Is quantum field theory deterministic or not

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Julian Schwinger, Quantum Mechanics

Book review written for

The author is known: He is one of the co-founders of modern quantum electrodynamics (1948-1950). He lived from 1918-1994, was born in NY and was in some ways an antipode of Richard Feynman in his physical thinking and work.

He was very formal and never used Feynman diagrams. However, he was also very fond of practically relevant calculations and was the first to calculate the radiation correction to the magnetic moment of the spin 1/2 particle (electron), i.e. the one-loop three-vertex in the minimum QED. He was involved in the development of the general renormalization theory and, independently of Tomonaga, developed the concept of "time-slicing", i.e. the introduction of a scalar time parameter, into the theory of relativity. Together with Tomonaga and Feynman he received the Nobel Prize for the founding of the modern relativistic quantum field theory (if I am not mistaken in the date, it was the Nobel Prize in 1965).

The present book has been edited very carefully by the editor and is a summary of several scripts written by Schwinger himself as an introduction to quantum theory. It is therefore for the basic lecture in the theory course (at most German universities this should be in the 5th semester, in Darmstadt, where I studied, there was already an introductory course in the 4th semester).

The formal content itself largely follows the classic scheme of such a course, albeit with some differences in the presentation and in the physical understanding of the formalism of quantum theory. As far as my lack of English allows, I can say that it is first and foremost a very carefully thought-out language that you can tell that every sentence is well-formulated without it appearing even remotely cumbersome.

Let's go through the contents briefly, as far as I could grasp on first reading:

The book begins with a lengthy prologue that does not contain a single formula. I don't usually read something like that, but this preface is an exception. It is a concise summary of Schwinger's view of the "interpretation" of the quantum theoretical formalism, which is already summarized in the subtitle of the book "Symbolism of Atomic Measurements".

He explains exactly what a measurement means in an atomistic world view of physics, namely the inevitability of disturbances of the system due to the measurement, which, as is well known, leads to the famous conclusion that not all possible observables of a quantum system can be assigned sharp measured values ​​simultaneously. If one measures an observable very precisely, the values ​​of other observables are all the less determined, and the experimental setup essentially determines which aspect of the system is currently being recorded.

He immediately emphasizes that quantum theory is completely causal, i.e. that if the initial state of the system is exactly given and if the forces acting on it are known, the state of the system is exactly known at all times, as in classical mechanics. The theory, however, is not deterministic because precisely because of the unavoidable, not negligible, but also unpredictable effect of the measuring apparatus on the object to be measured, not all observables can have sharp values ​​at the same time. According to Schwinger, quantum theory is nothing more than the formally idealized description of measurements on quantum systems: “This symbolization of atomic measurements is quantum mechanics, developed by Heisenberg, Born, Schrödinger and others, essentially in the years 1925 to 1927, still very distant from our present point of view. ''

The actual quantum theory course begins with a mathematical analysis of the presented concept of the measurement process. This chapter is, so to speak, "pre-quantum theory", because the Hilbert space is by no means simply postulated and then a posteriori proof is given that the Hilbert space formalism actually describes the phenomena exactly. Rather, starting from the principle of measuring a quantum system detailed in the prologue, the Bra-Ket formalism (and ultimately of course the Hilbert space formalism) is developed and illustrated using Stern-Gerlach experiments (SGE). Indeed, quantum theory as we all know it comes into being as the "Symbolism of Atomic Measurements". For me it is the fulfillment of the didactic dream to justify the quantum theory without heuristic aids from the so-called "wave mechanics" or any correspondence principles to classical physics, which, as is well known, lead to the "Copenhagen philosophy of confusion" (Einstein).

First, we work properly with discrete limited space grids with periodic boundary conditions (so that position and momentum are initially discrete variables) and later the continuum limit is carried out in a mathematically correct manner. Without unnecessary mathematical ballast, the non-relativistic quantum theory, to which the textbook is limited, is carried out mathematically clean, in particular the separability of the Hilbert space over states of stationary uncertainty (eigenstates of the harmonic oscillator). The first “quarter”, called “Quantum Kinematics”, concludes with a detailed analysis of the angular momentum. Thanks to the famous Schwinger oscillator model, which is derived from the well-known treatment of angular momentum algebra (Lie algebra of SO (3)), the otherwise difficult-looking calculations (e.g. for Clebsch-Gordan coefficients) are elegant and simple. The crowning glory is the complete explanation of the Galileo invariance and the justification of the "Hamilton Operator for a system of elementary particles".

The second quarter (the book is structured according to a lecture that apparently comprised 3 quarters at Schwinger) is headed "Quantum Dynamics" and named in a "Quantum Action Principle" (of course it is Schwinger's quantum effect principle, only was Schwinger so modestly not to call it that) formulated. This makes the various "images" of quantum theory easy to understand.

This is followed by some elementary examples, which, although they are the well-known problems (free particle, constant force, harmonic oscillator, WKB method, Rayleigh-Ritz's principle of variation), are presented in an exciting way.

The harmonic oscillator is also dealt with a lot further than one is used to, in particular, explicitly time-dependent driven oscillators are considered. The chapter closes with a breathtakingly beautiful treatment of the hydrogen problem (traced back to the 2-dimensional isotropic oscillator in polar coordinates). Then it is also converted into parabolic coordinates and the classical perturbation theory of the Stark effect and the Zeeman effect are discussed. The chapter closes with an exact treatment of the Rutherford scattering.

The third quarter deals with "Interacting Particles", that is, many-particle systems (starting with the two-particle problem, of course). Identical particles are introduced with the help of creation and annihilation operators, and many electron atoms are treated in the most important approximations (Hartree-Fock, Thomas-Fermi).

The work closes with a complete treatment of the so-called "non-relativistic" QED, i.e. the treatment of the radiation from atoms, in which the electrons may be treated non-relativistically, up to and including the Lambshift.

The book is rounded off by a comprehensive collection of 351 exercises. However, following the American tradition, there are no suggestions for a solution.

It is to be hoped that this book will find the most widespread use in university teaching. However, it is also easily suitable for self-study, but requires some previous mathematical knowledge. However, anyone who enjoys a clear modern introduction to quantum theory that avoids wrong turns will be well served with the book. One would have a continuation in a second volume, in which the relativistic quantum field theory is then presented in a didactic way. It would be interesting from a didactic perspective alone, because there would definitely be no Feynman diagrams. But for that, Steven Weinberg's three volumes "Quantum Theory of Fields" are already available.

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