DOG Discrete Order Geometry and Hilbert Spaces

Author: Zhang Suhang
(Luoyang, Henan)

Abstract

Discrete Order Geometry (DOG) takes finite discrete lattice points, nested order and local correlations as its ontology, without presupposing a continuous space or inner product structure. This paper investigates the correspondence between DOG discrete order states and Hilbert spaces. The DOG order state space, order inner product, order operators and order evolution maps are defined. It is proved that under the regular continuous limit of order homogenization, lattice densification and coefficient regularization, the DOG order state space can be embedded into a Hilbert space, order operators converge to self-adjoint operators, and discrete order evolution converges to unitary evolution. The conclusion is that the Hilbert space serves as an effective description of DOG order states under the regular continuous limit, rather than a foundational postulate of DOG.

Keywords: Discrete Order Geometry; Hilbert space; order state; order operator; continuous limit

1 Introduction

Discrete Order Geometry (DOG) adopts finite discrete lattice sets, nested order and local correlations as its underlying ontology. Within DOG, state fields, coupling relations and evolution rules are all defined by discrete order structures, with no presupposition of continuous spaces or inner products.

Quantum mechanics and quantum field theory take Hilbert spaces as their state spaces. This paper studies:

1. How the DOG order state space is defined;
2. How the DOG order inner product is constructed;
3. Under what regular limit conditions the DOG order state space embeds into a Hilbert space;
4. How order operators converge to self-adjoint operators;
5. How discrete order evolution converges to unitary evolution.

This paper does not claim that Hilbert space constitutes the ontology of DOG; it only establishes limiting correspondence relations.

2 DOG Order State Space

2.1 Lattice Point State Fields

Let the set of DOG lattice points be:
\mathcal{L}=\{p,q,r,\dots\}.
For each lattice point p, define an order state:
\psi_p\in \mathcal{S}_p,
where \mathcal{S}_p denotes the local state set at lattice point p.

The global order state is written as:
\Psi=(\psi_p)_{p\in\mathcal{L}}.

2.2 Order State Space

The DOG order state space is defined as:
\mathcal{V}_{\mathrm{DOG}}=\bigoplus_{p\in\mathcal{L}}\mathcal{S}_p,
the direct sum of all local lattice point states.

If \mathcal{S}_p=\mathbb{C}, then:
\mathcal{V}_{\mathrm{DOG}}\cong \mathbb{C}^{|\mathcal{L}|}.

2.3 Order Basis

For each lattice point p, take the basis vector:
e_p.
Any order state can be expressed as:
\Psi=\sum_{p\in\mathcal{L}}\psi_p e_p.

3 DOG Order Inner Product

3.1 Order Weights

For lattice point p, define an order weight:
w_p>0.
The weight is determined by the DOG order hierarchy, neighbourhood structure and coupling relations.

3.2 Order Inner Product

Define the order inner product:
\langle \Psi|\Phi\rangle_{\mathrm{DOG}}
=
\sum_{p\in\mathcal{L}} w_p\,\overline{\psi_p}\,\phi_p.

3.3 Properties of the Inner Product

Theorem 1: \langle\cdot|\cdot\rangle_{\mathrm{DOG}} satisfies the inner product axioms.

Proof:

1. Conjugate symmetry follows directly from the definition;
2. Linearity holds with respect to the second argument;
3. Positive definiteness: w_p>0, so \langle\Psi|\Psi\rangle_{\mathrm{DOG}}\ge0, with equality if and only if \Psi=0.

Thus \mathcal{V}_{\mathrm{DOG}} equipped with this inner product forms an inner product space.

4 DOG Order Operators

4.1 Definition of Order Operators

An order operator is defined by:
\hat A:\mathcal{V}_{\mathrm{DOG}}\to\mathcal{V}_{\mathrm{DOG}}.
Under the order basis, \hat A is represented by the matrix (A_{pq}).

4.2 Order Self-Adjoint Operators

If
\langle \hat A\Psi|\Phi\rangle_{\mathrm{DOG}}
=
\langle \Psi|\hat A\Phi\rangle_{\mathrm{DOG}},
then \hat A is called an order self-adjoint operator.

4.3 Order Unitary Operators

If
\hat U^\dagger \hat U=\hat U\hat U^\dagger=\hat I,
then \hat U is called an order unitary operator.

4.4 Correspondence between Order Operators and Matrices

Under the order basis, an order operator corresponds to a matrix:
\hat A \;\longleftrightarrow\; (A_{pq}).
Self-adjoint operators correspond to self-adjoint matrices, and unitary operators correspond to unitary matrices.

5 DOG Order Evolution

5.1 Discrete Order Evolution

DOG discrete order evolution is defined as:
\Psi(n+1)=\hat U_{\mathrm{DOG}}\,\Psi(n),
where \hat U_{\mathrm{DOG}} is an order unitary operator.

5.2 Continuous Limit

Let the time step \Delta t\to0, the lattice points undergo densification, and order coefficients are regularized.

Define the generator:
\hat H_{\mathrm{DOG}}
=
i\hbar\lim_{\Delta t\to0}
\frac{\hat U_{\mathrm{DOG}}-\hat I}{\Delta t}.

Then order evolution converges to:
i\hbar\partial_t\Psi=\hat H_{\mathrm{DOG}}\Psi.

5.3 Order Hamiltonian

If \hat U_{\mathrm{DOG}} is unitary, then \hat H_{\mathrm{DOG}} is self-adjoint.

6 Embedding of DOG Order States into Hilbert Spaces

6.1 Regular Limit Conditions

The regular limit conditions for embedding DOG into a Hilbert space are defined as:

1. The number of lattice points tends to infinity while the macroscopic scale remains finite;
2. Lattice densification: \Delta x\to0;
3. Order weights converge to a continuous measure: w_p\to w(x)\mathrm{d}x;
4. Sequences of order coefficients converge to smooth functions;
5. The order state space is completed under the order inner product.

6.2 Embedding Theorem

Theorem 2: Under the regular limit conditions, the DOG order state space \mathcal{V}_{\mathrm{DOG}} can be embedded into a Hilbert space \mathcal{H}.

Proof:

1. \mathcal{V}_{\mathrm{DOG}} constitutes an inner product space under the order inner product;
2. Under lattice densification and weight continuation, the order inner product converges to the continuous inner product:
\langle \Psi|\Phi\rangle_{\mathrm{DOG}}
\to
\int \overline{\psi(x)}\,\phi(x)\,w(x)\,\mathrm{d}x;
3. Completion yields the Hilbert space \mathcal{H};
4. The embedding map is:
\iota:\mathcal{V}_{\mathrm{DOG}}\hookrightarrow\mathcal{H}.

Therefore, \mathcal{V}_{\mathrm{DOG}} is embeddable into \mathcal{H}.

6.3 Convergence of Operators

Theorem 3: Under the regular limit conditions, DOG order self-adjoint operators converge to self-adjoint operators on the Hilbert space; order unitary operators converge to unitary operators.

Proof: Order operators correspond to matrices under the order basis. In the regular limit, matrix elements converge to the kernel of continuous operators, and self-adjointness and unitarity are preserved in the limit.

7 Examples

7.1 One-Dimensional Chain

\mathcal{L}=\{1,2,\dots,N\},\qquad w_p=1.
Order state space:
\mathcal{V}_{\mathrm{DOG}}\cong\mathbb{C}^N.
In the regular limit N\to\infty:
\mathcal{V}_{\mathrm{DOG}}\to L^2(\mathbb{R}),
the one-dimensional Hilbert space.

7.2 Uniform Order

If the order weight w_p=w is constant, the order inner product reduces to the standard inner product, and the embedding becomes an isometric embedding.

7.3 Non-Uniform Order

If w_p varies with hierarchy, the resulting Hilbert space after embedding carries a weighted measure, corresponding to non-uniform continuous systems.

8 Conclusion

This paper establishes the correspondence between order state spaces and Hilbert spaces within DOG:

1. The DOG order state space is constructed as the direct sum of lattice point states;
2. The order inner product is defined by order weights and satisfies the inner product axioms;
3. Order operators correspond to matrices under the order basis, preserving self-adjointness and unitarity;
4. Under the continuous limit of lattice densification, weight continuation and coefficient regularization, the DOG order state space embeds into a Hilbert space;
5. The Hilbert space is an effective description of DOG order states under the regular continuous limit, rather than a foundational postulate of DOG.

This framework provides an underlying interface for DOG to connect with quantum mechanics and quantum field theory.

References

(Omitted)