Spin Foam Quantization and Anomalies

TL;DR

The paper investigates anomalies in spin foam quantizations of gravity by examining how the continuum path-integral measure, constrained BF theory (notably the Plebanski action), and the discrete spin-foam amplitudes interact. It develops both passive (measure-derived) and active (background-independence) perspectives to derive consistency conditions on lower-dimensional amplitudes and demonstrates that many proposed normalizations fail anomaly-free criteria. A finite toy model and a Husain--Kuchař–like spin foam illustrate how gauge fixing and second-class constraints shape the measure and potentially regulate divergences, while also showing that naively finitary choices can be unphysical. The analysis provides concrete prescriptions for constructing anomaly-free, background-independent spin foam measures and clarifies the role of gauge fixing and constraint algebra in controlling bubble divergences, with implications for Barrett–Crane-type models and beyond.

Abstract

The most common spin foam models of gravity are widely believed to be discrete path integral quantizations of the Plebanski action. However, their derivation in present formulations is incomplete and lower dimensional simplex amplitudes are left open to choice. Since their large-spin behavior determines the convergence properties of the state-sum, this gap has to be closed before any reliable conclusion about finiteness can be reached. It is shown that these amplitudes are directly related to the path integral measure and can in principle be derived from it, requiring detailed knowledge of the constraint algebra and gauge fixing. In a related manner, minimal requirements of background independence provide non trivial restrictions on the form of an anomaly free measure. Many models in the literature do not satisfy these requirements. A simple model satisfying the above consistency requirements is presented which can be thought of as a spin foam quantization of the Husain--Kuchar model.

Figures (7)

• Figure 1: Vacuum bubbles to be physically equivalent in an anomaly free spin foam model. Their equivalence constrains the possible behavior of the face and edge amplitudes. We represent the sequence of moves that relate the bubble on the left with that on the right.
• Figure 2: Relevant 15j-symbols for $B_1(j)$, $B_2(j)$ and $B_3(j)$. Thin lines are trivial representations.
• Figure 3: Sequence of subdivision and piecewise linear transformation relating the tetrahedral bubble with the pyramidal one.
• Figure 4: Vertex contributions to the bubble amplitudes above (thin lines represent edges labeled with the trivial representation). From left to right their value is given by $(2j+1)^{-1}$, $(2j+1)^{-2}$ and $(2j+1)^{-1}(2l+1)^{-1}$ in the Riemannian Barrett--Crane model if we normalize the corresponding intertwiners.
• Figure 5: Two equivalent spin foam configurations. The dotted 2-cells on the left are labelled by the trivial representation.