Ultraviolet behavior in background independent quantum field theory

We describe a background independent quantization of the scalar field that provides an explicit realization of Fock-like states and associated operators in a polymer Hilbert space. The vacuum expectation values of the commutator and anticommutator of the creation and annihilation operators become energy dependent, and exhibit a surprising transition to fermionic behavior at high energy. Furthermore, the approach yields a modified dispersion relation with a leading correction proportional to the momentum cubed. These results suggests a fundamental change in the ultraviolet properties of quantum fields.

Figure 1

A pictorial depiction of the basis state $|{\mu }_1,\dots ,{\mu }_7⟩$.

Figure 2

The expectation value of the commutator, the anticommutator and the number operator in the state $|0_{\mathbf{k}}⟩$ as a function of $\ln ({\gamma }_{\mathbf{k}})$.

Figure 3

The expectation value of the commutator, the anticommutator and the number operator in the state $|1_{\mathbf{k}}⟩$ as a function of $\ln ({\gamma }_{\mathbf{k}})$.

Figure 4

The expectation value of the commutator, the anticommutator and the number operator in the state $|2_{\mathbf{k}}⟩$ as a function of $\ln ({\gamma }_{\mathbf{k}})$.

Figure 5

The expectation value of the commutator, the anticommutator and the number operator in the state $|3_{\mathbf{k}}⟩$ as a function of $\ln ({\gamma }_{\mathbf{k}})$.