SU(2) lattice gauge theory: Local dynamics on nonintersecting electric flux loops

We use Schwinger bosons as prepotentials for lattice gauge theory to define local linking operators and calculate their action on linking states for ($2+1$)-dimensional SU(2) lattice gauge theory. We develop a diagrammatic technique and associate a set of rules (lattice Feynman rules) to compute the entire loop dynamics diagrammatically. The physical loop space is shown to contain only nonintersecting loop configurations after solving the Mandelstam constraint. The smallest plaquette loops are contained in the physical loop space, and other configurations are generated by the action of a set of fusion operators on these basic loop states.

Figure 1

Prepotentials on a link.

Figure 2

A particular site on a two-dimensional lattice and associated prepotentials.

Figure 3

SU(2) fluxes: all possible linking at a site of a two-dimensional lattice.

Figure 4

The left- and right-hand sides of $=$ denote the respective sides of Eq. (21) with all the coefficients.

Figure 5

The left- and right-hand sides of $=$ denote the respective sides of Eq. (22) with all the coefficients.

Figure 6

The left- and right-hand sides of $=$ denote the respective sides of Eq. (24) with all the coefficients. Note that the usual vertex symbol denotes the unusual coefficient for the first term of the decomposition.

Figure 7

The coefficients are explicitly given in the last column.

Figure 8

Pictorial representation of the Mandelstam constraint in terms of prepotentials as given in (27).

Figure 9

Physical loops: nested and overlapped but nonintersecting ones.

Figure 10

Variables defined locally at dual site $(\tilde{x})$, on a two-dimensional lattice spanned by basis vectors $e_1$ along the $+X$ axis and $e_2$ along the $+Y$ axis.

Figure 11

${\mathrm{\Pi }}_{N_2}^{+}(\tilde{x}+\frac{e_1}{2})|L(\tilde{x})=1,L(\tilde{x}+e_1)=1⟩=|L(\tilde{x})=1$, $L(\tilde{x}+e_1)=1,N_2(\tilde{x}+\frac{e_1}{2})=1⟩$.

Figure 12

${\mathrm{\Pi }}_{D_2}^{-}{\mathrm{\Pi }}_{D_1}^{+}(\tilde{x}-\frac{e_1}{2}+\frac{e_2}{2})|L(\tilde{x})=1,L(\tilde{x}{e}_1+e_2)=1⟩=|L(\tilde{x})=1,L(\tilde{x}+e_1)=1,D_1(\tilde{x}-\frac{e_1}{2})=1,D_2(\tilde{x}-\frac{e_1}{2})=-1⟩$.

Figure 13

Pictorial representation of the fusion constraint as given in (41).

Figure 14

The Hamiltonian operator in terms of prepotential becomes a sum of 16 operators as shown above diagrammatically. The solid line along a link denotes the presence of a prepotential creation operator on that link, whereas a dotted line denotes the annihilation operators on that. Clearly, the whole set is rotationally symmetric and Hermitian. These sets of operators again can be subdivided in six classes of operators as shown in (a)–(f) denoted by (a) $\equiv H_{++++}$, (b) $\equiv H_{+++-}$, (c) $\equiv H_{++--}$, (d) $\equiv H_{+-+-}$, (e) $\equiv H_{+---}$, and (f) $\equiv H_{----}$.

Figure 15

Explicit action of type (a) or $H_{++++}$.

Figure 16

Explicit action of type (b) or $H_{+++-}$.

Figure 17

Explicit action of type (c) or $H_{++--}$.

Figure 18

Explicit action of type (d) or $H_{+-+-}$.

Figure 19

Explicit action of type (e) or $H_{+---}$.

Figure 20

Explicit action of type (f) or $H_{----}$.