2.2 Short-distance correlations

On the other hand, if the entropy (3View Equation) is non-vanishing, this shows that in the global system there exist correlations across the surface Σ between modes, which reside on different sides of the surface. In this review we shall consider the case in which the system in question is a quantum field. The short-distance correlations that exist in this system have two important consequences:

For a free massless scalar field the 2-point correlation function in d spacetime dimensions has the standard form

< Ļ• (x),Ļ•(y) >= ---Ωd-----, (5 ) |x − y|d− 2
where Γ (d−22) Ωd = 4πdāˆ•2. Correspondingly, the typical behavior of the entanglement entropy in d dimensions is
A (Σ) S ∼ -šœ–d−-2-, (6 )
where the exact pre-factor depends on the regularization scheme. Although the similarity between (5View Equation) and (6View Equation) illustrates well the field-theoretical origin of the entanglement entropy, the exact relation between the short-distance behavior of 2-point correlation functions in the field theory and the UV divergence of the entropy is more subtle, as we shall discuss later in the paper.
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