 The algebraic formulations of quantum mechanics that were developed by von Neumann and Segal did not change the way that quantum mechanics was done. Nevertheless, they did have a substantial impact in two related contexts: QFT and quantum statistical mechanics. The key difference leading to the impact has to do with the domain of applicability. The domain of quantum mechanics consists of finite quantum systems, meaning quantum systems that have a finite number of degrees of freedom. Whereas in QFT and quantum statistical mechanics, the systems of special interest – i.e., quantum fields and particle systems in the thermodynamic limit, respectively – are infinite quantum systems, meaning quantum systems that have an infinite number of degrees of freedom. Dirac (1927) was the first to recognize the importance of infinite quantum systems for QFT, which is reprinted in Schwinger (1958).