The Conceptual Relation between Standard Quantum Mechanics and Quantum Field Theory

Τ. Padmanabhan

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.

Prislac, Thomas, Echo, AI

Appendix Q – Quantum Mechanics and GUFT: A Structural Mapping of Coherence Fields Q.1. Scope and Caution Quantum mechanics (QM) is one of the most precisely tested scientific theories in history. Standard QM and quantum field theory (QFT) correctly describe an enormous range of phenomena, from atomic spectra to semiconductor devices to superconducting qubits. We are not proposing a new physics theory or “fixing” quantum mechanics. Instead, this appendix asks a narrower, safer question: How can the core structures of quantum theory—states, coherence, entanglement, measurement, decoherence—be mapped into the GUFT framework of coherence fields (Ψ = E × T, ΔS, etc.) in a way that is faithful to physics and conceptually useful? We proceed with three strict constraints: We take standard QM/QFT (as found in textbooks and reviewed sources) as non-negotiable. We treat GUFT as a descriptive overlay: a way to talk about structure and information, not an alternative dynamics. Where we draw analogies (e.g., empathy ↔ correlation), we note they are structural, not literal or magical.

B. K. Agarwal

Jordan François, Lucrezia Ravera

We treat the mechanics of point particles as a one-dimensional general-relativistic gauge field theory, which may be referred to as mechanical field theory (MFT), exploiting the bundle geometry of mechanical field space (MFS). The diffeomorphism covariance of MFT encodes its relational character, arising—as in all general-relativistic physics—via the conjunction of a hole and a point-coincidence argument. Any putative “boundary problem,” meaning the claim that “spacetime” boundaries break diffeomorphism and gauge symmetries, thereby dissolves. It is highlighted that the standard path integral (PI) on the MFS, the exact analog of the PI used in gauge field theory, is conceptually and technically from the standard PI of quantum mechanics. We then use the dressing field method to give a manifestly invariant and relational reformulation of MFT, which reproduces the standard textbook formulation when a clock field is chosen as a (natural) dressing field. The dressed, or basic, PI on the MFS, defining i.e., the quantization of invariant relational degree of freedom—is shown to reproduce the standard PI of quantum mechanics. This establishes the soundness of relational quantization as a general guiding principle: We outline it for general-relativistic gauge field theories.

Simon W. Saunders