This paper is written as a staged debate between two seasoned theorists, called Alice and Bob, about how to recognize a Bose–Einstein condensate. Alice defends the long‑standing Onsager–Penrose (OP) criterion. Bob explains a newer idea called the proper condensate (PC) notion, which uses a different mathematical probe of particle number. The aim is to compare the two viewpoints and to clarify what each one can — and cannot — detect.

The Onsager–Penrose criterion looks at how the average number of particles in a given single‑particle wavefunction f grows when the total particle number n increases. If this average grows proportional to n for some f, one says there is a macroscopic condensate in that mode. In plain terms: if a fixed fraction κ of all particles sits in the same wavefunction as the system grows, OP calls that a condensate and κ is its fraction.

The proper condensate idea instead studies the resolvents of the number operator. For a chosen wavefunction f and a small positive µ, one looks at the inverse (µ + N(f))−1, where N(f) counts particles in f. In sequences of n‑particle states that contain a PC, the expectation value of these resolvents scaled appropriately (for example n( n + N(f) )−1 ) tends to zero for large n. One then identifies the space of f for which the limit does not vanish; the orthogonal complement of that space gives the condensate wavefunction(s). These resolvents behave, in the limit, like sharp yes/no projectors, so the method gives a crisp mathematical separation between condensate and non‑condensate parts.

Bob shows a quantitative link between the two criteria: using inequalities such as Cauchy–Schwarz, small values of the scaled resolvents imply growth of the ordinary particle number and hence satisfy the OP condition for some positive κ. In other words, under the conditions discussed, the PC test can imply the OP signal. But the debate points out that the PC criterion can also be finer or simply different. Bob presents an explicit model of non‑interacting bosons in a container with soft boundaries. There the ground state is concentrated in a fixed ball and excited states form a countable set. By selecting sequences where many particles occupy distinct excited states, expectations of N(f) can grow only like n1/2 for some choices of f — too slowly for OP to call this a condensate — while the PC test still identifies a condensate because the resolvent limits behave differently.