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Cambridge University Press eBooks

Miscellany and Addenda to Part I

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter considers the treatment of a few topics, which are relevant to the general purposes of the book, but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Two topics are explicitly considered, which are relevant to a useful partition of the Dyson equation and to the Keldysh formalism.

DOI
10.1017/9781009411509.021

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This chapter considers the treatment of a few topics, which are relevant to the general purposes of the book, but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Two topics are explicitly considered, which are relevant to a useful partition of the Dyson equation and to the Keldysh formalism.
The content of the present book is the result of a work developed over a period of more than fifteen years, to which many students and collaborators, as well as renowned philosophers and physicists, have contributed in various aspects, as the references testify. The aim of this work has been to present the MHI in a systematic and organic way, incorporating recent discussions and making explicit its underlying motivations. I have tried to show that the proposal does not address isolated interpretative topics, but provides a broad, articulated, and consistent view to make sense of the quantum world. Throughout the chapters, I have argued in favor of the reasonableness and fruitfulness of the proposal. Of course, the reasons put forward are not directly empirical, but this does not make them any less sound. The main arguments presented in the text can be summarized in the following points: Three more general aspects can be added to these advantages:
The topics of this book are the mathematical foundations of non-relativistic quantum mechanics and the mathematical theory they require. The main characteristic of the book is that the mathematics is developed assuming familiarity with elementary analysis only. Moreover, all the proofs are carried out in detail. These features make the book easily accessible to readers with only the mathematical training offered by undergraduate education in mathematics or in physics, and also ideal for individual study. The principles of quantum mechanics are discussed with complete mathematical accuracy and an effort is made to always trace them back to the experimental reality that lies at their root. The treatment of quantum mechanics is axiomatic, with definitions followed by propositions proved in a mathematical fashion. No previous knowledge of quantum mechanics is required. This book is designed so that parts of it can be easily used for various courses in mathematics and mathematical physics, as suggested in the Preface. The book is of interest to researchers and graduate students in functional analysis, who can see how closely an important part of their chosen field is linked with quantum mechanics, and also to physicists, who can see how the abstract language of functional analysis brings unity to the apparently distinct approaches employed in quantum theory.
Similar to Chapter 20 of Part I and Chapter 31 of Part II, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book, but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it addresses a schematic derivation of the Lindblad Master equation (aimed at helping the reader in retracing and better identifying the essential steps made and approximations adopted in the more general derivation presented in Chapter 35), as well as the physical assumptions underlying the original Kadanoff–Baym ansatz.
In this chapter, we take a deeper look at a number of interesting and important models that contain one or two particles. We focus on examples, from a variety of settings and fields, with the goal of not only finding relevant physical phenomena occurring in these systems, but also giving the reader a place where they can actually practise much of what has been said previously throughout the book. In some cases, straight application of the methods explained earlier suffices, while in some others slightly different (yet related) methods are needed. Most of the models studied in this chapter are justified in chapters 4 and 5, sometimes in full. Other times, the justification from chapters 4 and 5 may be only partial, and the interested reader is encouraged to consult more specialized literature in the subject.
Similar to Chapter 20 of Part I, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t ₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.