Locality and entanglement in bandlimited quantum field theory

Abstract

We consider a model for a Planck scale ultraviolet cutoff which is based on Shannon sampling. Shannon sampling originated in information theory, where it expresses the equivalence of continuous and discrete representations of information. When applied to quantum field theory, Shannon sampling expresses a hard ultraviolet cutoff in the form of a bandlimitation. This introduces nonlocality at the cutoff scale in a way that is more subtle than a simple discretization of space: quantum fields can then be represented as either living on continuous space or, entirely equivalently, as living on any one lattice whose average spacing is sufficiently small. We explicitly calculate vacuum entanglement entropies in 1+1 dimension and we find a transition between logarithmic and linear scaling of the entropy, which is the expected 1+1 dimensional analog of the transition from an area to a volume law. We also use entanglement entropy and mutual information as measures to probe in detail the localizability of the field degrees of freedom. We find that, even though neither translation nor rotation invariance are broken, each field degree of freedom occupies an incompressible volume of space, indicating a finite information density.

Covariant bandlimitation from Generalized Uncertainty Principles

“…Analytic functions exhibit a severe non-locality since all of the degrees of freedom are contained in a small volume around any point in spacetime (as the Taylor series can be evaluated in a neighborhood). (For a related discussion, see also [26].) Hence it seems that anything akin to ordinary sampling theory would allow for superluminal signalling, thus would be inappropriate for the configuration space of a relativistic field theory.…”

Impact of relativity on particle localizability and ground state entanglement

“…As has been studied in [46], bandlimitation has a non-trivial effect on the structure of local degrees of freedom and entanglement of the field. We will return to explicitly discuss these effects in Subsection 4.3 and Section 5 (respectively).…”

“…It is straightforward to show that the above reconstruction formula extends to bandlimited quantum fields [46]:…”

“…The question of characterizing the spatial degrees of freedom of bandlimited fields was answered by Shannon sampling theory [55][56][57] for classical bandlimited functions, and can be extended to quantum field theory [46,[58][59][60]. ( [46] also contains a discussion about localizability in bandlimited quantum fields, albeit the context is a model for Planck-scale physics.) The basic result of sampling theory for a function, f , with a cutoff k ∞ < Λ, can be summarized by the following reconstruction formula:…”

“…Even though we made use of the latter cutoff in Subsection 4.3 to establish the form of the local degrees of freedom of the field in terms of sample points, if one restricts attention to finitely many sample points then the entanglement calculations can be performed in either case. For a discussion of a similar situation, see [46].…”