This book introduces mathematicians, physicists, and philosophers to a new, coherent approach to the theory and interpretation of quantum physics, in which classical and quantum thinking live peacefully side by side and jointly fertilize the intuition. From a mathematical point of view, the focus on the similarities between quantum mechanics and classical mechanics leads to the new framework of coherent spaces. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, see also

https://www.physicsoverflow.org/42177/foundations-of-quantum-physics-v-coherent-foundations

https://www.physicsoverflow.org/41115/introduction-to-coherent-spaces

https://www.physicsoverflow.org/41116/introduction-to-coherent-quantization.

To relate this mathematical formalism to the actual informal practice of using quantum mechanics in the scientific culture is the purpose of the still unsettled subject of quantum interpretation. In this book, the thermal interpretation is presented. For further reading see

https://www.physicsoverflow.org/41989/foundations-of-quantum-physics-i-critique-of-the-tradition

https://www.physicsoverflow.org/41990/foundations-of-quantum-physics-the-thermal-interpretation

https://www.physicsoverflow.org/41991/foundations-of-quantum-physics-iii-measurement

https://www.physicsoverflow.org/42149/foundations-quantum-physics-more-thermal-interpretation

The book is not meant to be a textbook on quantum mechanics, further technical details are given in the references.

In the first part of the book, the mathematical concepts needed for quantum physics are presented. The axiomatic core is chosen such as to emphasize the similarity of quantum mechanics and classical mechanics, to give the most direct access to statistical mechanics, and to avoid any allusions to measurement. The notion of uncertainty is introduced, leading to the concept of probability and the weak law of large numbers which is important for taking the classical limit. Classical probability theory is presented in terms of Whitle’s approach using expectations. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined. Like ordinary averages, q-expectations become more accurate by averaging over many similar items. Averaging over macroscopic spacetime regions produces macroscopic quantities with negligible uncertainty, and leads to classical physics. The Ehrenfest picture is applied to obtain classical equations of motion for the q-expectations. Euclidean spaces, needed to rigorously formulate practical quantum physics and to prepare the introduction of coherent spaces, are discussed in detail. Coherent spaces themselves are then gently introduced as a nonlinear version of the notion of a complex Euclidean space. The focus is set on symmetries which are the transformations that can be quantized. Coherent quantum physics is introduces as physics in terms of a coherent space. The kinematical structure of quantum physics and the meaning of the fundamental quantum observables are given by the symmetries of this coherent space, their infinitesimal generators, and associated operators on the quantum space of the coherent space. To upgrade to quantum field theory, one has to consider in place of position and momentum operators for finitely many particles in quantum mechanics an infinite number of operators for fields. The basic quantities in quantum field theory as usually considered in textbooks addressed ad physicists are smeared fields.

The second part about the interpretation of quantum mechanics starts with the request that good foundations of a theory have to be independent of the notion of measurement. Callens criterion, which relates the formal properties of a system to its measured quantities, is introduced. In the thermal interpretation, objective properties are q-expectations of products of quantum fields and what is computable from these. States of a system are then characterized by such q-expectations and their appropriately normally ordered or time-ordered products, some of which can be approximately measured. The basic dynamics for the q-expectations, given by the Ehrenfest equations, has the structure of deterministic Hamiltonian mechanics. The dynamics of the whole universe is given by the Ehrenfest equations for q-expectations too, and defines the dynamics of every physical subsystem by restriction. All physical systems are treated as subsystems of the universe. Measurement outcomes are functions of q-expectations of the detector, hence are deterministically predicted by the state of the universe. Some standard quantum mechanics experiments, such as Stern-Gerlach and entanglement-experiments, are re-explained in the light of the thermal interpretation.

A main point in the critique of the tradition as issued in the last part of the book, is that Born’s rule only holds for idealized measurements. These thought experiments don’t include real-world experimental issues that occur in the laboratory. They are usually taken as the starting point for introducing quantum mechanics. In this book, the merits of even talking about such idealized measurements, is questioned. The concepts of states and ensembles are critically reviewed. Among other things, it is shown that if the state of every composite quantum system contains all information that can be known about a system, states cannot be pure states. Some of the traditional interpretations are reconsidered from the perspective of the thermal interpretation.

Part I of the book contains a nice exposition of partly new and rigorous mathematical concepts for quantum mechanics not usually found in other texts addressed at physicists. In particular, the coherent spaces provide an interesting new mathematical framework for doing quantum physics. If one is satisfied with the tradition, this is all that is needed. Indeed, if one clearly separates theoretical and experimental quantum physics as two different subfields, there is no problem with the conventional teaching of and thinking about quantum mechanics. However, the issues presented here arise if one stretches the conventional microscopic domain of most importance of quantum mechanics to encompass its macroscopic impacts and applications usually described by statistical mechanics, thermodynamics, up to everyday observable mechanical issues, astrophysics, and even cosmological issues concerning the whole universe. Theoretical physicists usually don’t think in terms of such an IOE (interpretation of everything as the thermal interpretation aspires to be), but describe the range of physical phenomena from the smallest to the largest scales by a succession of effective theories with their own domains of validity, which are (at least in principle) linked by coarse graining or renormalization. Another rather controversial discussion is concerned with the existence of particles and the “resolution” of the particle-wave duality in favor of the field point of view. Particles are considered to emergence only as asymptotic entities from the more fundamental quantum field point of view. For well known experiments, such as the photoelectric effect and particle tracks, the existence of particles is questioned. By applying the thermal interpretation, which deals with q-expectation as basic properties and rejecting any scientific value of single measurements, the conventional notion that quantum mechanics predicts just probabilities (for single measurements) as its defining property is contradicted. Instead, any probabilistic behavior of quantum mechanics is explained by tracking only a number of the relevant variables of a quantum system, which introduces a chaotic or probabilistic behavior. The discreteness of measurement results in quantum mechanics is also ascribed rather to the discrete response of the detector instead of the quantum system under investigation.

Nevertheless, the book is an immensely rich and detailed source of interesting and unusual considerations not found elsewhere. In particular, detailed explanations of experimental practices and concerns as they appear in the laboratory are given throughout the book.