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.
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.
The nature and properties of the vacuum as well as the meaning and localization properties of one or many particle states have attracted a fair amount of attention and stirred up sometimes heated debate in relativistic quantum field theory over the years. I will review some of the literature on the subject and will then show that these issues arise just as well in non-relativistic theories of extended systems, such as free bose fields. I will argue they should as such not have given rise either to surprise or to controversy. They are in fact the result of the misinterpretation of the vacuum as ``empty space'' and of a too stringent interpretation of field quanta as point particles. I will in particular present a generalization of an apparently little known theorem of Knight on the non-localizability of field quanta, Licht's characterization of localized excitations of the vacuum, and explain how the physical consequences of the Reeh-Schlieder theorem on the cyclicity and separability of the vacuum for local observables are already perfectly familiar from non-relativistic systems of coupled oscillators.
Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Additionally, relativistic quantum mechanics is the basis for the standard model of elementary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels. However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to require the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent profound work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of recent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.
The rampant success of quantum theory is the result of applications of the 'new' quantum \nmechanics of Schrödinger and Heisenberg (1926-7), the Feynman-Schwinger-Tomonaga Quantum \nElectrodynamics (1946-51), the electro-weak theory of Salaam, Weinberg, and Glashow (1967-9), \nand Quantum Chromodynamics (1973-); in fact, this success of `the' quantum theory has depended \non a continuous stream of brilliant and quite disparate mathematical formulations. In this carefully \nconcealed ferment there lie plenty of unresolved difficulties, simply because in churning out fabulously \naccurate calculational tools there has been no sensible explanation of all that is going on. It is even \nargued that such an understanding is nothing to do with physics. A long-standing and famous \nillustration of this is the paradoxical thought-experiment of Einstein, Podolsky and Rosen (1935). \nFundamental to all quantum theories, and also their paradoxes, is the location of sub-microscopic \nobjects; or, rather, that the specification of such a location is fraught with mathematical inconsistency. \nThis project encompasses a detailed, critical survey of the tangled history of Position \nwithin quantum theories. The first step is to show that, contrary to appearances, canonical quantum \nmechanics has only a vague notion of locality. After analysing a number of previous attempts at a \n`relativistic quantum mechanics', two lines of thought are considered in detail. The first is the work \nof Wan and students, which is shown to be no real improvement on the usual `nonrelativistic' theory. \nThe second is based on an idea of Dirac's - using backwards-in-time light-cones as the hypersurface \nin space-time. There remain considerable difficulties in the way of producing a consistent scheme here. \nTo keep things nicely stirred up, the author then proposes his own approach - an adaptation \nof Feynman's QED propagators. \nThis new approach is distinguished from Feynman's since the propagator or Green's function \nis not obtained by Feynman's rule. The type of equation solved is also different: instead of an \ninitial-value problem, a solution that obeys a time-symmetric causality criterion is found for an \ninhomogeneous partial differential equation with homogeneous boundary conditions. \nTo make the consideration of locality more precise, some results of Fourier transform theory are \npresented in a form that is directly applicable. \nSomewhat away from the main thrust of the thesis, there is also an attempt to explain the \nmanner in which quantum effects disappear as the number of particles increases in such things as \nexperimental realisations of the EPR and de Broglie thought experiments.
Problems with the extension to relativity. The Klein–Gordon equation The attempt to extend the causal interpretation to cover relativistic quantum mechanics raises several problems of technique and principle. These relate not only to difficulties peculiar to the quantum theory of motion but have to do with still unresolved issues in the underlying quantum mechanical formalism (such as the appearance of infinities in the computation of physical quantities and the lack of a relativistic theory of measurement). Some problems are common to both, such as the difficulty of defining positive densities. The subject has not been extensively investigated and in this chapter we shall discuss only some of the relevant issues. There are two particularly important, overlapping problems: Can the trajectory concept be retained in the relativistic quantum domain? And: how do we interpret relativistic quantum fields? The unresolved issue underlying both questions is the extent to which we should require a quantum theory to be ‘relativistic’. It is known that Lorentz covariant wave equations can have consequences in conflict with relativity. This, as we shall see, is indeed the case with the current relativistic formalism as interpreted, or completed, by the quantum theory of motion. Especially in field theory, the individual processes that build up and explain the covariant statistical predictions of the formalism exhibit features that are in conflict with our entrenched ideas regarding covariance and locality as necessary concomitants of relativity theory. That is, relativity is statistically valid but the individual events do not have an intrinsically relativistic character.
Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.
Abstract It is a widespread belief, at least within the physics community, that there is no relativistic quantum theory of (localizable) particles; and, thus, that the only relativistic quantum theory is a theory of fields. This belief has received much support in recent years in the form of rigorous nogo theorems by Malament (1996) and Hegerfeldt (1998a,b). In particular, Hegerfeldt shows that in a generic quantum theory (relativistic or nonrelativistic), if there are states with localized particles, and if there is a lower bound on the system’s energy, then superluminal spreading of the wavefunction must occur. Similarly, Malament shows the inconsistency of a few intuitive desiderata for a relativistic quantum mechanics of (localizable) par