Abstract
Quantum field theory defined on a noncommutative space is a useful toy model of quantum gravity and is known to have several intriguing properties, such as nonlocality and UV/IR mixing. They suggest novel types of correlation among the degrees of freedom of different energy scales. In this paper, we investigate such correlations by the use of entanglement entropy in the momentum space. We explicitly evaluate the entanglement entropy of scalar field theory on a fuzzy sphere and find that it exhibits different behaviors from that on the usual continuous sphere. We argue that these differences would originate in different characteristics; non-planar contributions and matrix regularizations. It is also found that the mutual information between the low and the high momentum modes shows different scaling behaviors when the effect of a cutoff becomes important.
Highlights
How to formulate quantum geometries properly, the NC geometry is believed to be a good toy model of it
There still arises an important distinction between QFT on the usual and the fuzzy sphere; the contribution to a two-point function from planar and non-planar Feynman diagrams are different in the fuzzy sphere case and this difference remains after we take a continuum limit
We focus on the difference in the behavior of entanglement entropy (EE) between on the NC space and on the commutative counterpart with respect to an energy scale
Summary
EE in the momentum space is evaluated based on conventional perturbation theory of quantum mechanics in [3]. Theory, when the perturbation λHint is considered, the ground state will be a superposition of all energy eigenstates of H0 as. The EE is defined as the summation of F (l1, m1; l2, m2; l3, m3) with suitable choices of (li, mi); for example, l1, l2 ≤ x and l3 > x and so on These summations are summarized and mi summation part can be performed (see appendix B.2 for details), and the leading order contribution of the EE is expressed as λ2 ln(λ2) 16R2. F is m1,m2,m3 F . f (l1, l2, l3) is symmetric under the exchange of any two of li, but f(l1; l2) is not symmetric under l1 ↔ l2. fterms appear when two of (li, mi) coincide
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