Despite its unparalleled empirical success, the standard formalism of quantum mechanics, strictly bound to complex Hilbert spaces, raises profound foundational questions regarding its axiomatic origins. Among the most enduring mysteries is why quantum theory permits pairwise, second-order interference, yet strictly forbids genuine higher-order multi-path interference. While Rafael Sorkin's quantum measure theory formalized this absence of higher-order interference mathematically, a comprehensive physical justification rooted in thermodynamics has remained elusive. This report provides an exhaustive analysis bridging Sorkin's hierarchy of interference with operational thermodynamics within the framework of Generalized Probabilistic Theories (GPTs). Building upon Krumm et al.'s seminal Theorem 12—which establishes a strict equivalence between the absence of third-order interference and the coincidence of measurement and decomposition max-entropies—this paper generalizes the Mamun, SM (2026) ……………………………….1 Higher-Order Interference and Thermodynamic Entropy in Generalized Hilbert Spaces Bridging Sorkin's Framework with Operational Thermodynamics (Vol-III) equivalence to arbitrary-order Rényi entropies across non-Hilbert state spaces. Also available on Academia.edu: https://www.academia.edu/attachments/132551016/download_file?s=portfolio
Despite its unparalleled empirical success, the standard formalism of quantum mechanics, strictly bound to complex Hilbert spaces, raises profound foundational questions regarding its axiomatic origins. Among the most enduring mysteries is why quantum theory permits pairwise, second-order interference, yet strictly forbids genuine higher-order multi-path interference. While Rafael Sorkin's quantum measure theory formalized this absence of higher-order interference mathematically, a comprehensive physical justification rooted in thermodynamics has remained elusive. This report provides an exhaustive analysis bridging Sorkin's hierarchy of interference with operational thermodynamics within the framework of Generalized Probabilistic Theories (GPTs). Building upon Krumm et al.'s seminal Theorem 12—which establishes a strict equivalence between the absence of third-order interference and the coincidence of measurement and decomposition max-entropies—this paper generalizes the Mamun, SM (2026) ……………………………….1 Higher-Order Interference and Thermodynamic Entropy in Generalized Hilbert Spaces Bridging Sorkin's Framework with Operational Thermodynamics (Vol-III) equivalence to arbitrary-order Rényi entropies across non-Hilbert state spaces. Also available on Academia.edu: https://www.academia.edu/attachments/132551016/download_file?s=portfolio
We investigate the extent to which thermodynamic reasonableness, formalized as two operational postulates on entropy and reversible dynamics within the framework of Generalized Probabilistic Theories (GPTs), uniquely determines the complex Hilbert space formalism of standard quantum mechanics. Building on Krumm, Müller, and Masanes (2017), we provide an extended classification of state spaces satisfying thermodynamic consistency for systems of (N \ge 3) distinguishable states, proving that self-duality of the state cone, spectral entropy compatibility, and continuous reversible dynamics jointly eliminate real and quaternionic quantum theories as viable candidates. We show that augmenting the thermodynamic postulates with local tomographic closure and a purification principle is sufficient to recover full complex Hilbert space structure, deriving analogues of the Born rule via entropy maximization arguments. A central novel result is a no-go theorem establishing that non-complex state spaces cannot support a continuous, reversible, and entropy-preserving dynamics compatible with thermodynamic equilibration in composite systems. We further propose experimentally falsifiable signatures of post-quantum deviations using higher-order interference, generalized Landauer erasure, and multi-slit protocols. Also available on Academia.edu: https://www.academia.edu/attachments/132550729/download_file?s=portfolio
Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio
Despite its enormous empirical success, the formalism of quantum theory still raises fundamental questions: why is nature described in terms of complex Hilbert spaces, and what modifications of it could we reasonably expect to find in some regimes of physics? Here we address these questions by studying how compatibility with thermodynamics constrains the structure of quantum theory. We employ two postulates that any probabilistic theory with reasonable thermodynamic behaviour should arguably satisfy. In the framework of generalised probabilistic theories, we show that these postulates already imply important aspects of quantum theory, like self-duality and analogues of projective measurements, subspaces and eigenvalues. However, they may still admit a class of theories beyond quantum mechanics. Using a thought experiment by von Neumann, we show that these theories admit a consistent thermodynamic notion of entropy, and prove that the second law holds for projective measurements and mixing procedures. Furthermore, we study additional entropy-like quantities based on measurement probabilities and convex decomposition probabilities, and uncover a relation between one of these quantities and Sorkin’s notion of higher-order interference.
The 2017 paper by Marius Krumm, Howard Barnum, Jonathan Barrett, and Markus P. Müller, published in New Journal of Physics, represents a landmark contribution to the foundations of quantum mechanics within the framework of generalised probabilistic theories (GPTs). By imposing two minimal thermodynamic postulates—reversible transformations between distinguishable pure states and the existence of a semi-permeable membrane distinguishing them—the authors demonstrate that key structural features of quantum theory, including self-duality of the state space, analogues of projective measurements, subspaces, and eigenvalues, emerge naturally from thermodynamic compatibility. Yet these postulates leave room for a Mamun, SM (2026)……………………………………………1 Thermodynamic Sculpting of Quantum Foundations: An Anatomical Review of Krumm et al. (2017) in the Context of Advances in Generalized Probabilistic Theories and Pathways for Hilbert Space Evolution (Vol-I) restricted class of non-quantum theories consistent with a von Neumann-style thermodynamic entropy and the second law. This review provides a thorough anatomical dissection of the paper's arguments, contrasts its findings with post-2017 developments in GPTs, quantum thermodynamics, and experimental tests of higher-order interference, and explores plausible evolutionary pathways for assessing and potentially extending the Hilbert space formalism. Also available on Academia.edu: https://www.academia.edu/attachments/132550634/download_file?s=portfolio
As first noted by Rafael Sorkin, there is a limit to quantum interference. The interference pattern formed in a multi-slit experiment is a function of the interference patterns formed between pairs of slits; there are no genuinely new features resulting from considering three slits instead of two. Sorkin has introduced a hierarchy of mathematically conceivable higher-order interference behaviours, where classical theory lies at the first level of this hierarchy and quantum theory theory at the second. Informally, the order in this hierarchy corresponds to the number of slits on which the interference pattern has an irreducible dependence. Many authors have wondered why quantum interference is limited to the second level of this hierarchy. Does the existence of higher-order interference violate some natural physical principle that we believe should be fundamental? In the current work we show that such principles can be found which limit interference behaviour to second-order, or “quantum-like”, interference, but that do not restrict us to the entire quantum formalism. We work within the operational framework of generalised probabilistic theories, and prove that any theory satisfying Causality, Purity Preservation, Pure Sharpness, and Purification—four principles that formalise the fundamental character of purity in nature—exhibits at most second-order interference. Hence these theories are, at least conceptually, very “close” to quantum theory. Along the way we show that systems in such theories correspond to Euclidean Jordan algebras. Hence, they are self-dual and, moreover, multi-slit experiments in such theories are described by pure projectors.