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Postępy Fizyki

Postawić kwantową teorię pola z głowy na nogi część 1

Piotr Chankowski, Instytut Fizyki Teoretycznej,Wydział Fizyki UW

Published Sep 20, 2023• 1 citation
Publisher

Abstract

Despite the impression that can be gained from most of the standard textbooks, Dirac, Klein–Gordon and other wave equations do not constitute the basis of relativistic quantum field theory. In this article I attempt to show how it should be formulated properly and discuss some of its aspects which usually are presented unsatisfactorily. My aim is to cause the change in the way quantum field theory is taught.The text is split into three parts. In the first one I briefly recall the quantum field theory historical development and present its formulation as a quantum theory of interacting particles (relativistic or nonrelativistic).

DOI
10.61947/uw.pf.2023.74.3.5-14

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Despite the impression that can be gained from most of the standard textbooks, Dirac, Klein–Gordon and other wave equations do not constitute the basis of relativistic quantum field theory. In this article I attempt to show how it should be formulated properly and discuss some of its aspects which usually are presented unsatisfactorily. My aim is to cause the change in the way quantum field theory is taught.The text is split into three parts. In the first one I briefly recall the quantum field theory historical development and present its formulation as a quantum theory of interacting particles (relativistic or nonrelativistic).
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincaré group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincaré group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.
Quantum field theory is the study of quantum systems with an infinite number of degrees of freedom. The simplest quantum field theories are the free bose fields, which are essentially assemblies of an infinite number of coupled oscillators. Examples include the quantized electromagnetic field, lattice vibrations in solid state physics, and the Klein-Gordon field, some of which are relativistic while others are not. Particles show up in quantum field theory as “field quanta.” As we will see, these do not quite have all the properties of the usual particles of Newtonian physics or of Schrödinger quantum mechanics. I will in particular show (Section 3) that the quanta of free bose fields, relativistic or not, can not be perfectly localized in a bounded subset of space. This, in my opinion, shows conclusively that the difficulties encountered when attempting to define a position observable for the field quanta of relativistic fields, that continue to be the source of regular debate in the literature [20] [21] [22] [3] [10] [15] [18] [19] [48] [47], do not find their origin in any form of causality violation, as seems to be generally thought. Instead, they result from an understandable but ill-fated attempt to force too stringent a particle interpretation on the states of the quantum field containing a finite number of quanta.