The identification of physical subsystems in quantum mechanics as compared to classical mechanics poses significant conceptual challenges, especially in the context of quantum gravity. Traditional approaches associate quantum systems with classical ones localized in spacetime, using either Hilbert space factors for finite-dimensional systems or local operator algebras in algebraic quantum field theory. These methods ensure statistical independence for state preparations and measurements. However, covariant linearized quantum gravity disrupts this framework by preventing the formation of gauge-invariant local algebras, thereby undermining statistical independence. This presents a major obstacle for modeling early universe cosmology and gravity-induced entanglement experiments, and poses a significant roadblock toward a comprehensive theory of quantum gravity. A pivotal shift is proposed: the identification between classical and quantum systems should be dynamically evolving rather than fixed, opening the possibility of a single-world unitary quantum mechanics. This perspective aligns with the broader aim of understanding how classical spatiotemporal existence emerges from quantum mechanics and connects the measurement problem with quantum gravity.
The identification of physical subsystems in quantum mechanics as compared to classical mechanics poses significant conceptual challenges, especially in the context of quantum gravity. Traditional approaches associate quantum systems with classical ones localized in spacetime, using either Hilbert space factors for finite-dimensional systems or local operator algebras in algebraic quantum field theory. These methods ensure statistical independence for state preparations and measurements. However, covariant linearized quantum gravity disrupts this framework by preventing the formation of gauge-invariant local algebras, thereby undermining statistical independence. This presents a major obstacle for modeling early universe cosmology and gravity-induced entanglement experiments, and poses a significant roadblock toward a comprehensive theory of quantum gravity. A pivotal shift is proposed: the identification between classical and quantum systems should be dynamically evolving rather than fixed, opening the possibility of a single-world unitary quantum mechanics. This perspective aligns with the broader aim of understanding how classical spatiotemporal existence emerges from quantum mechanics and connects the measurement problem with quantum gravity.
In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.
Quantum measurement is a fundamental concept in the field of quantum mechanics. The action of quantum measurement, leading the superposition state of the measured quantum system into a definite output state, not only reconciles contradictions between quantum and classical mechanics but also facilitates quantum state manipulations, including reading and resetting. Despite its significance, four fundamental issues -- randomness, instantaneousness, irreversibility, and preferred-basis -- continue to pose significant challenges to the broader application of quantum measurement and our overall understanding of quantum mechanics. In this work, we employ a new space-time approach to relativistic quantum mechanics to address these issues systematically. Our approach provides a comprehensive elucidation of the intricate connections between quantum measurement and quantum unitary evolution, as well as an in-depth analysis for the interdependence of non-local correlations and relativistic theories. We thereby reveal a more fundamental dynamical theory, beyond the traditional time-evolution equation in quantum mechanics, where the axioms of quantum measurement naturally emerge as a corollary. These findings contribute to the advancement of related fields, and our work holds potential implications for future research and applications in the realm of quantum mechanics.
Quantum systems are viewed as emergent systems from the fundamental degrees of freedom. The laws and rules of quantum mechanics are understood as an effective description, valid for the emergent systems and specially useful to handle probabilistic predictions of observables. After introducing the geometric theory of Hamilton-Randers spaces and reformulating it using Hilbert space theory, a Hilbert space structure is constructed from the Hilbert space formulation of the underlying Hamilton-Randers model and associated with the space of wave functions of quantum mechanical systems. We can prove the emergence of the Born rule from ergodic considerations. A geometric mechanism for a natural spontaneous collapse of the quantum states based on the concentration of measure phenomena as it appears in metric geometry is discussed.We show the existence of stable vacua states for the quantized matter Hamiltonian. Another consequence of the concentration of measure is the emergence of a weak equivalence principle for one of the dynamics of the fundamental degrees of freedom. We suggest that the reduction of the quantum state is driven by a gravitational type interaction. Such interaction appears only in the dynamical domain when localization of quantum observables happens, it must be a classical interaction. We discuss the double slit experiment in the context of the framework proposed, the interference phenomena associated with a quantum system in an external gravitational potential, a mechanism explaining non-quantum locality and also provide an argument in favour of an emergent interpretation of every macroscopic time parameter. Entanglement is partially described in the context of Hamilton-Randers theory and how naturally Bell's inequalities should be violated.
Quantum gravity suggests that spacetime may not be fundamental, and it has been argued that we can understand a theory without a fundamental spacetime if we are able to claim that spacetime 'emerges' from some non-spatiotemporal entities. In this sense, strategies like functionalism have been deployed to claim that this emergence is possible and plausible, both in principle and in practice for current approaches to quantum gravity. In this article I argue that this analysis is incomplete, as it tends to overlook the way the dynamics of these theories is 'quantum' in a way that differs from standard quantum theory. The challenge for the emergentist (and for the quantum gravity theorist) is to give an interpretation not only to the kinematical and classical aspects of these theories, but to the dynamical and quantum ones, and to show how the spacetime roles can be fulfilled, if possible at all. Therefore, I argue that some current approaches to quantum gravity seem to fail to provide meaningful theories, that spacetime functionalism is of no help, and that the position of the spacetime emergentists is weakened, as they lack any example of a successful reduction of spacetime to some truly quantum non-spatiotemporal stuff.
Quantum information has become a powerful tool for probing the structure of quantum field theories, yet its application to gauge theories remains subtle. On the one hand, quantum information theory assumes subsystem locality, i.e.~the factorization of the total Hilbert space into subsystems. On the other hand, gauge constraints prevent the total Hilbert space to decompose into a spacetime-local tensor product structure. Because the Hilbert space structure of gauge theories does not accommodate the subsystem decomposition used in quantum information theory, standard information-theoretic results, such as the Local Operations and Classical Communication (LOCC) theorem, cannot be used straightforwardly in the context of gauge theories. In this work, we bridge this gap in the case of a two-dimensional lattice gauge model that captures key features of electromagnetism. In particular, we construct gauge-invariant local algebras and derive a physically meaningful decomposition of the Hilbert space, providing an operationally consistent notion of locality in the absence of a local tensor-product structure. We apply this framework to field-mediated entanglement protocols relevant to proposed tests of the quantum nature of gravity. We show that the discretized version of electromagnetism satisfies an analogue of the LOCC theorem: entanglement cannot be generated without genuine quantum field interactions, even in the absence of a spacetime-local tensor product factorization of the Hilbert space. This may point towards an operational way to define a subsystem structure for gauge theories.
Quantum information has become a powerful tool for probing the structure of quantum field theories, yet its application to gauge theories remains subtle. On the one hand, quantum information theory assumes subsystem locality, i.e.~the factorization of the total Hilbert space into subsystems. On the other hand, gauge constraints prevent the total Hilbert space to decompose into a spacetime-local tensor product structure. Because the Hilbert space structure of gauge theories does not accommodate the subsystem decomposition used in quantum information theory, standard information-theoretic results, such as the Local Operations and Classical Communication (LOCC) theorem, cannot be used straightforwardly in the context of gauge theories. In this work, we bridge this gap in the case of a two-dimensional lattice gauge model that captures key features of electromagnetism. In particular, we construct gauge-invariant local algebras and derive a physically meaningful decomposition of the Hilbert space, providing an operationally consistent notion of locality in the absence of a local tensor-product structure. We apply this framework to field-mediated entanglement protocols relevant to proposed tests of the quantum nature of gravity. We show that the discretized version of electromagnetism satisfies an analogue of the LOCC theorem: entanglement cannot be generated without genuine quantum field interactions, even in the absence of a spacetime-local tensor product factorization of the Hilbert space. This may point towards an operational way to define a subsystem structure for gauge theories.