We explore different implementations of the quantum Bell causality (QBC) condition in the quantum sequential growth (QSG) dynamics of causal set quantum gravity, for non-commuting transition operators. Assuming a non-singular dynamics we show that for the two most natural choices of operator orderings for the QBC, the transition operator algebra reduces to a commutative one. As a third choice, we take the operator ordering to depend on the size of the precursor set. We find several new commutation relations which further constrain the algebra but do not imply commutativity. On the other hand, if any of the generators of the ``antichain subalgebra'' belongs to its center, then this implies commutativity of the full algebra. The complexity of the algebra prevents us from obtaining a general form for the transition operators, which hinders computability. In an attempt to construct the simplest non-trivial d=2 representation, we find that a Pauli matrix representation of the generators of the antichain subalgebra leads to inconsistencies, implying that if a non-trivial representation exists, it must be higher dimensional. Our work can be viewed as a first step towards finding a non-commutative realisation of QSG.
We explore different implementations of the quantum Bell causality (QBC) condition in the quantum sequential growth (QSG) dynamics of causal set quantum gravity, for non-commuting transition operators. Assuming a non-singular dynamics we show that for the two most natural choices of operator orderings for the QBC, the transition operator algebra reduces to a commutative one. As a third choice, we take the operator ordering to depend on the size of the precursor set. We find several new commutation relations which further constrain the algebra but do not imply commutativity. On the other hand, if any of the generators of the ``antichain subalgebra'' belongs to its center, then this implies commutativity of the full algebra. The complexity of the algebra prevents us from obtaining a general form for the transition operators, which hinders computability. In an attempt to construct the simplest non-trivial d=2 representation, we find that a Pauli matrix representation of the generators of the antichain subalgebra leads to inconsistencies, implying that if a non-trivial representation exists, it must be higher dimensional. Our work can be viewed as a first step towards finding a non-commutative realisation of QSG.
This article presents a simplified version of the author's previous work. We first construct a causal growth process (CGP). We then form path Hilbert spaces using paths of varying lengths in the CGP. A sequence of positive operators on these Hilbert spaces that satisfy certain normalization and consistency conditions is called a quantum sequential growth process (QSGP). The operators of a QSGP are employed to define natural decoherence functionals and quantum measures. These quantum measures are extended to a single quantum measure defined on a suitable collection of subsets of a space of all paths. Continuing our general formalism, we define curvature operators and a discrete analogue of Einstein's field equations on the Hilbert space of causal sets. We next present a method for constructing a QSGP using an amplitude process (AP). We then consider a specific AP that employs a discrete analogue of a quantum action. Finally, we consider the special case in which the QSGP is classical. It is pointed out that this formalism not only gives a discrete version of general relativity, there is also emerging a discrete analogue of quantum field theory. We therefore have discrete versions of these two theories within one unifying framework.
Abstract The classical sequential growth model for causal sets provides a template for the dynamics in the deep quantum regime. This growth dynamics is intrinsically temporal and causal, with each new element being added to the existing causal set without disturbing its past. In the quantum version, the probability measure on the event algebra is replaced by a quantum measure, which is Hilbert space valued. Because of the temporality of the growth process, in this approach, covariant events (or observables) are measurable only if the quantum measure extends to the associated sigma algebra of events. This is not always guaranteed. In this work we find a criterion for extension (and thence covariance) in complex sequential growth models for causal sets. We find a large family of models in which the measure extends, so that all covariant events/observables are measurable.
We present a bounded analysis of the finite, nonsingular, occurrence-ON quantum sequential-growth presentation at dimension d=2 and source stages n<=4. The preprint records five ledger-bound results: a frozen strong/strong commutativity baseline; an exact rational fixed-vector/general-covariance and reachable-state/martingale witness of noncommutativity; a source-native Eq. (120) lemma; commutativity on a proper reconstruction slice; and conditional statewise-to-operator recovery lemmas. Its purpose is to make the distinction between statewise and operator semantics auditable, rather than to classify all finite semantics. In particular, it does not claim the full 955 or 721 profiles, a complete finite occurrence-ON classification, or a reachable-visible witness. The bounded U2 auxiliary-ideal investigation is retained as a reproducibility and resource-boundary record, not as a unit-ideal theorem.
Quantum theory challenges classical causal explanations for physical phenomena, calling for a quantum generalisation of causal structure. This thesis advances the study of causal decompositions, a research programme that aims to bridge two prominent, yet conceptually distinct types of structural assumption found across the quantum causality literature. The first is in terms of the global dependency relations between input and output systems of quantum transformations, which in the unitary case we shall refer to as causal structure; the second is in terms of the compositional structure of quantum circuits or networks. Concretely, a causal decomposition is a circuit representation of a unitary transformation in which no-influence relations that hold between its inputs and outputs are made manifest in the absence of paths through the circuit connecting them. Applications include translating between different quantum causal modelling frameworks; equipping a framework based on unitary causal structure with a local, bottom-up description via generalised autonomous causal mechanisms; and relating notions of compatibility between quantum transformations and relativistic causality. Whether causal decompositions always exist remains an open problem. The contributions of this thesis are threefold. First, we develop an order-theoretic account of circuit syntax and demonstrate that every circuit with a given input-output connectivity can be rewritten into a one of a canonical form given by the concept lattice construction from lattice theory. This systematises the search for causal decompositions by providing the shape of the quantum circuit one should look for. Second, we establish a combinatorial condition that fully characterises when causal decompositions into unitary quantum circuits exist without requiring the generalisation to routed quantum circuits prompted by earlier research. This solves a well-defined subproblem of the broader quest, and our proof, an interplay of order theory and operator algebra, lays a foundation for future extensions. Third, we detail the application to relativistic causality. Within a multipartite framework generalising existing literature, we emphasise that the mere lack of superluminal signalling through a quantum channel is not sufficient to conclude its realisability by dynamics that are locally consistent with relativity theory. We conjecture that, on the other hand, a lack of superluminal causal influences through its unitary dilations is sufficient—at least up to arbitrarily small errors. We demonstrate this in various special cases and show that the general statement reduces to the existence of a weak version of causal decompositions.
Bell’s theorem is a foundational result in quantum theory, demonstrating that no local hidden-variable model satisfying certain assumptions can reproduce quantum correlations. While mathematically correct within its formal framework, the theorem implicitly relies on a specific representational choice: measurement outcomes are modeled as scalar, commutative variables (±1). This assumption, though natural for flat and static configurations, is not mandated by physical necessity. In this work, we examine the consequences of relaxing this representational constraint when measurement outcomes arise as projections of states with nontrivial topology. We show that when physical correlations are naturally organized as elements of an S³-valued structure — such as those associated with spin and rotational degrees of freedom — the induced measurement algebra becomes noncommutative. Under these conditions, the key algebraic factorizations employed in Bell-type derivations no longer apply. We construct an explicit local, realistic model in which observable correlations reproduce the quantum expectation E(a, b) = − cos θ and violate the CHSH bound S ≤ 2, without invoking non-local interactions or superluminal influences. The effective S³×I structure arises not as a fundamental substrate, but as the natural geometric form taken by the actualization of physical processes within an underlying relational framework. Importantly, this structure was not introduced to address quantum entanglement specifically; it emerged independently in the study of cosmological expansion, emergent gravity, and mass generation. Its relevance to Bell-type correlations follows as a consequence of geometry, not as an ad hoc modification. These results suggest that quantum entanglement may reflect topological features of physical realization rather than intrinsic non-locality. Possible experimental implications and connections to geometric and substrate-based approaches to fundamental physics
We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded sl(2,C) subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.