Locality and entanglement in bandlimited quantum field theory

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 $1+1$ dimensions 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.

Figure 1

Spatial profile (in the horizontal axis) of five equally spaced degrees of freedom, as given by the function (17), as a function of their spacing (vertical axis). Both axes are scaled so that the cutoff length, or Nyquist spacing, $\pi /\mathrm{\Omega }$, is equal to one. The sample points are centered at $x=0$. We see that for spacings above the Nyquist spacing, the degrees of freedom occupy a region consisting of five disjoint intervals of length $\sim 1$ surrounding each sample point. Below the Nyquist spacing, the degrees of freedom merge to occupy a single interval of length $\sim 5$. This interval does not decrease in size even as the sampling points are taken on top of one another.

Figure 2

$\varphi -\varphi$ correlations as a function of their separation. The horizontal axis is scaled by $\mathrm{\Omega }/\pi$ so that integer values correspond to Nyquist spacings. The bandlimited correlations are blue with the Nyquist spacings indicated by red dots. The black dashed line shows the ultraviolet-divergent correlations without the ultraviolet bandlimit. We see that for points on the Nyquist lattice, the bandlimited correlators are in closer agreement to the correlation functions without the bandlimit. A counterterm of $\log (\omega )/(2\pi )$ is added to $⟨\varphi \varphi ⟩$ to cancel the infrared divergence.

Figure 3

$\pi \text{−}\pi$ correlations as a function of their separation. The horizontal axis is scaled by $\mathrm{\Omega }/\pi$ so that integer values correspond to Nyquist spacings. The vertical axis is scaled by $1/{\mathrm{\Omega }}^2$. The bandlimited correlations are blue with the Nyquist spacings indicated by red dots. The black dashed line shows the ultraviolet-divergent correlations without the ultraviolet bandlimit.

Figure 4

Entropy of a set of sample points for a thermally distributed classical bandlimited Klein-Gordon field for several temperatures. We see that the magnitude of the entropy grows linearly with the number of points, illustrating a volume law. Notice also that this figure shows negative entropy at low temperatures, which simply reflects the fact that continuous probability distributions can have negative entropy. We fix the inherent ambiguity in these entropies by requiring that it matches the quantum entropy at high temperature.

Figure 5

The entropy of a harmonic oscillator as a function of the symplectic eigenvalue, both classically and quantum mechanically. In quantum mechanics, the uncertainty principle requires $d\ge 1/2$, which is saturated by the vacuum state for which the entropy is zero. Classically, the symplectic eigenvalue can be any positive number, but the entropy becomes negative for small $d$. This again is a consequence of the fact that the entropy of a continuous probability distribution is not bounded from below.

Figure 6

Classical and quantum $\varphi -\varphi$ correlations as a function of their separation at various temperatures. The horizontal axis is scaled by $\mathrm{\Omega }/\pi$ so that integer values correspond to Nyquist spacings. The vertical axis is scaled by $\mathrm{\Omega }/T$. The classical correlator is plotted as black dots. The scaling of the vertical axis absorbs all of the temperature dependence of the classical correlator (since it grows proportionally to $T/\mathrm{\Omega }$), thus the single graph of the classical correlator completely characterizes its behavior. The quantum correlators are shown as lines for temperatures up to $T/\mathrm{\Omega }=1$, at which point the quantum correlator converges to the classical correlator.

Figure 7

Classical and quantum $\pi \text{−}\pi$ correlations as a function of their separation at various temperatures. The horizontal axis is scaled by $\mathrm{\Omega }/\pi$ so that integer values correspond to Nyquist spacings. The vertical axis is scaled by $\mathrm{\Omega }/T$. The classical correlator is plotted as black dots. The scaling of the vertical axis absorbs all of the temperature dependence of the classical correlator (since it grows proportionally to $T/\mathrm{\Omega }$), thus the single graph of the classical correlator completely characterizes its behavior. The quantum correlators are shown as lines for temperatures approaching $T/\mathrm{\Omega }=1$, at which point the quantum correlator converges to the classical correlator.

Figure 8

Entropy for $N$ Nyquist-spaced sample points for a thermal state of a quantum Klein-Gordon field. The plot illustrates the transition between the logarithmic behavior at low temperature to linear behavior at higher temperatures where the entropy becomes an extensive variable.

Figure 9

The dependence of the entanglement entropy on the number of sample points, for sample spacings between 1 and 2 Nyquist spacings. The $\mathrm{\Delta }x$ labels in the legend are scaled by $\mathrm{\Omega }/\pi$ so that the Nyquist spacing is 1. We see that once the adjacent point spacing has reached twice the Nyquist spacing, the entropy has transitioned from the logarithmic scaling law to a volume law. To confirm this, we performed a fit to $S(N)=aN+b\log N+c$ for values of $\mathrm{\Delta }x\in [1,2]$. We find that the coefficient of the linear term increases from $a=0$ at $\mathrm{\Delta }x=1$, and the coefficient of the logarithmic term decreases to $b=0$ at $\mathrm{\Delta }x=2$. The infrared scale is $\omega /\mathrm{\Omega }=10^{-300}$.

Figure 10

Entanglement entropy of five points as a function of the spacing between them. The horizontal axis is scaled by $\mathrm{\Omega }/\pi$ so that the Nyquist spacing is 1. For spacings below the Nyquist spacing, we see a plateauing effect indicating that the entanglement entropy is not sensitive to the spacing between the points for spacings below the Nyquist spacing. Above the Nyquist spacing the entropy tends to increase as the spacing increases. In this plot the infrared to ultraviolet ratio is $\omega /\mathrm{\Omega }=10^{-5}$.

Figure 11

Entanglement entropy of five points as a function of the spacing between them at various temperatures. The horizontal axis is scaled by $\mathrm{\Omega }/\pi$ so that the Nyquist spacing is 1. We see here a plateau in the entropy at small spacings, similar to the plateau in the ground state entropy. The analogous classical calculation is also shown as black dots for temperature $T/\mathrm{\Omega }=1$. At temperatures $T/\mathrm{\Omega }>1$, the entropy behaves the same as for $T/\mathrm{\Omega }=1$ but shifted vertically by a constant $\sim \log (T/\mathrm{\Omega })$. In this plot the infrared to ultraviolet ratio is $\omega /\mathrm{\Omega }=10^{-5}$.

Figure 12

Entanglement entropy dependence of number of derivatives traced out at a single point. Fitted curve to Nyquist-spaced points is $S(N)=0.334\log (N)+3.25$, where $N$ is the number of sampling points. Fitted curve to derivative sampling points is $S(N)=\frac{1}{3}\log (N)+3.14$, where $N-1$ is the number of derivatives sampled (which corresponds to $N$ sampled points). We see that both curves differ by a constant $\approx 0.11$. The infrared to ultraviolet ratio is $\omega /\mathrm{\Omega }=10^{-300}$.

Figure 13

Mutual information between subsystem $A_N$ and probe point $P$. The spatial axes are scaled so that the Nyquist spacing is 1. The variable $x$ denotes the position of the probe point $P$, with $x=0$ at the center of the subsystem $A_N$. For spacings much larger than the Nyquist spacing, the degrees of freedom occupy independent intervals of the order of the Nyquist spacing in size. For spacings at or below the Nyquist spacing, the points in the interval describe a fixed volume of size $N$.