Abstract
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.
Highlights
There is no guarantee that the standard (Hamiltonian or path integral) procedures of quantization will allow you to construct a quantum theory – in terms of the same dynamical variables – if you try to impose some extra constraints, like for e.g. Lorentz invariance,1 general covariance, or the notion of relativistic causality, which exist in the classical theory
To make a seamless transition you need to describe non-relativistic quantum mechanics (NRQM) in a language which is closer to that of Quantum Field Theory (QFT); not the other way around
One main conclusion – which we have reached from several different perspectives – is that, to make a seamless transition from QFT to NRQM, you need to describe NRQM in a language which is closer to that of QFT and not the other way around
Summary
Given the classical theory of a non-relativistic particle, there is a systematic way of obtaining its quantum version (NRQM), using either a Hamiltonian approach or one based on path integrals. There is no guarantee that the standard (Hamiltonian or path integral) procedures of quantization will allow you to construct a quantum theory – in terms of the same dynamical variables – if you try to impose some extra constraints, like for e.g. Lorentz invariance, general covariance, or the notion of relativistic causality, which exist in the classical theory. The equations of motion describing a relativistic particle does go over to those describing a non-relativistic particle when you take the limit c → ∞ This suggests that, in the corresponding quantum avatars, one should be able to get NRQM from QFT by taking the limit c → ∞. The symbol ≡ in an equation tells you that the equation is used to define some quantity
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