Compact Non-Abelian Finite-Lattice Regulator

Adopt the following pure-gauge Euclidean regulator. Assume full AC for normalized Haar-probability existence and for the L² transfer-operator framework. Let $N_s,N_t\ge 3$ be integers, $a>0$ a common lattice spacing, and $g_0>0$ a dimensionless bare coupling. Use natural units $\mathrm{\hslash }=c=1$. The finite periodic lattice is

$\mathrm{\Lambda }=(\mathbb{Z}\mathrm{/}{N}_s\mathbb{Z}{)}^3\times (\mathbb{Z}\mathrm{/}{N}_t\mathbb{Z}),$

with spatial extent $L_s=N_{s}a$ in each spatial direction and Euclidean time extent $L_t=N_{t}a$. The global gauge group supplied by the finite non-Abelian lattice datum is

$SU(2)=\{U\in M_2(\mathbb{C}):U^{\ast }U=I_2,\det U=1\},$

the compact Hausdorff non-Abelian Lie group established in the preceding local lemma, with its smooth defining two-dimensional unitary representation. Use that representation; $U^{\ast }$ denotes conjugate transpose and ${\mathrm{Tr}}_2$ its matrix trace. All link variables are dimensionless.

For each positive spatial direction $i\in \{1,2,3\}$, assign a link matrix $U_{x,i,t}\in SU(2)$ from $(x,t)$ to $(x+\widehat{i},t)$. Assign a temporal link matrix ${\eta }_{x,t}\in SU(2)$ from $(x,t)$ to $(x,t+1)$, with time understood modulo $N_t$. A reversed link carries the inverse matrix. Vertex gauge transformations are all functions $\mathrm{\Omega }:\mathrm{\Lambda }\to SU(2)$, acting by

$U_{x,i,t}\mapsto {\mathrm{\Omega }}_{x,t}{U}_{x,i,t}{\mathrm{\Omega }}_{x+\widehat{i},t}^{-1},{\eta }_{x,t}\mapsto {\mathrm{\Omega }}_{x,t}{\eta }_{x,t}{\mathrm{\Omega }}_{x,t+1}^{-1}\mathrm{.}$

Let $E_{\mathrm{\Lambda }}^{+}$ be the finite set of these positively oriented spatial and temporal links, let ${\mathcal{C}}_{\mathrm{\Lambda }}:=SU(2{)}^{E_{\mathrm{\Lambda }}^{+}}$ be the full link-configuration space with its finite-product topology, and equip it with the product Borel sigma-algebra ${\mathcal{B}}_{\mathrm{\Lambda }}:={⨂}_{\mathrm{\ell }\in E_{\mathrm{\Lambda }}^{+}}\mathcal{B}(SU(2))$.

For $i<j$, define the spatial plaquette product

$U_{x,ij,t}=U_{x,i,t}{U}_{x+\widehat{i},j,t}{U}_{x+\widehat{j},i,t}^{-1}{U}_{x,j,t}^{-1},$

and define the temporal-spatial plaquette product

$Q_{x,i,t}={\eta }_{x,t}{U}_{x,i,t+1}{\eta }_{x+\widehat{i},t}^{-1}{U}_{x,i,t}^{-1}\mathrm{.}$

Set $\beta =4\mathrm{/}{g}_0^2$, a positive dimensionless parameter. For a spatial slice $u=(U_{x,i})$, set

$S_s(u)=\beta {\sum }_{x,i<j}(1-\frac{1}{2}{\mathrm{ReTr}}_2{U}_{x,ij}),$

and for adjacent slices $u^{\mathrm{\prime }},u$ and temporal links $\eta =({\eta }_x)$, set

$S_t(u^{\mathrm{\prime }},u;\eta )=\beta {\sum }_{x,i}(1-\frac{1}{2}{\mathrm{ReTr}}_2({\eta }_x{U}_{x,i}^{\mathrm{\prime }}{\eta }_{x+\widehat{i}}^{-1}{U}_{x,i}^{-1}))\mathrm{.}$

The adopted Wilson action is $S_{\mathrm{\Lambda }}={\sum }_{t=0}^{N_t-1}(S_s(u_t)+S_t(u_{t+1},u_t;{\eta }_t))$, with $u_{N_t}=u_0$. The compact Hausdorff group $SU(2)$ has the unique normalized left Haar probability measure $dU$, which is also right invariant under the assumed Axiom of Choice. On each positively oriented link use this measure; $d{\mu }_{\mathrm{\Lambda }}$ is its finite product over the link set. For a bounded ${\mathcal{B}}_{\mathrm{\Lambda }}$-measurable function $O:{\mathcal{C}}_{\mathrm{\Lambda }}\to \mathbb{C}$, call $O$ gauge-invariant when $O(\mathrm{\Omega }\cdot U)=O(U)$ for every vertex transformation $\mathrm{\Omega }:\mathrm{\Lambda }\to SU(2)$ and every $U\in {\mathcal{C}}_{\mathrm{\Lambda }}$, where $\mathrm{\Omega }\cdot U$ is the link configuration whose coordinates are the transformed links displayed above. The exact finite-volume partition function and the expectation of each such observable are

$Z_{\mathrm{\Lambda }}=\int e^{-S_{\mathrm{\Lambda }}}d{\mu }_{\mathrm{\Lambda }},⟨O{⟩}_{\mathrm{\Lambda }}=Z_{\mathrm{\Lambda }}^{-1}\int O{e}^{-S_{\mathrm{\Lambda }}}d{\mu }_{\mathrm{\Lambda }}\mathrm{.}$

If $O$ represents a physical observable with assigned units, $⟨O{⟩}_{\mathrm{\Lambda }}$ has those same units; this postulate assigns no unit to the generic mathematical function $O$.

For the transfer convention, let $V_s=(\mathbb{Z}\mathrm{/}{N}_s\mathbb{Z}{)}^3$ be the finite vertex set of one slice and $E_s=V_s\times \{1,2,3\}$ its positive spatial-link set. Define $d{\mu }_s:={⨂}_{(x,i)\in E_s}dU$ on $SU(2{)}^{E_s}$, where each factor is the normalized Haar Borel probability; let ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=L^2(SU(2{)}^{E_s},d{\mu }_s;\mathbb{C})$ be the complex $L^2$ space. Define ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\{\psi \in {\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}:\mathcal{U}(h)\psi =\psi \text{ for every }h\in SU(2{)}^{V_s}\}$, where $\mathcal{U}(h)\psi (u)=\psi (h^{-1}\text{\hspace{0.17em}⁣}\cdot u)$ in $L^2$. The finite-lattice well-definedness result establishes that this fixed subspace is closed. It is the group-invariant finite-slice Gauss sector. For one transfer step let $\eta =({\eta }_x{)}_{x\in V_s}\in SU(2{)}^{V_s}$ denote its temporal-link tuple, and define $d{\mu }_V:={⨂}_{x\in V_s}dU$ on those coordinates, with each factor the normalized Haar Borel probability $dU$. Thus $d{\mu }_V$ is the finite product probability over all temporal links in one time step. Define

$K(u^{\mathrm{\prime }},u)=e^{-\frac{1}{2}{S}_s(u^{\mathrm{\prime }})-\frac{1}{2}{S}_s(u)}{\int }_{SU(2{)}^{V_s}}{e}^{-S_t(u^{\mathrm{\prime }},u;\eta )}d{\mu }_V(\eta ),(T\psi )(u^{\mathrm{\prime }})=\int K(u^{\mathrm{\prime }},u)\psi (u)d{\mu }_s(u)\mathrm{.}$

The finite Euclidean time direction is assembled by iterating this one-step transfer operator, with the periodic boundary identification stated above. This is a Euclidean transfer convention only: no self-adjoint or positive-spectrum claim for $T$, no Lorentzian Hamiltonian identification, and no continuum or infinite-volume limit are postulated. The model has no dynamical matter fields, so it makes no matter-screening, confinement, or string-breaking claim. The regulator is adopted, not derived from a continuum theory or empirical evidence.

The quantities $a,L_s,L_t$ have units of length; $g_0,\beta ,U_{x,i,t},{\eta }_{x,t},S_{\mathrm{\Lambda }},Z_{\mathrm{\Lambda }}$ are dimensionless in the stated natural-unit convention. No measured parameter is supplied.

Creutz, Quarks, Gluons and Lattices, Chapter 7, pp. 34–38

Creutz, Quarks, Gluons and Lattices, Chapter 8, pp. 39–43

Creutz, Quarks, Gluons and Lattices, Chapter 15, pp. 101–107

Tong, Gauge Theory, Chapter 4, §§4.2.1–4.2.3, pp. 206–215

Axiom of Choice for normalized Haar-probability existence

Normalized Haar probability on a compact group

Compact non-Abelian Lie-group structure and smooth defining representation of SU(2)

Finite non-Abelian lattice-data formulation and face-potential interface

Finite-volume integral and transfer-operator well-definedness