This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.
This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.
"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
A graph-theoretical framework for quantum mechanics is developed in which observables and symmetry operators are incorporated into a single complete bi-coloured graph. The proposed Symmetry/Observables Quantum Graph (SOQG) treats Hamiltonians, observables, conserved quantities, and unitary symmetry transformations on equal footing as operators acting on a common Hilbert space. The vertices of the graph represent operators, while the colours of the edges are determined by their commutation relations: commuting operators are connected by maroon edges and non-commuting operators by teal edges. The resulting structure constitutes a finite Ramsey graph that provides a combinatorial representation of operator compatibility. The formalism is applied to several representative quantum systems, including a particle in a central potential, a charged particle in a uniform magnetic field, and a particle in a homogeneous external field. In each case, monochromatic cliques identify maximal compatibility sectors containing both observables and symmetry operators, thereby revealing the interplay between conserved quantities and dynamical symmetries. It is further shown that coloured-graph automorphisms generally differ from the physical symmetry group of the underlying quantum system, leading to the notion of second-order graph symmetry. The SOQG is proved to be invariant under simultaneous unitary transformations of all operators and therefore represents intrinsic properties of the operator algebra rather than a particular Hilbert-space representation. The graph-theoretical structure is analysed using Ramsey theory and the Mantel–Turán extremal theorem, which provide rigorous conditions for the appearance of mutually compatible and mutually incompatible operator subsets. The framework is subsequently generalized to parameter-dependent Symmetry/Observables Quantum Graphs (PSOQGs), whose compatibility structure changes as external parameters modify the underlying commutation relations. For magnetic translation operators, the magnetic flux through the elementary translation cell induces a flux-controlled colour-switching transition between commuting and non-commuting compatibility sectors, accompanied by changes in the graph automorphism group. The proposed framework establishes a direct connection between operator theory, symmetry, graph theory, and extremal combinatorics, providing a new structural perspective on quantum systems.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
Abstract Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.
We provide a mechanism by which, from a background-independent model with no quantum mechanics, quantum theory arises in the same limit in which spatial properties appear. Starting with an arbitrary abstract graph as the microscopic model of spacetime, our ansatz is that the microscopic dynamics can be chosen so that (i) the model has a low-energy limit which reproduces the nonrelativistic classical dynamics of a system of N particles in flat spacetime, (ii) there is a minimum length, and (iii) some of the particles are in a thermal bath or otherwise evolve stochastically. We then construct simple functions of the degrees of freedom of the theory and show that their probability distributions evolve according to the Schr\"odinger equation. The nonlocal hidden variables required to satisfy the conditions of Bell's theorem are the links in the fundamental graph that connect nodes adjacent in the graph but distant in the approximate metric of the low-energy limit. In the presence of these links, distant stochastic fluctuations are transferred into universal quantum fluctuations.
Abstract We propose a Ramsey—theoretic framework for quantum mechanics in which observables are represented as vertices of a complete graph and their commutation relations are encoded by a bi-coloring of the edges. Edges connecting strongly commuting operators are colored red, while non-commuting pairs are colored green. This construction transforms the algebra of quantum observables into a combinatorial object—a quantum Ramsey graph. Because commutation relations are preserved under unitary conjugation, the resulting coloring and its Ramsey substructures are unitary invariants, providing a new class of unitarily invariant combinatorial characteristics of quantum systems. When the Hamiltonian is included as a vertex, the graph encodes dynamical information through its commutation relations, revealing conserved quantities and symmetry constraints. Universal results from classical Ramsey theory then impose unavoidable constraints on quantum systems. In particular, the Ramsey number R ( 3 , 3 ) = 6 guarantees that any set of six observables contains a monochromatic triangle. Physically, this implies that every sufficiently rich operator set contains either a classical subcontext of three mutually commuting observables or an irreducibly incompatible triple. We analyze representative systems, including free particles, particles in central potentials, harmonic oscillators, and interacting spin systems, illustrating how symmetries and dynamics determine which Ramsey-enforced structures are realized. Hamiltonian-centered star-shaped graphs visualize conserved quantities and the propagation of incompatibility, while cyclic compatibility subgraphs connect naturally to contextuality scenarios (e.g. Klyachko–Can–Binicioğlu–Shumovsky-type structures). We discuss algebraic constraints (e.g. Jacobi identity effects under Lie-closure) on non-commuting triangles and extend the framework to infinite operator sets, where infinite monochromatic cliques arise. Overall, the work shows that compatibility, incompatibility, and contextuality reflect not only physical postulates but also universal combinatorial constraints induced by the structure of commuting observables.