Localization and diffusion in polymer quantum field theory

Polymer quantization is a nonstandard approach to quantizing a classical system inspired by background-independent approaches to quantum gravity such as loop quantum gravity. When applied to field theory it introduces a characteristic polymer scale at the level of the fields’ classical configuration space. Compared with models with space-time discreteness or noncommutativity, this is an alternative way in which a characteristic scale can be introduced in a field theoretic context. Motivated by this comparison we study here localization and diffusion properties associated with polymer field observables and dispersion relation in order to shed some light on the novel physical features introduced by polymer quantization. While localization processes seems to be only mildly affected by polymer effects, we find that polymer diffusion differs significantly from the “dimensional reduction” picture emerging in other Planck-scale models beyond local quantum field theory.

Figure 1

Expectation value of the number operator in the vacuum state and in a one-particle state as a function of ${\gamma }_{\overrightarrow{p}}$.

Figure 2

Polymer correction to the ordinary one-particle dispersion relation.

Figure 3

Numerical evaluation of $⟨E{⟩}_{\text{poly}}$ for $\ell =1$ and $0.01<\sigma <2$, $\mathrm{\Delta }\sigma =0.01$.

Figure 4

Expectation value of the energy for a smoothed state in the ordinary case (continuous black line) and in the polymer case (continuous blue line and dotted line).

Figure 5

Spectral dimension in units $\ell =1$ (the continuous line, the dashed line is for ordinary Euclidean space).