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: $\mathcal{W}_t^{\mathrm{raw}} := (V,\;w_t,\;g_t)$

• $V$: finite node set ($n:=|V|$),
• $w_t:V\times V\to\mathbb{R}_{\ge0}$: 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). $\mathcal{G} := G^V = \{h:V\to G\}.$ Pointwise multiplication: $(h_1\cdot h_2)(i):=h_1(i)\cdot h_2(i)$.

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

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

3.3 Properties of the Gauge Action#

Lemma 3.4 (Group action). The action of $\mathcal{G}$ 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 $h\cdot g := g^{h^{-1}}$. $\square$

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 $W_t$. $\square$

3.4 Relational Field#

Definition 3.6 (Relational field). The relational field at time $t$: $\mathcal{W}_t := [\mathcal{W}_t^{\mathrm{raw}}]_{\mathcal{G}} = \bigl\{(V,w_t,g_t^h) : h\in\mathcal{G}\bigr\}.$ The gauge orbit---the equivalence class of all raw fields connected by gauge transformation.

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

Proof. Reflexivity ($h=e$), symmetry ($h\mapsto h^{-1}$), transitivity ($h_1,h_2\mapsto h_2\cdot h_1$). $\square$

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 $\gamma=(i_0,i_1,\ldots,i_k,i_0)$, the holonomy $\mathrm{Hol}_t(\gamma) := g_t(i_0,i_1)\,g_t(i_1,i_2)\cdots g_t(i_k,i_0)\in G$ has gauge-invariant conjugacy class $[\mathrm{Hol}_t(\gamma)]_{\mathrm{conj}}$.

Proof. (i) Lemma 3.5. (ii) Under gauge $h$, the telescoping product gives $\mathrm{Hol}_t^h(\gamma) = h(i_0)\,\mathrm{Hol}_t(\gamma)\,h(i_0)^{-1}$. The conjugacy class is therefore invariant. $\square$

3.6 Discrete Curvature#

Corollary 3.9 (Discrete curvature). For a triangle $\triangle=(i,j,k)$: $\Omega_t(\triangle) := g_t(i,j)\,g_t(j,k)\,g_t(k,i)\in G$ is the length-3 holonomy. Its conjugacy class is gauge-invariant.

Scalar curvature: for bi-invariant distance $d_G$, $\omega_t(\triangle) := d_G\bigl(\Omega_t(\triangle),e\bigr)^2$ is fully gauge-invariant.

Proof. $\Omega_t^h = h(i)\,\Omega_t\,h(i)^{-1}$. Bi-invariance: $d_G(h\Omega h^{-1},e) = d_G(\Omega,e)$. $\square$

Interpretation.

• $\Omega_t(\triangle)=e$: the triangle is flat---three transits are perfectly compatible.
• $\Omega_t(\triangle)\ne e$: non-zero curvature---a twist or inconsistency in the transits.
• $\omega_t(\triangle)$ 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 $\mathcal{W}_t$ 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 $\mathcal{W}_t$. $\square$