Momentum-space entanglement and mutual information in Kaluza-Klein reduction

Abstract

In theories with compact extra dimensions, the four-dimensional effective description is obtained by Kaluza-Klein reduction. The state available to a four-dimensional observer is the reduced density matrix obtained by tracing the higher-dimensional vacuum over the tower of massive modes, and it need not be pure. In this paper we study the momentum-space entanglement of this state for a five-dimensional scalar on $M_4\times S^1$ with a cubic or quartic self-interaction. We compute, at leading order in the coupling, the mutual information $I_n$ between the zero modes below a resolution scale $μ$ and a single Kaluza-Klein level $n$. When the zero modes couple linearly and without derivatives to pairs of tower modes, $I_n$ falls as $1/m_n$, where $m_n$ is the mass of level $n$. In the cubic theory, this tail is independent of the ultraviolet cutoff, with coefficient $(μR)^2/1024π^3$ for compactification radius $R$, and the mutual information with the whole tower grows logarithmically with the number of levels. In the quartic theory, the tail is masked by states containing additional hard quanta, which make $I_n$ depend on the cutoff. Compared with the corrections the tower makes to low-energy amplitudes, which are dominated by the lightest levels and captured by local operators, the mutual information with the tower is not dominated by any finite set of levels and arises from the time-nonlocal part of the effective action. The local four-dimensional effective theory therefore describes the low-energy observables of the reduced system but not its state. Unbroken supersymmetry cancels the vacuum energy of the compactification but not the mutual information, which at leading order is a sum of squared amplitudes.

Kaluza–Klein reduction reproduces low-energy amplitudes after integrating out massive modes, but what correlations with those modes remain in the reduced vacuum? This paper studies a real scalar on $M_4\times S^1$ with cubic or quartic interactions, partitioning the Hilbert space into zero modes with $|\mathbf k|\le\mu$ and the remaining modes. It computes entropy densities and mutual information with a KK level $\pm n$ using perturbative vacuum amplitudes: squared amplitudes determine the order-coupling-squared Rényi-2 quantities and the leading coupling logarithm of von Neumann quantities. For the cubic interaction, the per-level entropy kernel obeys $J_n^{(3)}\sim A_3/(nR)$, where $A_3=(\mu R)^2/(1024\pi^3)$, for $\mu R\ll n\ll\Lambda_{\rm UV}R$. The momentum integral is cutoff independent in this regime. Level mutual informations are additive at this order, yielding logarithmic growth with the number of levels, unlike the convergent heavy-level corrections to renormalised low-energy amplitudes. Analytic phase-space estimates are checked numerically. Quartic interactions contain a similar tail, but additional hard-quanta channels dominate: the fixed-level mutual information is cutoff dependent, and the level-summed kernel grows as $\mu^2R^2\Lambda_{\rm UV}^3$. The paper associates the correlations with time-nonlocal terms omitted by a local derivative expansion. Supersymmetry cannot cancel the leading positive contributions if the bosonic amplitudes remain unchanged. Limitations include leading-order perturbation theory, an assumed extensive continuation beyond the finite-box expansion, and the unbounded cubic potential. Gauge/gravitational couplings, higher-dimensional compactifications, de Sitter dynamics and ultraviolet completions are not calculated.