Armin Wachter

Relativistic Quantum Mechanics

Relativistic Quantum Mechanics Springer Netherlands
ISBN: 978-90-481-3644-5

1. edition: April 2010
Hard cover, 390 pages


Which problems do arise within relativistic enhancements of the Schrödinger theory, especially if one adheres to the usual one-particle interpretation, and to what extent can these problems be overcome? And what is the physical necessity of quantum field theories?

In many books, answers to these fundamental questions are given highly insufficiently by treating the relativistic quantum mechanical one-particle concept very superficially and instead introducing field quantization as soon as possible. By contrast, this monograph emphasizes relativistic quantum mechanics in the "narrow sense": it extensively discusses relativistic one-particle concepts and reveals their problems and limitations, therefore motivating the necessity of quantized fields in a physically comprehensible way.

The first chapters contain a detailed presentation and comparison of the Klein-Gordon and Dirac theories, always in view of the non-relativistic theory. In the third chapter, we consider relativistic scattering processes and develop the Feynman rules from propagator techniques. This is where the impossibility to get around a quantum field theoretical reasoning is discussed and basic quantum field theoretical concepts are introduced. This book addresses undergraduate and graduate physics students who are interested in a clearly arranged and structured presentation of relativistic quantum mechanics in the "narrow sense" and its connection to quantum field theories. Each section contains a short summary and exercises with solutions. A mathematical appendix rounds up this excellent introductory book on relativistic quantum mechanics.


It is more important to repair errors than to prevent them. This is the quintessence of the philosophy of human cognition known as critical rationalism which is perhaps at its most dominant in modern natural sciences. According to it insights are gained through a series of presumptions and refutations, through preliminary solutions that are continuously, rigorously, and thoroughly tested. Here it is of vital importance that insights are never verifiable but, at most, falsifiable. In other words: a natural scientific theory can at most be regarded as "not being demonstrably false" until it can be proven wrong. By contrast, a sufficient criterion to prove its correctness does not exist.

Newtonian mechanics, for example, could be regarded as "not being demonstrably false" until experiments with the velocity of light were performed at the end of the 19th century that were contradictory to the predictions of Newton's theory. Since, so far, Albert Einstein's theory of special relativity does not contradict physical reality (and this theory being simple in terms of its underlying assumptions), relativistic mechanics is nowadays regarded as the legitimate successor of Newtonian mechanics. This does not mean that Newton's mechanics has to be abandoned. It has merely lost its fundamental character as its range of validity is demonstrably restricted to the domain of small velocities compared to that of light.

In the first decade of the 20th century the range of validity of Newtonian mechanics was also restricted with regard to the size of the physical objects being described. At this time, experiments were carried out showing that the behavior of microscopic objects such as atoms and molecules is totally different from the predictions of Newton's theory. The theory more capable of describing these new phenomena is nonrelativistic quantum mechanics and was developed in the subsequent decade. However, already at the time of its formulation, it was clear that the validity of this theory is also restricted as it does not respect the principles of special relativity.

Today, about one hundred years after the advent of nonrelativistic quantum mechanics, it is quantum field theories that are regarded as "not being demonstrably false" for the description of microscopic natural phenomena. They are characterized by the facts that
  • they can be Lorentz-covariantly formulated, thus being in agreement with special relativity,
  • they are many-particle theories with infinitely many degrees of freedom and account very precisely for particle creation and annihilation processes.
Naturally, the way toward these modern theories proceeded through some intermediate steps. One began with nonrelativistic quantum mechanics – in conjunction with its one-particle interpretation – and tried to extend this theory in such a way that it becomes Lorentz-covariant. This initially led to the Klein-Gordon equation as a relativistic description of spin-0 particles. However, this equation contains a basic flaw because it leads to solutions with negative energy. Apart from the fact that they seem to have no reasonable interpretation, their existence implies quantum mechanically that stable atoms are not possible as an atomic electron would fall deeper and deeper within the unbounded negative energy spectrum via continuous radiative transitions. Another problem of this equation is the absence of a positive definite probability density which is of fundamental importance for the usual quantum mechanical statistical interpretation. These obstacles are the reason that for a long time, the Klein-Gordon equation was not believed to be physically meaningful.

In his efforts to adhere to a positive definite probability density, Dirac developed an equation for the description of electrons (more generally: spin-1/2 particles) which, however, also yields solutions with negative energy. Due to the very good accordance of Dirac's predictions with experimental results in the low energy regime where negative energy solutions can be ignored (e.g. energy spectrum of the hydrogen atom or gyromagnetic ratio of the electron), it was hardly possible to negate the physical meaning of this theory completely.

In order to prevent electrons from falling into negative energy states, Dirac introduced a trick, the so-called hole theory. It claims that the vacuum consists of a completely occupied "sea" of electrons with negative energy which, due to Pauli's exclusion principle, cannot be filled further by a particle. Additionally, this novel assumption allows for an (at least qualitatively acceptable) explanation of processes with changing particle numbers. According to this, an electron with negative energy can absorb radiation, thus being excited into an observable state of positive energy. In addition, this electron leaves a hole in the sea of negative energies indicating the absence of an electron with negative energy. An observer relative to the vacuum interprets this as the presence of a particle with an opposite charge and opposite (i.e. positive) energy. Obviously, this process of pair creation implies that, besides the electron, there must exist another particle which distinguishes itself from the electron just by its charge. This particle, the so-called positron, was indeed found a short time later and provided an impressive confirmation of Dirac's ideas. Today it is well-known that for each particle there exists an antiparticle with opposite (not necessarily electric) charge quantum numbers.

The problem of the absence of a positive definite probability density could finally be circumvented in the Klein-Gordon theory by interpreting the quantities ρ and j as charge density and charge current density (charge interpretation). However, in this case, the transition from positive into negative energy states could not be eliminated in terms of the hole theory, since Pauli's exclusion principle does not apply here and, therefore, a completely filled sea of spin-0 particles with negative energy cannot exist.

The Klein-Gordon as well as the Dirac theory provides experimentally verifiable predictions as long as they are restricted to low energy phenomena where particle creation and annihilation processes do not play any role. However, as soon as one attempts to include high energy processes both theories exhibit deficiencies and contradictions. Today the most successful resort is – due to the absence of contradictions with experimental results – the transition to quantized fields, i.e. to quantum field theories.

This book picks out a certain piece of the cognitive process just described and deals with the theories of Klein, Gordon, and Dirac for the relativistic description of massive, electromagnetically interacting spin-0 and spin-1/2 particles excluding quantum field theoretical aspects as far as possible (relativistic quantum mechanics in the "narrow sense"). Here the focus is on answering the following questions:
  • How far can the concepts of nonrelativistic quantum mechanics be applied to relativistic quantum theories?
  • Where are the limits of a relativistic one-particle interpretation?
  • What similarities and differences exist between the Klein-Gordon and Dirac theories?
  • How can relativistic scattering processes, particularly those with pair creation and annihilation effects, be described using the Klein-Gordon and Dirac theories without resorting to the formalism of quantum field theory and where are the limits of this approach?
Unlike many books where the "pure theories" of Klein, Gordon, and Dirac are treated very quickly in favor of an early introduction of field quantization, the book in hand emphasizes this particular viewpoint in order to convey a deeper understanding of the accompanying problems and thus to explicate the necessity of quantum field theories.

This textbook is aimed at students of physics who are interested in a concisely structured presentation of relativistic quantum mechanics in the "narrow sense" and its separation from quantum field theory. With an emphasis on comprehensibility and physical classification, this book ranges on a middle mathematical level and can be read by anybody who has attended theoretical courses of classical mechanics, classical electrodynamics, and nonrelativistic quantum mechanics.

This book is divided into three chapters and an appendix. The first chapter presents the Klein-Gordon theory for the relativistic description of spin-0 particles. As mentioned above, the focus lies on the possibilities and limits of its one-particle interpretation in the usual nonrelativistic quantum mechanical sense. Additionally, extensive symmetry considerations of the Klein-Gordon theory are made, its nonrelativistic approximation is developed systematically in powers of v/c, and, finally, some simple one-particle systems are discussed.

In the second chapter we consider the Dirac theory for the relativistic description of spin-1/2 particles where, again, emphasis is on its one-particle interpretation. Both theories, emanating from certain enhancements of nonrelativistic quantum mechanics, allow for a very direct one-to-one comparison of their properties. This is reflected in the way that the individual sections of this chapter are structured like those of the first chapter – of course, apart from Dirac-specific issues, e.g. the hole theory or spin that are considered separately.

The third chapter covers the description of relativistic scattering processes within the framework of the Dirac and, later on, Klein-Gordon theory. In analogy to nonrelativistic quantum mechanics, relativistic propagator techniques are developed and considered together with the well-known concepts of scattering amplitudes and cross sections. In this way, a scattering formalism is created which enables one-particle scatterings in the presence of electromagnetic background fields as well as two-particle scatterings to be described approximately. Considering concrete scattering processes to lowest orders, the Feynman rules are developed putting all necessary calculations onto a common ground and formalizing them graphically. However, it is to be emphasized that these rules do not, in general, follow naturally from our scattering formalism. Rather, to higher orders they contain solely quantum field theoretical aspects. It is exactly here where this book goes for the first time beyond relativistic quantum mechanics "in the narrow sense". The subsequent discussion of quantum field theoretical corrections (admittedly without their deeper explanation) along with their excellent agreement with experimental results may perhaps provide the strongest motivation in this book to consider quantum field theories as the theoretical fundament of the Feynman rules.

Important equations and relationships are summarized in boxes to allow the reader a well-structured understanding and easy reference. Furthermore, after each section there are a short summary as well as some exercises for checking the understanding of the subject matter. The appendix contains a short compilation of important formulae and concepts.

Finally, we hope that this book helps to bridge over the gap between nonrelativistic quantum mechanics and modern quantum field theories, and explains comprehensibly the necessity for quantized fields by exposing relativistic quantum mechanics "in the narrow sense".

Cologne, April 2010
Armin Wachter