Exact Mutual Information Difference: Scalar vs. Maxwell Fields

We compute, for any Rényi index $n$, the exact difference between the mutual Rényi informations of a pair of free massless scalars and that of a Maxwell field in $d=4$ dimensions. Using the standard dimensional reduction method in polar coordinates, the problem is mapped to that of a single scalar field in $d=2$ with Dirichlet boundary conditions, which in turn can be conveniently related to the algebra of a chiral current on the full line. This latter identification, which maps algebras on an interval to two-interval algebras, yields exact results that clarify the structure of the long-distance OPE perturbative expansion of the mutual information. We find that this series has a finite radius of convergence only for integer $n>1$, while it becomes only asymptotical for $n=1$ and general non-integer values of $n$.

💡 Research Summary

The paper presents an exact calculation of the difference between the Rényi mutual informations (RMIs) of two free massless scalar fields and a free Maxwell field in four dimensions, for any Rényi index (n). The authors begin by recalling that the mutual information (I(A,B)) quantifies both classical and quantum correlations between disjoint regions (A) and (B), and that in a conformal field theory (CFT) it depends only on the conformally invariant cross‑ratio (\eta) when the regions are spherical. While exact results for (I_n(\eta)) are known in 1+1 dimensions, higher‑dimensional cases have remained inaccessible except for holographic large‑(N) limits.

In Section 2 the Maxwell field is expanded in vector spherical harmonics (Y^{\ell m}_s) ((s=r,e,m)). Each angular momentum sector ((\ell,m)) decouples into an independent one‑dimensional radial system. For (\ell\ge1) the Hamiltonian coincides with that of a massless scalar mode after a suitable rescaling, whereas the (\ell=0) sector is absent because the Gauss constraints force the radial electric and magnetic components to vanish. Consequently, the Maxwell theory is equivalent to two copies of a four‑dimensional scalar field with the (\ell=0) mode removed. The missing mode behaves exactly like a two‑dimensional massless scalar defined on the half‑line (