Benjamin Feintzeig: Classical limits of particle concepts in quantum field theory

The speaker begins by expressing a personal struggle with the negativity often found in quantum field theory, particularly regarding the limitations of particle interpretations. They aim to address the issue of localized particles, mentioning three key problems: localization in space, the incompatibility of different particle concepts, and issues with interactions. The speaker acknowledges the existence of particles despite theoretical limitations and introduces David Wallace's unpublished work on the emergence of particles from bosonic quantum field theory as a potential solution to understanding particle structures as emergent phenomena.

Shifting focus to the classical limit of quantum field theories, the speaker finds motivation in understanding quantum mechanics through the lens of classical theories. They propose using the classical limit to examine particle concepts, specifically number operators, and their presence in different representations. The speaker credits a group of undergraduate students for their significant contributions to the technical work and calculations related to this research, while clarifying that the interpretive aspects are their own responsibility.

The speaker delves into the no-go theorems that suggest the impossibility of having localized particles in quantum field theory, discussing the problems with position operators and local number operators. They mention the concept of 'almost local' number operators as a potential workaround, introduced by R. Clifton and H. Halverson, which allows for the interpretation of particles in scattering theory. However, the speaker expresses dissatisfaction with these operationalistic descriptions, seeking a deeper understanding of particle interpretations.

The speaker discusses the issue of inequivalent representations, such as the Minkowski and Rindler representations, and the challenges they pose to a fundamental particle interpretation. They mention the work of Clifton and Halverson, which argues against the possibility of translating between different particle interpretations due to the lack of unitary equivalence. The speaker acknowledges the context-dependent nature of interpretations but continues to seek a more comprehensive understanding of particle content in quantum theories.

Introducing the concept of strict quantization, the speaker outlines the technical tools used for taking the classical limit (h-bar goes to 0) in quantum theories. They discuss the construction of a family of C*-algebras and quantization maps that allow for the approximation of quantum theories by classical ones. The speaker highlights the innovation in extending these tools to unbounded quantities, such as field operators and number operators, which are relevant for understanding particle concepts.

The speaker describes the setup for the classical Klein-Gordon theory, focusing on the real scalar field and its initial data. They explain the use of a real vector space for field configurations and conjugate momenta, the role of the symplectic form in defining the algebra of the quantum theory, and the importance of the space of test functions. The speaker also discusses the definition of the strict quantization in this context, including the choice of complex structure and its implications for the one-particle Hilbert space.

The speaker presents the Berezin quantization map, emphasizing its continuity and positivity properties, which allow for a unique extension to the completions of the algebra in the weak topology. They discuss the relevance of this map for field theories with infinitely many degrees of freedom and its ability to define unbounded operators without a Hilbert space representation. The speaker also addresses the approximation properties of the quantization map and the possibility of improving the notion of approximation.

The speaker provides a framework for analyzing the classical limits of unbounded quantities, such as field operators and number operators, in free field theories. They discuss the definition of number operators in a representation-independent way and the comparison of these operators through the classical limit. The speaker highlights the approximation of the Minkowski number operator to the classical total energy of the Klein-Gordon field, suggesting a deeper understanding of localization in quantum field theory.

The speaker focuses on the classical limit of the Minkowski number operator, revealing that it corresponds to the total energy in the classical theory. They discuss the implications of this result for the understanding of particle localization and the comparison between different number operators. The speaker raises questions about the classical limits of the Rindler number operator and how it compares to the classical energy, indicating areas for further research.

The speaker concludes with a discussion on non-standard interpretations of quantum theories, advocating for an approach that considers relations between theories and mathematical rigor. They differentiate their approach from those who suggest giving up on algebraic methods or mathematical rigor in favor of more contextual interpretations. The speaker emphasizes the importance of maintaining a rigorous mathematical framework while seeking to understand quantum theories through their relations and approximations.

In the final paragraph, the speaker engages in a dialogue about the relationship between quantum and classical theories, the nature of limits, and the interpretation of physical quantities. They address concerns about the dimensionality of parameters, the scale of physical changes, and the technical requirements for field operators. The speaker also discusses the distinction between discrete quantum quantities and continuous classical fields, highlighting the challenges in understanding how particles appear to be localized in certain contexts.

The speaker explores the concept of almost local operators and their role in quantum theory, challenging the operationalistic view presented by others. They discuss the quasi-local algebra and the approximation of local measurements, suggesting that there is a narrative within quantum theory that explains almost localization. The speaker also addresses the difference between the classical and quantum definitions of local number operators and the implications of these differences for the understanding of localization.

In the final summary, the speaker reflects on the classical theory's approach to localization, discussing the differences between the classical and quantum definitions of local operators. They emphasize the importance of understanding the classical limits of quantum theories and the potential contributions of these results to the broader understanding of quantum field theory. The speaker also addresses questions about the completeness of the labeling of states and the implications of their research for the field.