# Entanglement entropy calculation

I have been trying to understand this paper:

Susskind. “Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory.” [1402.1128] Long Short-Term Memory Based Recurrent Neural Network Architectures for Large Vocabulary Speech Recognition, 14 July 1994, arxiv.org/abs/hep-th/9401070

The calculation is quite straightforward using the replica trick. However,there is one bit that is confusing me.

In Section 2, they calculate the entropy contribution from a free field by computing the thermal partition function. Since it is a free field, the partition function is a product over different modes, and hence upon taking a logarithm to find the free energy, it becomes a sum over different modes.

However, in Section 3, they claim more generally that "The functional integral can be represented in terms of first quantized particle paths, according to a standard prescription. In higher orders of perturbation theory, the paths branch to form Feynman diagrams." I would like to understand how precise this is. How does one go from the second quantized partition function to the first quantized sum over paths? In momentum space, I see that the thermal partition function computes transition elements for each mode, and hence the sum. However, in position space, the sum should now be over all possible paths with any number of loops. When one takes the logarithm, does this give a sum over single particle paths only?

• After looking for some references, I found "Path integral approach to the heat kernel-Fiorenzo Bastianelli" which discusses how using the heat kernel trick, the effective action of a free scalar field can be written as a sum over first quantized paths. – Pratik Rath Jul 29 '18 at 15:57