Large N vector models in the Hamiltonian framework

We present a fluctuating $N$ formalism, based on second-quantization, to describe large $N$ vector models from field theory using Hamiltonian methods. We first present the method in the simpler setting of a quantum mechanical system with quartic interactions, and then apply these techniques to the $O(N)$ model in $2+1$ and $3+1$ dimensions. We recover various known results, such as the gap equation determining the ground state of the system, the presence of bound states at negative coupling and the leading order contribution to critical exponents, and provide an interpretation of the large $N$ path integral saddle point as a Bose-Einstein condensate of extended objects in the presence of a non-local interaction. In the large $N$ limit, this formalism leads naturally to a description of elementary $O(N)$ symmetric excitations in terms of bilocal fields, which are at the basis of $\text{AdS}_4/\text{CFT}_3$ studies of the $O(N)$ model and Vasiliev gravity.