Definition. Spatial link fields and their transfer kernel [fdip-007Z]
AGENTDRAFTED
Let \(G\) be a finite group, \(U(1)\) or \(SU(2)\), with
normalized Haar measure. Fix a
finite incidence presentation with vertex set
\(V_s\), oriented edge set \(E_s\) and source and target maps
\(s,t:E_s\to V_s\). The spatial configuration space and
link Hilbert space are
\[X=G^{E_s},\qquad
\mathcal H_{\mathrm {link}}=L^2(X,dU;\mathbb C),\]
where \(dU\) is product Haar probability measure. The
gauge action of \(K=G^{V_s}\) is
\[(h\cdot U)_e=h_{t(e)}U_eh_{s(e)}^{-1},\qquad
(R_hf)(U)=f(h^{-1}\cdot U).\]
Haar invariance makes every \(R_h\) unitary.
Choose a real continuous central kernel \(k\) satisfying the
positive coefficient conditions:
\[\begin {gathered}
k(g)>0,\qquad k(g^{-1})=k(g),\\
k(g)=\sum _{\rho \in \widehat G}d_\rho a_\rho \chi _\rho (g),
\qquad a_\rho >0,\qquad
\sum _{\rho \in \widehat G}d_\rho ^2a_\rho <\infty .
\end {gathered}\]
All irreducible representations here are complex and unitary.
The expansion is uniformly absolutely convergent. Pointwise
positivity and coefficient positivity are separate hypotheses.
Let \(V:X\to \mathbb R\) be continuous and gauge invariant,
and set \(b(U)=e^{-V(U)/2}\). Compactness makes \(b\) and
\(b^{-1}\) bounded. If \(M_b\) denotes multiplication by \(b\),
define
\[\begin {aligned}
c(U,U')&=\prod _{e\in E_s}k(U_eU_e'^{-1}),\\
(C_kf)(U)&=\int _Xc(U,U')f(U')\,dU',\\
T&=M_bC_kM_b.
\end {aligned}\]
This is the pure-gauge transfer operator for the
specified data. Its integral kernel is \(b(U)c(U,U')b(U')\).
The spatial potential may be a sum of Wilson plaquette terms,
but the definition only needs its stated invariance and continuity.
The factorization separates the two spatial half-weights from
the temporal convolution. It parallels the gauge factors in
[luscher1977transfer, Eqs. (17), (20)–(21), p. 287];
the source comparison distinguishes this pure-gauge
model from that paper's gauge and fermion construction.