Spinor Representation for Loop Quantum Gravity

TL;DR

This work develops a spinor-based quantization of loop quantum gravity that recasts edge degrees of freedom from SU(2) group variables to holomorphic spinor variables in Bargmann space. By constructing a unitary map that sends SU(2) representation matrices to holomorphic spinor polynomials, it demonstrates unitary equivalence between the standard LQG edge Hilbert space and a spinor Hilbert space, and shows that Haar measure becomes a product of Gaussian measures. The formalism extends to arbitrary graphs with cylindrical consistency, enabling a continuum spinor Hilbert space that is unitarily equivalent to the LQG continuum space, and provides a straightforward characterization of SU(2)-invariant states via spinor invariants. Overall, the spinor representation offers geometric clarity (polyhedra faces via spinors), simplifies calculations by replacing group integrals with Gaussian integrals, and opens prospects for spinor-based group field theory and spinfoam amplitudes.

Abstract

We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.

Figures (6)

• Figure 1: When considering the spinors $| z \rangle$ as fundamental variables one can follow different paths: on the one hand one can first divide out the ${\mathrm U}(1)$-gauge invariance on the edges, which leads back to the standard phase space of loop gravity with group variables $g$ and Lie-algebra variables $X$. Then one still has to divide out by ${\rm SU}(2)$. On the other hand one can also first divide out the ${\rm SU}(2)$-gauge invariance at the vertices, which leads to the ${\mathrm U}(N)$-framework and a characterization of the intertwiner-space in terms of $E$- and $F$-variables. These spaces then have to be glued together in an appropriate way to respect ${\mathrm U}(1)$-gauge invariance at the edges.
• Figure 2: The one-loop graph. Gauge invariant spin network functions on this graph can be mapped unitarily onto polynomials in $F$.
• Figure 3: The two vertex graph. Gauge invariant spin network functions on this graph can be mapped unitarily onto traces of polybinomials such as ${\rm Tr}(F \tilde{F}), {\rm Tr}(F \tilde{F} F \tilde{F}), \dots$
• Figure 4: A graph with five vertices ($v_1, v_2, v_3, v_4, v_5$) and nine edges ($a,b,c,d,e,f,g,h,i$). Let $N_i$ be the valence of vertex $v_i$. Then at each vertex the ${\rm SU}(2)$-invariant, holomorphic part of the classical phase space is $\frac{N_i^2 - N_i}{2}$-dimensional. For example, at vertex $v_5$ (the highlighted region of the above graph) this space consists of ${^5F}_{ab} = [ {^5z}_a \mid {^5z}_b \rangle , {^5F}_{ac} = [ {^5z}_a \mid {^5z}_c \rangle , {^5F}_{bc} = [ {^5z}_b \mid {^5z}_c \rangle$, where $| {^5z}_a \rangle, | {^5z}_b \rangle, | {^5z}_c \rangle$ are the spinors at vertex $v_5$ into direction of the vertices $a,b,c$ respectively.
• Figure 5: Taking care of ${\mathrm U}(1)$-invariance on each edge: the highlighted region depicts a part of the total graph $\gamma$ consisting of two vertices $v_1$ and $v_2$ connected by an edge $a$. In order to ensure that (\ref{['generic_function']}) for this graph is ${\mathrm U}(1)$-invariant one must demand that ${^1J}_{ab} + {^1J}_{ac} + {^1J}_{ad} + {^1J}_{ae} = {^2J}_{af} + {^2J}_{ag} + {^2J}_{ah} + {^2J}_{ak} + {^2J}_{al}$.