symmetry variable in a multidimensional representation, is not affected by subsequent observations. A position variable and the canonical commutator with momentum, which are basic elements of nonrelativistic quantum mechanics, emerge from spacetime symmetry, but require the link between space and time of relativistic invariance. The transition to the classical regime is analyzed in terms of a quenching of nonlocality in the state of affairs of the multidimensional symmetry variables. While the elementary variables constitute individual quanta in irreducible representations, product representations of spacetime symmetry describe systems of bosons and fermions, which form local fields with canonical properties. The discussion is focused on spacetime invariance (noninteracting quanta), but gauge invariance is itself a primary manifestation of symmetry and is as such encompassed by the theory of symmetry variables.
We present a group-theoretic reconstruction of quantum mechanics in which the standard formalism emerges from the representation theory of symmetry groups rather than being introduced axiomatically. Starting from physically motivated symmetry principles, we show how the conventional postulates of quantum mechanics, including the Hilbert space structure of states, observables as generators of symmetry transformations, the Born probability rule, unitary time evolution, the composition of multipartite systems, and the measurement framework, can be recovered within a unified representation-theoretic framework. We then examine the measurement problem by interpreting wave-function collapse as the scattering of quantum states, thereby accounting for the apparent loss of information. This framework offers a conceptually unified perspective on the foundations of quantum mechanics, connecting its postulates, measurement theory, and the emergence of quantum phenomena to a common symmetry-based origin.
Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. All axiomatization attempts suffer at least one postulate of a mathematical nature. Here I will analyze the possibility of deriving QM as the mathematical representation of a "fair operational framework", i.e. a set of rules which allows the experimenter to make predictions on future "events" on the basis of suitable "tests", e.g. without interference from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: "no-signaling from the future"--for the possibility of making predictions on the basis of past tests; PFAITH: "existence of a preparationally faithful state"--for the possibility of preparing any state and calibrating any test. I will show that all theories satisfying NSF admit a C*-algebra representation of events as linear transformations of effects. Based on a very general notion of dynamical independence, it is easy to see that all such probabilistic theories are "non-signaling without interaction" ("non-signaling" for short)--another requirement for a fair operational framework. Postulate PFAITH then implies the "local observability principle", the tensor-product structure for the linear spaces of states and effects, the impossibility of bit commitment and additional features, such an operational definition of transpose, a scalar product for effects, weak-selfduality of the theory, and more. Dual to Postulate PFAITH an analogous postulate for effects would give additional quantum features, such as teleportation. However, all possible consequences of these postulates still need to be investigated, and it is not clear yet if we can derive QM from the present postulates only. [CONTINUES on manuscript]
Hamiltonian mechanics describes the evolution of a system through its Hamiltonian. The Hamiltonian typically also represents the energy observable, a Noether-conserved quantity associated with the time invariance of the law of evolution. In both quantum and classical mechanics, Hamiltonian mechanics demands a precise relationship between time evolution and observable energy, albeit using slightly different terminology. We distill basic conditions satisfied in both quantum and classical mechanics, including canonical coordinate symmetries and inner product invariance. We express these conditions in the framework of generalized probabilistic theories, which includes generalizing the definition of energy eigenstates in terms of time-invariant properties of the Hamiltonian system. By postulating these conditions to hold, we derive a unified Hamiltonian system model. This unified framework describes quantum and classical mechanics in a consistent language, facilitating their comparison. We moreover discuss alternative theories: an equation of motion given by a mixture of commutation relations, an information-restricted version of quantum theory, and Spekken's toy theory. The findings give a deeper understanding of the Hamiltonian in quantum and classical theories and point to several potential research topics.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.