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Uniformly bounded finite-range quantum spin interaction — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Analyticity of Holomorphic Functions; Liouville and Morera
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Banach Alaoglu Goldstine and Krein Milman
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Bounded Variation and the Riemann–Stieltjes Integral
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Norms and Realification
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Continuous Functional Calculus for Self Adjoint and Normal Operators
- Contour Integration
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Foundations of the Real Numbers for Analysis
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gelfand Theory and Commutative C Star Algebras
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group C Star Algebras and the Fell Unitary Dual
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Local spin matrix algebras and isometric inclusions
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal density states and spectral moments
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Partitions of Unity and Paracompactness
- pi: the Equivalent Characterizations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Pure and Mixed States, Ensembles, and Schmidt Decomposition
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Splitting Fields
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorems of Calculus
- The Galois Correspondence
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Uniformly bounded finite-range quantum spin interaction
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
The nearest-neighbor diagonal spin interaction gives an explicit finite-range example, nonzero when . Its support size, range, and per-site interaction norm are checked directly from the two-site Pauli matrix.
3 · Logical flowchart
4 · Definitions, postulates, theorems and proofs
None yet.
5 · Examples, experiments and counterexamples
Nearest-neighbor Ising interaction satisfies the finite-range bounds
Example
Take , , , and let For every , let and set for a fixed ; set for every other finite nonempty support . If interpreted as an energy coefficient, has units J, but no value or measurement is assumed. This is a nearest-neighbor diagonal spin interaction of the kind described by Uniformly bounded finite-range quantum spin interaction.
Facts & Assumptions
Given: The displayed Pauli matrix, , and the local tensor-factor conventions of Local spin matrix algebras and their factor inclusions.
Local spin matrix algebras and their factor inclusions: the two-site algebra is the finite matrix algebra with the ordinary operator norm.
Uniformly bounded finite-range quantum spin interaction: an interaction term must be self-adjoint, have uniformly bounded support size and diameter, and have finite per-site norm sum.
Verification
Given: [F1]–[F2] and the interaction in the Example.
Since and , the tensor product is a self-adjoint unitary in and has norm one. Therefore every nonzero term is self-adjoint and has norm .
Each support has cardinality two and max-metric diameter one. Each site belongs to exactly two such edges, so . All other terms vanish. Hence the interaction satisfies the definition with , , and , including the case .