Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts (1712.06605v2)

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 non-trivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually non-trivial. 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 description between NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as it 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. (Abridged)

Overview of Non-Relativistic Quantum Mechanics and Quantum Field Theory Transition

The paper "Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores, and Facts" by T. Padmanabhan addresses the challenging transition from quantum field theory (QFT) to non-relativistic quantum mechanics (NRQM). It explores why a seamless transition is non-trivial and often glossed over in standard literature. The implications for both theoretical foundations and practical applications in combining quantum mechanics with relativity are considered.

The central focus is the non-relativistic limit of QFT, examining how anti-particles and relativistic effects influence this transition. The paper reviews different approaches, such as Hamiltonian dynamics and path integrals, and scrutinizes the conceptual difficulties arising from their application to relativistic particles, contrasting the seamless translation observed in NRQM. The complications stem largely from the intrinsic presence of anti-particles in maintaining relativistic causality.

Key Technical Insights

1. Propagators and Particle Localization: The transition from QFT to NRQM is not straightforward due to the disparate roles of space and time in these theories. In NRQM, particle positions are well-localized, whereas in QFT, no proper position operator is defined due to Lorentz invariance. The propagator in QFT, especially when associated with position states, does not facilitate the seamless propagation compatible with NRQM due to this localization issue.
2. Symmetry and Dynamics: While NRQM handles quantum conditions with energy boundedness and spatial dynamics seamlessly via well-localized states, QFT requires positions to act as field labels rather than operators, which leads to inherent conflicts when seeking non-relativistic limits.
3. Path Integral and Regularization: The analysis extends to path integral formalism, comparing Hamiltonian and Lagrangian approaches. While Lagrangian path integrals are intuitive, the standard method falters in relativistic contexts due to measure ambiguities, whereas Hamiltonian path integrals can achieve a semblance of Lorentz invariance with adjusted measures. Lattice regularization emerges as a promising alternative but highlights the inherent differences in handling spacetime paths between NRQM and QFT.
4. Field Operator Transition: The paper establishes a framework where free particles in QFT, manifesting as excitations described by field operators, afford a bridge to NRQM. Here, the distinct field operators for particles and anti-particles in QFT transform seamlessly into NRQM's operator framework.
5. Contradictory Claims and Interpretations: Some claims in literature are addressed, specifically regarding the NRQM limits of real scalar fields and the roles of anti-particles, pointing out ambiguities and misinterpretations typically introduced by simplified textbook approaches.

Implications and Future Directions

This research paper underscores the complex interaction between SR principles and quantum mechanics, particularly illustrating the challenges in integrating QFT concepts into the NRQM framework. By dissecting the theoretical underpinnings and employing rigorous mathematical scrutiny, this paper demystifies the assumptions and folklores prevalent in theoretical physics.

The implications are profound in that they not only clarify common misconceptions but also highlight a need for deeper exploration into the foundational structures governing quantum theory in relativistic contexts. This work opens pathways for further inquiry into quantum gravity interactions and has potential applications in designing quantum technologies responsive to both NRQM and relativistic requirements.

Conclusion

Padmanabhan's examination of the conundrum faced in obtaining NRQM from QFT reveals crucial insights into the foundational principles of quantum mechanics under special relativity. The paper advocates for a paradigm where NRQM is conceived through the language of QFT, adhering to reality’s relativistic demands, thereby potentially influencing broader theories aimed at unifying quantum mechanics with general relativity.