John Bain: Intertheoretic Implications of Non Relativistic Quantum Field Theories

The paragraph discusses the debate surrounding the interpretation of particles in relativistic quantum field theories. It introduces the receipt view, which argues that relativistic quantum field theories do not admit particle interpretations due to the non-uniqueness of particle concepts informed by classical spacetime intuitions. The speaker proposes that this concept should be abandoned and reviews Newtonian quantum gravity as an alternative non-relativistic monofield theory that includes gravity. The paragraph also touches on the implications of such theories for our understanding of the relationship between physical theories and reality.

This paragraph delves into the differences between classical and relativistic spacetime structures, focusing on Lorentzian and classical spacetimes. It explains that Lorentzian spacetimes are characterized by a pseudo-Riemannian metric with a specific signature and a unique curvature tensor, while classical spacetimes have a unique way to separate time from space due to their metrical structure. The paragraph also discusses the symmetry groups associated with these spacetimes, such as the Poincaré group for Lorentzian and the Galilei group for classical spacetimes, and how these structures influence the formulation of quantum field theories.

The speaker addresses the challenges in defining particle interpretations within relativistic quantum field theories, specifically focusing on the conditions that must be met for a theory to admit a particle interpretation. These conditions include the existence of local number operators and a unique total number operator. The paragraph discusses how these conditions fail in relativistic theories due to issues such as the non-uniqueness of representations and the lack of local number operators in Minkowski spacetime. It also touches on the separability corollary and its implications for the vacuum state in quantum field theories.

This paragraph explores how the spatial and temporal distinction between relativistic and non-relativistic theories affects the debate on particle interpretations in quantum field theory. It discusses the conditions for particle interpretations, such as localizability and countability, and argues that these are informed by non-relativistic concepts of time. The speaker suggests that these conditions are inappropriate for interpreting relativistic quantum field theories and proposes that the presence of an absolute temporal metric in classical spacetimes guarantees the local simplicity of the vacuum state, which is not a generic feature of relativistic theories.

The paragraph introduces Newtonian quantum gravity as a non-relativistic quantum field theory that incorporates gravity and is set in a curved classical spacetime. It contrasts this with the challenges faced in constructing a fully relativistic quantum field theory of gravity. The speaker reviews different types of classical Newtonian gravitation theories, including those formulated with a fixed classical spacetime and those that geometrize gravity by incorporating the gravitational potential into the spacetime connection. The paragraph also discusses the implications of these theories for the structure of spacetime and the interpretation of particles.

The speaker discusses Newton-Cartan gravity (NCG), a theory that geometrizes gravity and eliminates the gravitational potential, leading to a generalized Poisson equation and a two-metric expression. The paragraph explores different versions of NCG, depending on the curvature constraints imposed, and how these relate to the non-relativistic limit of general relativity. It also introduces Newtonian quantum gravity as a theory that arises from imposing certain curvature conditions, leading to a unique one-particle structure and a Fock-space representation, which is not typically possible in relativistic quantum field theories.

This paragraph examines the inter-theoretical relations between various fundamental theories of physics, including classical mechanics, special relativity, and quantum mechanics. The speaker presents a diagrammatic representation, referred to as the 'great dimensional monolith of physics,' which illustrates the relationships between these theories. The paragraph discusses the challenges in constructing a fully relativistic quantum theory of gravity and the potential routes to achieve this, including quantizing general relativity, turning on gravity in a relativistic quantum field theory, and relativizing Newtonian quantum gravity.

The speaker explores the nature of limits in the context of the 'great dimensional monolith of physics,' such as the non-relativistic limit, the g (Newtonian gravitational constant) goes to zero limit, and the h-bar (reduced Planck constant) goes to zero limit. The paragraph discusses the complexities and potential misunderstandings associated with these limits, including the distinction between Riemannian and Ricci flatness and the challenges in defining the non-relativistic limit of general relativity. It also touches on the implications of these limits for the structure of the monolith and the relationships between theories.

The paragraph introduces a modified model of theoretical physics, represented as a hypercube, which includes an additional axis to distinguish between particle and field theories, as well as between relativistic and non-relativistic theories, gravitational and non-gravitational theories, and classical and quantum theories. The speaker discusses the potential ways of turning on fully relativistic quantum gravity, including quantizing general relativity, turning on gravity in a relativistic quantum field theory, relativizing Newtonian quantum gravity, and considering a thermodynamic limit. The paragraph also explores the implications of this model for understanding the relationships between different theories and the potential for a relativistic quantum particle theory of gravity.

This final paragraph engages in a philosophical discussion on the interpretation of particles in quantum field theories. It considers the possibility of adding a third condition to the existing criteria for particle interpretations and explores the implications of the non-uniqueness of the total number operator in relativistic quantum field theories. The paragraph also touches on the practical approach of working physicists, who often rely on conserved quantities to infer the number of constituents in a system, rather than focusing on the number of particles themselves. The discussion highlights the ongoing debates and the need for a deeper understanding of the conceptual apparatus for particles in quantum mechanics.