Relational Field

Prerequisites: Chapter 2 (Relation).

3.1 Raw Relational Field

Definition 3.1 (Raw relational field). The raw relational field at time $t$ is the triple: $W_{t}^{raw} := (V, w_{t}, g_{t})$

$V$ : finite node set ( $n := ∣ V ∣$ ),

$w_{t} : V \times V \to R_{\geq 0}$ : weight function,

$g_{t} : V \times V \to G$ : transit function.

This is the "coordinate-fixed" collection of all relations.

3.2 Gauge Symmetry

Motivation

The same relational structure can look different in different coordinate systems. A gauge transformation is a change of viewpoint, not a change of physics.

Each node $i$ carries a local frame. Re-framing by $h (i) \in G$ at every node transforms the transit $g_{t} (i, j)$ into $h (i) g_{t} (i, j) h (j)^{- 1}$ , while the scalar weight is unaffected.

Definitions

Definition 3.2 (Gauge group). $G := G^{V} = {h : V \to G} .$ Pointwise multiplication: $(h_{1} \cdot h_{2}) (i) := h_{1} (i) \cdot h_{2} (i)$ .

Since $G$ is compact and $V$ is finite, $G = G^{n}$ is compact (Fact B.2).

Definition 3.3 (Gauge action). For $h \in G$ : $g_{t}^{h} (i, j) := h (i) g_{t} (i, j) h (j)^{- 1}, w_{t}^{h} (i, j) := w_{t} (i, j) .$ Transits transform; weights are invariant.

3.3 Properties of the Gauge Action

Lemma 3.4 (Group action). The action of $G$ on raw relational fields is a valid group action:

(i) $g_{t}^{e} = g_{t}$ .

(ii) $(g_{t}^{h_{1}})^{h_{2}} = g_{t}^{h_{2} \cdot h_{1}}$ (right action).

Proof. (i) $g_{t}^{e} (i, j) = e g_{t} (i, j) e^{- 1} = g_{t} (i, j)$ .

(ii) $(g_{t}^{h_{1}})^{h_{2}} (i, j) = h_{2} (i) [h_{1} (i) g_{t} (i, j) h_{1} (j)^{- 1}] h_{2} (j)^{- 1} = (h_{2} \cdot h_{1}) (i) g_{t} (i, j) (h_{2} \cdot h_{1}) (j)^{- 1} = g_{t}^{h_{2} \cdot h_{1}} (i, j)$ .

Note: this is a right action. To get a left action, set $h \cdot g := g^{h^{- 1}}$ . $□$

Lemma 3.5 (Scalar invariants). Under gauge transformation:

(i) $W_{t}^{h} (i, j) = W_{t} (i, j)$ , (ii) $d_{t}^{h} (i) = d_{t} (i)$ .

Proof. Since $w_{t}^{h} = w_{t}$ , we have $W_{t}^{h} = W_{t}$ . Degree is a sum of $W_{t}$ . $□$

3.4 Relational Field

Definition 3.6 (Relational field). The relational field at time $t$ : $W_{t} := [W_{t}^{raw}]_{G} = {(V, w_{t}, g_{t}^{h}) : h \in G} .$ The gauge orbit---the equivalence class of all raw fields connected by gauge transformation.

Proposition 3.7 (Equivalence relation). $W^{raw} \sim W^{' raw} ⟺ \exists h \in G : g^{'} = g^{h}$ and $w^{'} = w$ is an equivalence relation.

Proof. Reflexivity ( $h = e$ ), symmetry ( $h \mapsto h^{- 1}$ ), transitivity ( $h_{1}, h_{2} \mapsto h_{2} \cdot h_{1}$ ). $□$

3.5 Invariants of the Relational Field

Proposition 3.8 (Classification of invariants).

Scalar invariants (fully invariant):

(i) $w_{t} (i, j)$ , $W_{t} (i, j)$ , $d_{t} (i)$ ---all weight- and degree-related quantities.

Conjugacy-class invariants:

(ii) For a closed loop $γ = (i_{0}, i_{1}, \dots, i_{k}, i_{0})$ , the holonomy $Hol_{t} (γ) := g_{t} (i_{0}, i_{1}) g_{t} (i_{1}, i_{2}) \dots g_{t} (i_{k}, i_{0}) \in G$ has gauge-invariant conjugacy class $[Hol_{t} (γ)]_{conj}$ .

Proof. (i) Lemma 3.5. (ii) Under gauge $h$ , the telescoping product gives $Hol_{t}^{h} (γ) = h (i_{0}) Hol_{t} (γ) h (i_{0})^{- 1}$ . The conjugacy class is therefore invariant. $□$

3.6 Discrete Curvature

Corollary 3.9 (Discrete curvature). For a triangle $△ = (i, j, k)$ : $Ω_{t} (△) := g_{t} (i, j) g_{t} (j, k) g_{t} (k, i) \in G$ is the length-3 holonomy. Its conjugacy class is gauge-invariant.

Scalar curvature: for bi-invariant distance $d_{G}$ , $ω_{t} (△) := d_{G} (Ω_{t} (△), e)^{2}$ is fully gauge-invariant.

Proof. $Ω_{t}^{h} = h (i) Ω_{t} h (i)^{- 1}$ . Bi-invariance: $d_{G} (h Ω h^{- 1}, e) = d_{G} (Ω, e)$ . $□$

Interpretation.

$Ω_{t} (△) = e$ : the triangle is flat---three transits are perfectly compatible.
$Ω_{t} (△) \neq = e$ : non-zero curvature---a twist or inconsistency in the transits.
$ω_{t} (△)$ is the discrete analogue of $∥ F_{A} ∥^{2}$ in Yang--Mills theory.

3.7 Completeness of Invariants

Proposition 3.10 (Completeness---connected case). If $(V, E_{t})$ is connected, the relational field $W_{t}$ is uniquely determined (up to gauge) by:

(i) all edge weights ${w_{t} (i, j)}$ ,

(ii) the holonomies of fundamental loops with respect to a fixed spanning tree $T \subset E_{t}$ .

Proof sketch. Fix a root $i_{0} \in V$ . The unique $T$ -path from $i_{0}$ to each node $i$ determines a gauge $h$ that trivialises all tree edges ( $g^{h} ∣_{T} = e$ ). The residual transits on non-tree edges are then determined by the fundamental-loop holonomies. This gauge fixing is unique up to a global constant $k \in G$ , so the conjugacy classes of holonomies determine $W_{t}$ . $□$

Prerequisites: Chapter 2 (Relation).

3.1 Raw Relational Field

Definition 3.1 (Raw relational field). The raw relational field at time $t$ is the triple: $W_{t}^{raw} := (V, w_{t}, g_{t})$

$V$ : finite node set ( $n := ∣ V ∣$ ),

$w_{t} : V \times V \to R_{\geq 0}$ : weight function,

$g_{t} : V \times V \to G$ : transit function.

This is the "coordinate-fixed" collection of all relations.

3.2 Gauge Symmetry

Motivation

The same relational structure can look different in different coordinate systems. A gauge transformation is a change of viewpoint, not a change of physics.

Each node $i$ carries a local frame. Re-framing by $h (i) \in G$ at every node transforms the transit $g_{t} (i, j)$ into $h (i) g_{t} (i, j) h (j)^{- 1}$ , while the scalar weight is unaffected.

Definitions

Definition 3.2 (Gauge group). $G := G^{V} = {h : V \to G} .$ Pointwise multiplication: $(h_{1} \cdot h_{2}) (i) := h_{1} (i) \cdot h_{2} (i)$ .

Since $G$ is compact and $V$ is finite, $G = G^{n}$ is compact (Fact B.2).

Definition 3.3 (Gauge action). For $h \in G$ : $g_{t}^{h} (i, j) := h (i) g_{t} (i, j) h (j)^{- 1}, w_{t}^{h} (i, j) := w_{t} (i, j) .$ Transits transform; weights are invariant.

3.3 Properties of the Gauge Action

Lemma 3.4 (Group action). The action of $G$ on raw relational fields is a valid group action:

(i) $g_{t}^{e} = g_{t}$ .

(ii) $(g_{t}^{h_{1}})^{h_{2}} = g_{t}^{h_{2} \cdot h_{1}}$ (right action).

Proof. (i) $g_{t}^{e} (i, j) = e g_{t} (i, j) e^{- 1} = g_{t} (i, j)$ .

Note: this is a right action. To get a left action, set $h \cdot g := g^{h^{- 1}}$ . $□$

Lemma 3.5 (Scalar invariants). Under gauge transformation:

(i) $W_{t}^{h} (i, j) = W_{t} (i, j)$ , (ii) $d_{t}^{h} (i) = d_{t} (i)$ .

Proof. Since $w_{t}^{h} = w_{t}$ , we have $W_{t}^{h} = W_{t}$ . Degree is a sum of $W_{t}$ . $□$

3.4 Relational Field

Definition 3.6 (Relational field). The relational field at time $t$ : $W_{t} := [W_{t}^{raw}]_{G} = {(V, w_{t}, g_{t}^{h}) : h \in G} .$ The gauge orbit---the equivalence class of all raw fields connected by gauge transformation.

Proposition 3.7 (Equivalence relation). $W^{raw} \sim W^{' raw} ⟺ \exists h \in G : g^{'} = g^{h}$ and $w^{'} = w$ is an equivalence relation.

Proof. Reflexivity ( $h = e$ ), symmetry ( $h \mapsto h^{- 1}$ ), transitivity ( $h_{1}, h_{2} \mapsto h_{2} \cdot h_{1}$ ). $□$

3.5 Invariants of the Relational Field

Proposition 3.8 (Classification of invariants).

Scalar invariants (fully invariant):

(i) $w_{t} (i, j)$ , $W_{t} (i, j)$ , $d_{t} (i)$ ---all weight- and degree-related quantities.

Conjugacy-class invariants:

(ii) For a closed loop $γ = (i_{0}, i_{1}, \dots, i_{k}, i_{0})$ , the holonomy $Hol_{t} (γ) := g_{t} (i_{0}, i_{1}) g_{t} (i_{1}, i_{2}) \dots g_{t} (i_{k}, i_{0}) \in G$ has gauge-invariant conjugacy class $[Hol_{t} (γ)]_{conj}$ .

Proof. (i) Lemma 3.5. (ii) Under gauge $h$ , the telescoping product gives $Hol_{t}^{h} (γ) = h (i_{0}) Hol_{t} (γ) h (i_{0})^{- 1}$ . The conjugacy class is therefore invariant. $□$

3.6 Discrete Curvature

Corollary 3.9 (Discrete curvature). For a triangle $△ = (i, j, k)$ : $Ω_{t} (△) := g_{t} (i, j) g_{t} (j, k) g_{t} (k, i) \in G$ is the length-3 holonomy. Its conjugacy class is gauge-invariant.

Scalar curvature: for bi-invariant distance $d_{G}$ , $ω_{t} (△) := d_{G} (Ω_{t} (△), e)^{2}$ is fully gauge-invariant.

Proof. $Ω_{t}^{h} = h (i) Ω_{t} h (i)^{- 1}$ . Bi-invariance: $d_{G} (h Ω h^{- 1}, e) = d_{G} (Ω, e)$ . $□$

Interpretation.

$Ω_{t} (△) = e$ : the triangle is flat---three transits are perfectly compatible.
$Ω_{t} (△) \neq = e$ : non-zero curvature---a twist or inconsistency in the transits.
$ω_{t} (△)$ is the discrete analogue of $∥ F_{A} ∥^{2}$ in Yang--Mills theory.

3.7 Completeness of Invariants

Proposition 3.10 (Completeness---connected case). If $(V, E_{t})$ is connected, the relational field $W_{t}$ is uniquely determined (up to gauge) by:

(i) all edge weights ${w_{t} (i, j)}$ ,

(ii) the holonomies of fundamental loops with respect to a fixed spanning tree $T \subset E_{t}$ .

3.1 Raw Relational Field#

3.2 Gauge Symmetry#

Motivation#

Definitions#

3.3 Properties of the Gauge Action#

3.4 Relational Field#

3.5 Invariants of the Relational Field#

3.6 Discrete Curvature#

3.7 Completeness of Invariants#

3.1 Raw Relational Field#

3.2 Gauge Symmetry#

Motivation#

Definitions#

3.3 Properties of the Gauge Action#

3.4 Relational Field#

3.5 Invariants of the Relational Field#

3.6 Discrete Curvature#

3.7 Completeness of Invariants#

3.1 Raw Relational Field

3.2 Gauge Symmetry

Motivation

Definitions

3.3 Properties of the Gauge Action

3.4 Relational Field

3.5 Invariants of the Relational Field

3.6 Discrete Curvature

3.7 Completeness of Invariants

3.1 Raw Relational Field

3.2 Gauge Symmetry

Motivation

Definitions

3.3 Properties of the Gauge Action

3.4 Relational Field

3.5 Invariants of the Relational Field

3.6 Discrete Curvature

3.7 Completeness of Invariants