# Identities and Many-Body Approaches in Bose-Einstein Condensates

## TL;DR

This paper reviews exact relations in Bose-Einstein condensates and discusses many-body approximation methods that satisfy these relations, bridging theoretical exact identities and practical approximation approaches.

## Contribution

It introduces key exact relations in BECs and discusses approximation methods that preserve these relations, connecting fundamental theory with practical modeling.

## Key findings

- Identifies key exact relations like Hugenholtz--Pines and Nepomnyashchii identities.
- Reviews many-body approximation methods that satisfy these exact relations.
- Serves as a bridge between theoretical identities and approximation techniques.

## Abstract

This paper discusses exact relations in Bose--Einstein condensates (BECs), starting from basic properties of an ideal Bose gas. In particular, focused on are the Hugenholtz--Pines relation, Nepomnyashchii--Nepomnyashchii identity, and identities for the density response function. After introducing these exact relations, a few approaches of many-body approximations are discussed, which satisfy the exact relations in BECs. This paper will serve as a bridge between theories on exact relations and those on approximations in BECs.