2.12 Entropy in theories with a modified propagator

In certain physically-interesting situations the propagator of a quantum field is different from the standard 1∕(p2 + m2 ) and is described by some function as 1∕F (p2). The quantum field in question then satisfies a modified Lorentz invariant field equation
2 𝒟 ψ = F(∇ )ψ = 0. (33 )
Theories of this type naturally arise in models with extra dimensions. The deviations from the standard form of propagator may be both in the UV regime (large values of p) or in the IR regime (small values of p). If the function 2 F(p ) for large values of p grows faster than 2 p this theory is characterized by improved UV behavior.

The calculation of the entanglement entropy performed in Section 2.9 can be generalized to include theories with operator (33View Equation). This example is instructive since, in particular, it illuminates the exact relation between the structure of 2-point function (the Green’s function in the case of free fields) and the entanglement entropy [183Jump To The Next Citation Point].

In d spacetime dimensions one has

∫ K (s,X, Y ) = --1--- ddp eipμ(Xμ− Yμ) e− sF (p2). (34 ) (2π )d
Note that we consider Euclidean theory so that 2 p ≥ 0. The Green’s function
G (X, Y ) = < ψ (X ),ψ(Y ) > (35 )
is a solution to the field equation with a delta-like source
𝒟 G (X, Y ) = δ(X, Y ) (36 )
and can be expressed in terms of the heat kernel as follows
∫ ∞ G (X, Y ) = dsK (s,X, Y ). (37 ) 0
Obviously, the Green’s function can be represented in terms of the Fourier transform in a manner similar to Eq. (34View Equation),
1 ∫ d ip (Xμ−Y μ) 2 2 2 G (X, Y) = ----d- d p e μ G (p ), G(p ) = 1∕F (p ). (38 ) (2 π)
The calculation of the trace of the heat kernel for operator (33View Equation) on a space with a conical singularity goes along the same lines as in Section 2.9. This was performed in [184Jump To The Next Citation Point] and the result is
---1--- TrK α(s) = (4π)d∕2(αV Pd(s) + 2παC2 (α)A (Σ)Pd− 2(s )) , (39 )
where the functions Pn (s) are defined as
∫ ∞ Pn (s) = --2-- dp pn−1e− sF (p2). (40 ) Γ (n2 ) 0
The entanglement entropy takes the form (we remind the reader that for simplicity we take the surface Σ to be a (d − 2)-dimensional plane) [184Jump To The Next Citation Point]
A (Σ) ∫ ∞ ds S = 12 ⋅-(4-π)(d−2)∕2 2 -s-Pd−2(s) . (41 ) 𝜖
It is important to note that [184Jump To The Next Citation Point, 183Jump To The Next Citation Point]

(i) the area law in the entanglement entropy is universal and is valid for any function 2 F(p );

(ii) the entanglement entropy is UV divergent independently of the function 2 F (p ), with the degree of divergence depending on the particular function 2 F (p );

(iii) in the coincidence limit, X = Y, the Green’s function (38View Equation)

2 1 ∫ ∞ G (X, X ) = ---d-----d- dp pd−1G (p2) (42 ) Γ ( 2)(4π)2 0
may take a finite value if G (p2) = 1∕F (p2) is decaying faster than 1∕pd. However, even for this function 2 F (p ), the entanglement entropy is UV divergent.

As an example, consider a function, which grows for large values of p as F(p2) ∼ p2k. The 2-point correlation function in this theory behaves as

1 < ϕ (X ),ϕ (Y) > ∼ ------------ (43 ) |X − Y|d−2k
and for k > d∕2 it is regular in the coincidence limit. On the other hand, the entanglement entropy scales as
A(Σ ) S ∼ -d−2-- (44 ) 𝜖 k
and remains divergent for any positive value of k. Comparison of Eqs. (43View Equation) and (44View Equation) shows that only for k = 1 (the standard form of the wave operator and the propagator) the short-distance behavior of the 2-point function is similar to the UV divergence of the entanglement entropy.
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