Appendix A: Mathematical Foundations

Appendix A: Mathematical Foundations

The physical theory constructed in this book spans multiple domains from microscopic discrete lattices to macroscopic continuous spacetime, and further to logic and computation. To maintain narrative fluency in the main text, many deep mathematical definitions and theorems are only cited physically. This appendix aims to provide a self-consistent, standardized mathematical tool reference manual, covering core concepts of Hilbert spaces, operator algebras, information geometry, and graph theory, serving as the mathematical skeleton of the entire book.

A.1 Hilbert Space Structure of Discrete Quantum Mechanics

In Axiom $\Omega$, we define physical entities as vectors in Hilbert space. For discrete ontology, we primarily focus on finite-dimensional spaces and their tensor products.

A.1.1 Local Space and Tensor Product

• Local Space:

  Let each cell (lattice point) $x$ be associated with a $d$-dimensional complex vector space ${\mathcal{H}}_x\cong {\mathbb{C}}^d$. For qubit systems, $d=2$, with basis denoted as $\{|0⟩,|1⟩\}$.

  Vectors $|\psi ⟩\in {\mathcal{H}}_x$ in the space satisfy the normalization condition $⟨\psi |\psi ⟩=1$.

• Global Space:

  The state space of the entire system is the tensor product of all local spaces:

  $${\mathcal{H}}_{\text{total}}=\underset{x\in \Lambda }{⨂}{\mathcal{H}}_x$$

  Note: For infinite lattices $\Lambda$, this tensor product must be understood in the sense of the algebraic tensor product of von Neumann algebras, typically restricted to the separable Hilbert space spanned by finite excitation states (states where all nodes except finitely many are in the ground state $|0⟩$).

A.1.2 Entanglement and Schmidt Decomposition

For a composite system ${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$, any pure state $|\Psi ⟩$ can be uniquely decomposed as:

$$|\Psi ⟩=\sum _{i=1}^k{\lambda }_i|a_i{⟩}_A\otimes |b_i{⟩}_B$$

where ${\lambda }_i>0$ are Schmidt coefficients satisfying $\sum {\lambda }_i^2=1$. $k\le \min (\dim {\mathcal{H}}_A,\dim {\mathcal{H}}_B)$ is the Schmidt rank.

• Entanglement Entropy: $S({\rho }_A)=-\sum {\lambda }_i^2\ln {\lambda }_i^2$. This is the core quantity used in Chapter 4 of this book to derive gravitational geometry.

A.2 Operator Algebra & Spectral Theory

Many physical intuitions in this book—such as the holographic principle, modular Hamiltonians, and the entanglement structure of information—rely on operator algebras (particularly von Neumann algebras) as their rigorous mathematical language. In discrete QCA ontology, physical systems are modeled as tensor products of local Hilbert spaces ${\mathcal{H}}_x$, and physical quantities (observables) on them constitute specific algebraic structures. This section briefly introduces relevant core concepts.

A.2.1 Von Neumann Algebras and Factors

Let $\mathcal{H}$ be a complex Hilbert space (possibly infinite-dimensional, corresponding to the limit $N\to \infty$). $\mathcal{B}(\mathcal{H})$ is the algebra of bounded linear operators on it.

Definition A.2.1 (Von Neumann Algebra):

A $\ast$-subalgebra $\mathcal{M}$ of $\mathcal{B}(\mathcal{H})$ is called a von Neumann algebra if it contains the identity operator $\mathbb{I}$ and satisfies the Bicommutant Theorem:

$$\mathcal{M}={\mathcal{M}}^{\prime \prime }$$

where ${\mathcal{M}}^{\prime }=\{T\in \mathcal{B}(\mathcal{H}):TA=AT,\forall A\in \mathcal{M}\}$ is the commutant algebra of $\mathcal{M}$. This means $\mathcal{M}$ is closed under the weak operator topology.

In quantum field theory and holographic theory, we particularly focus on factors, i.e., von Neumann algebras with trivial center:

$$\mathcal{Z}(\mathcal{M})=\mathcal{M}\cap {\mathcal{M}}^{\prime }=\mathbb{CI}$$

Von Neumann algebras are classified into three types based on the properties of their projection operators:

• Type I: Isomorphic to $\mathcal{B}(\mathcal{H})$. This is the standard quantum mechanics algebra describing systems with finite degrees of freedom (such as spin chains, single QCA cells). The trace on it is well-defined.

• Type II: No minimal projections exist, but a semifinite trace exists. This appears in certain statistical mechanics models.

• Type III: The most mysterious and important type. All non-zero projections are equivalent to the identity. Local algebras $\mathcal{A}(O)$ in local quantum field theory are typically Type III${}_1$ factors. This means that in the continuous limit, entanglement entropy within local regions diverges (requiring ultraviolet cutoff), which is precisely the mathematical root of our discussion of “the necessity of discrete ontology” in Chapter 1.

A.2.2 Modular Theory (Tomita-Takesaki Theory)

For a general von Neumann algebra $\mathcal{M}$ and a cyclic separating vector $|\Omega ⟩$, we cannot define a standard density matrix $\rho$ and trace as in Type I algebras. To define “states” and “evolution,” we need Tomita-Takesaki modular theory.

Define the operator $S$:

$$SA|\Omega ⟩=A^{†}|\Omega ⟩,\forall A\in \mathcal{M}$$

The polar decomposition of $S$ is $S=J{\Delta }^{1/2}$.

• $J$: Modular conjugation operator, an antilinear operator. It establishes an isomorphism between algebra $\mathcal{M}$ and its commutant ${\mathcal{M}}^{\prime }$: $J\mathcal{M}J={\mathcal{M}}^{\prime }$. Physically, this corresponds to CPT symmetry in holographic duality.

• $\Delta$: Modular operator, $\Delta =S^{†}S$. This is a positive self-adjoint operator.

Theorem A.2.2 (Tomita-Takesaki Theorem):

${\Delta }^{it}$ generates a one-parameter automorphism group ${\sigma }_t$ of $\mathcal{M}$:

$${\sigma }_t(A)={\Delta }^{it}A{\Delta }^{-it}\in \mathcal{M},\forall A\in \mathcal{M}$$

This is called the modular flow.

Physical Meaning:

For the vacuum state $|\Omega ⟩$, the modular Hamiltonian is defined as $H_{mod}=-\ln \Delta$.

$$\rho \sim e^{-H_{mod}}$$

This shows that any entangled state intrinsically defines a “thermal time flow.” For observers in Rindler wedges, the modular flow exactly corresponds to Lorentz boosts, and the modular Hamiltonian is the Hamiltonian generating such accelerated motion. This directly connects quantum entanglement with spacetime geometry, serving as the mathematical foundation for deriving the origin of gravity in Chapter 4 of this book.

A.2.3 Relative Entropy and Fisher Information

In the operator algebra framework, the “distance” between two states $\psi$ and $\varphi$ is measured by relative entropy, defined as:

$$S(\psi ||\varphi )=⟨\psi |\ln {\Delta }_{\psi ,\varphi }|\psi ⟩$$

where ${\Delta }_{\psi ,\varphi }$ is the relative modular operator.

For parameterized state families $\rho (\theta )$, the second-order expansion of relative entropy gives the Quantum Fisher Information Metric (QFIM):

$$S(\rho (\theta )||\rho (\theta +d\theta ))\approx \frac{1}{2}{g}_{ij}d{\theta }^{i}d{\theta }^j$$

This is precisely the microscopic source we use in Chapters 3 and 4 to construct the spacetime metric $g_{\mu \nu }$. Spacetime geometry is essentially the information geometry of quantum state space.

A.3 Category Theory Foundations

In the physical theory of this book, category theory is not merely an abstract mathematical language, but the “meta-language” describing deep connections between physical processes, logical structures, and computational operations. Particularly in the axiomatic definition of quantum cellular automata (QCA), classification of topological orders, and structured description of holographic entanglement, category theory provides indispensable tools.

A.3.1 Basic Definitions: Categories and Functors

Definition A.3.1 (Category $\mathcal{C}$):

A category consists of:

1. Object set $\text{Obj}(\mathcal{C})$: In physics, objects typically represent state spaces of physical systems (such as Hilbert space $\mathcal{H}$).

2. Morphism set $\text{Hom}(A,B)$: For any two objects $A,B$, there exists a morphism set from $A$ to $B$. In physics, morphisms represent physical processes (such as evolution operators $\hat{U}$, measurement channels $\mathcal{E}$).

3. Composition operation $\circ$: Morphisms satisfy associativity $(f\circ g)\circ h=f\circ (g\circ h)$.

4. Identity morphism $1_A$: Each object has a “do nothing” operation.

Definition A.3.2 (Functor $\mathcal{F}$):

A functor is a structure-preserving map between categories. It maps objects of category $\mathcal{C}$ to objects of category $\mathcal{D}$, maps morphisms to morphisms, and preserves composition relations.

• Physical meaning: The holographic principle can be formalized as a functor (or dual equivalence) from the “boundary conformal field theory category” to the “bulk gravitational theory category.”

A.3.2 Monoidal Categories & Tensor Networks

To describe composite systems (${\mathcal{H}}_A\otimes {\mathcal{H}}_B$) in quantum mechanics, we need to introduce monoidal categories.

Definition A.3.3 (Monoidal Category $(\mathcal{C},\otimes ,I)$):

This is a category equipped with a tensor product functor $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ and a unit object $I$.

• Associativity constraint: $(A\otimes B)\otimes C\cong A\otimes (B\otimes C)$ (via natural isomorphism $\alpha$).

• Unit constraint: $I\otimes A\cong A\cong A\otimes I$.

Graphical Calculus:

Morphisms in monoidal categories can be represented using string diagrams.

• Objects are wires.

• Morphisms are boxes connecting wires.

• Tensor product is parallel placement of wires.

• Composition is serial connection of wires.

Physical applications: Evolution processes of QCA, tensor network states (such as MPS, PEPS), and quantum circuit diagrams are essentially graphical calculus in strict monoidal categories. This language makes complex tensor contraction operations intuitive and easily verifiable.

A.3.3 Dagger Compact Categories

To describe unitarity and entanglement (such as preparation and measurement of Bell states) in quantum mechanics, we need richer structures.

Definition A.3.4 (Dagger Category $†$-Category):

This is a category equipped with a contravariant functor $†:{\mathcal{C}}^{op}\to \mathcal{C}$ satisfying $f^{††}=f$.

• Physical meaning: Corresponds to Hermitian conjugation in Hilbert space. Unitary operator $U$ is defined as $U^{†}\circ U=1_A$.

Definition A.3.5 (Compact Closed Category):

This is a monoidal category with “dual objects” $A^{\ast }$ and unit morphism $\eta :I\to A^{\ast }\otimes A$ and counit morphism $ϵ:A\otimes A^{\ast }\to I$, satisfying snake equations.

• Physical meaning:

  • $\eta$ corresponds to preparation of maximally entangled states (such as Bell state $|{\Phi }^{+}⟩$).

  • $ϵ$ corresponds to measurement (projection) of maximally entangled states.

  • Snake equations correspond to the geometric essence of quantum teleportation protocols: curved spacetime lines (entanglement) can transmit information from one end to the other.

Conclusion:

The axiomatic system of quantum mechanics (QM) can be extremely elegantly reconstructed as: Physical processes constitute a dagger compact category (DCC) on Hilbert space.

In the theoretical framework of this book, the QCA defined by Axiom $\Omega$ is essentially an algorithm running in a discrete DCC. This categorical perspective not only unifies quantum logic with spacetime geometry (topological quantum field theory TQFT), but also provides the most general mathematical template for possible future reconstruction of physical laws.

A.4 Information Geometry

The key to deriving special and general relativity in this book lies in geometrizing physical processes. Here we introduce the geometric structure of quantum state manifolds.

A.4.1 Projective Hilbert Space $\mathbb{C}{P}^{N-1}$

Physical states correspond to rays in Hilbert space, i.e., $|\psi ⟩\sim e^{i\theta }|\psi ⟩$. The manifold of all physical states is complex projective space $\mathbb{C}{P}^{N-1}$.

A.4.2 Fubini-Study Metric

This is the natural Riemannian metric defined on quantum state space, used to measure the “distance” between two quantum states.

For two closely spaced states $|\psi ⟩$ and $|\psi +d\psi ⟩$, their distance $d{s}^2$ is defined as:

$$d{s}_{FS}^2=4\left(⟨d\psi |d\psi ⟩-|⟨\psi |d\psi ⟩{|}^2\right)=4(\Delta H{)}^{2}d{t}^2$$

• Geometric meaning: This is the rate at which quantum states orthogonalize with other states during evolution.

• Physical application: In Chapter 3 of this book, we define the total information rate $c$ as the Fubini-Study velocity $d{s}_{FS}/dt$ along the evolution trajectory. The Light Path Conservation Theorem $v_{ext}^2+v_{int}^2=c^2$ is precisely the Pythagorean decomposition of this metric on the position subspace and internal subspace.

A.4.3 Berry Curvature

When system parameters adiabatically evolve in parameter space $\mathcal{M}$, the wave function acquires a geometric phase $\gamma$. This corresponds to a gauge field (Berry connection) on $\mathcal{M}$:

$$\mathcal{A}=-i⟨\psi |\nabla |\psi ⟩$$

The corresponding curvature tensor $\mathcal{F}=\nabla \times \mathcal{A}$ describes the geometric properties of parameter space.

In Chapter 6 of this book, we interpret gauge fields (electromagnetic fields, Yang-Mills fields) as Berry connections caused by local basis transformations in QCA networks.

A.5 Graph Theory and Discrete Topology

A.5.1 Cayley Graph

If QCA space has translational symmetry, the lattice $\Lambda$ can be viewed as a Cayley graph of some discrete group $G$ (such as ${\mathbb{Z}}^D$).

• Vertices: Group elements $g\in G$.

• Edges: Connect $g$ and $g^{\prime }$ if $g^{\prime }=g\cdot s$ (where $s$ is an element of generating set $S$).

This structure ensures homogeneity of physical laws.

A.5.2 Discrete Differential Forms

On discrete lattices, we cannot use standard differentials $dx$. Instead, we use cochains.

• 0-form (scalar field): Defined on vertices, $\varphi (x)$.

• 1-form (gauge field): Defined on edges (links), $U(x,x+\mu )$.

• 2-form (curvature/magnetic field): Defined on faces (plaquettes), $U_{\square }$.

Discrete version of Stokes’ Theorem:

$$\sum _{\text{boundary}}A=\sum _{\text{bulk}}dA$$

This plays a key role in deriving the discrete form of Maxwell’s equations (Chapter 6).

A.5.3 Topological Winding Number

For a Hamiltonian map $H(k):T^D\to G$ defined on the Brillouin zone (torus $T^D$), its homotopy class is characterized by integer topological invariants (such as Chern numbers, winding numbers).

$$\mathcal{W}=\frac{1}{2\pi i}\oint \text{Tr}(H^{-1}dH)$$

This is the mathematical foundation for explaining mass stability and fermion statistics in Chapter 5 of this book. Non-trivial winding number ($\mathcal{W}\ne 0$) means the system is in a topological phase and cannot be continuously deformed to a massless (trivial) state.