Abstract A common justification for replacing quantum mechanics with quantum field theory (QFT) is that the appearance or disappearance of particles cannot be described using quantum mechanics. We show that this justification for QFT is not generally true by presenting a counterexample: parametrized relativistic quantum mechanics (pRQM). We begin by outlining a pioneering formulation of QFT that includes an invariant evolution parameter. The introduction of an invariant evolution parameter helped guide the development of QFT and is a characteristic feature of pRQM. We then present a probabilistic formulation of pRQM that highlights features of the theory that make it suitable for modelling particle stability. Two examples of particle stability are then presented within the context of pRQM to show that a quantum mechanical theory can be applied to particle stability. The examples considered in this paper are exponential particle decay and neutrino oscillations.
Abstract A common justification for replacing quantum mechanics with quantum field theory (QFT) is that the appearance or disappearance of particles cannot be described using quantum mechanics. We show that this justification for QFT is not generally true by presenting a counterexample: parametrized relativistic quantum mechanics (pRQM). We begin by outlining a pioneering formulation of QFT that includes an invariant evolution parameter. The introduction of an invariant evolution parameter helped guide the development of QFT and is a characteristic feature of pRQM. We then present a probabilistic formulation of pRQM that highlights features of the theory that make it suitable for modelling particle stability. Two examples of particle stability are then presented within the context of pRQM to show that a quantum mechanical theory can be applied to particle stability. The examples considered in this paper are exponential particle decay and neutrino oscillations.
We study the time evolution of decaying particles in renormalizable models of Relativistic Quantum Field Theory. Significant differences between the latter and Non Relativistic Quantum Mechanics are found —in particular, the Zeno effect seems to be absent in such RQFT models. Conventional renormalization yields finite time behaviour in some cases but fails to produce finite survival probabilities in others. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at short times in non relativistic quantum mechanics (QM henceforth); see, for instance, [1] for a comprehensive review.
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.
Quantum field theory is the traditional solution to the problems inherent in melding quantum mechanics with special relativity. However, it has also long been known that an alternative first-quantized formulation can be given for relativistic quantum mechanics, based on the parametrized paths of particles in spacetime. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such a spacetime approach has proved particularly useful in the study of quantum gravity and cosmology. This paper shows how a spacetime path formalism can be considered to arise naturally from the fundamental principles of the Born probability rule, superposition, and Poincaré invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches in the literature, relating, in particular, “off-shell” theories to traditional on-shell quantum field theory. It reproduces the results of perturbative quantum field theory for free and interacting particles, but provides intriguing possibilities for a natural program for regularization and renormalization. Further, an important consequence of the formalism is that a clear probabilistic interpretation can be maintained throughout, with a natural reduction to nonrelativistic quantum mechanics.
According to classical physics, particles are basic constituents of the physical world. Quantum theory is much less friendly to particles; in particular, relativistic quantum field theory (RQFT) creates serious obstacles for the idea that particles are fundamental. Apparently, when moving from the domain of RQFT to that of classical mechanics (CM), particles have to emerge at some stage. It is standard to assume that this emergence has been completed at the level of quantum mechanics, halfway between RQFT and CM, even though particles of the same kind in many-particle quantum mechanics have the curious feature of being ‘entities without identity’. Against this ‘Received View’ about the nature of quantum particles we outline and defend an Alternative View (AV), in which the emergent character of particles is emphasized. According to this AV, the step to a particle theory has not yet been made in quantum mechanics: conditions have still to be satisfied in order to make the particle concept applicable. If these conditions are met, the quantum particles that emerge are distinguishable individuals possessing physically defined identities, in stark contrast to what the Received View asserts. We will compare and contrast the two Views, both from a physical and a logical/conceptual point of view. This article is part of the theme issue ‘Identity, individuality and indistinguishability in physics and mathematics’.
We study the time evolution of decaying particles in the framework of certain renormalizable models of relativistic quantum field theory. Significant differences between the latter and the non-relativistic quantum mechanical case are found —in particular, regarding the so-called quantum Zeno effect. In some cases, the finite time behaviour is well defined after having performed a conventional renormalization. However, in other situations, the latter does not suffice to obtain finite survival probabilities. In all cases, the crucial consistency with Fermi’s golden rule holds. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at