Doreen Fraser: How analytic continuation relativistic QFT models relate to non-relativistic models

The speaker opens the discussion by expressing excitement about the conference and introduces the topic of analytic continuation in quantum field theory. They provide background on Wick rotation, explaining its role in transforming Lorentz invariant vector products into Euclidean vector products, and the relevance of Minkowski space and Lorentz symmetries in quantum field theory. The speaker highlights the historical interplay between condensed matter physics and particle physics and argues that the analogies used in quantum field theory are often purely formal, focusing on mathematical structures rather than physical similarities.

The speaker addresses objections to the use of Wick rotation and analytic continuation in mapping between non-relativistic and relativistic models. They acknowledge the concern that such mappings undermine physical interpretations and explain that analytic continuation often relies on physical facts, suggesting a physical basis for the analogies. The speaker, however, maintains that these analogies are formal and plans to defend this position by responding to the objection and outlining their approach to the topic.

The speaker outlines the philosophical framework for the discussion, emphasizing the need to sharpen the question of why analytic continuation works in quantum field theory. They plan to explore the technique's role in both perturbative and non-perturbative contexts, beginning with the historical development in quantum electrodynamics (QED) by Dyson and Schwinger. The speaker aims to distinguish between mathematical tricks and physical explanations, setting the stage for a deeper analysis of analytic continuation.

The speaker discusses Dyson's 1949 work on the S-matrix in quantum electrodynamics, explaining how Dyson used analytic continuation to manage ultraviolet divergences by transforming integration paths away from singularities. They then move to Schwinger's work, which focused on the properties of correlation functions obtained through Wick rotation, and Symanzik's influential 1966 proposal to use Euclidean quantum field theory as a non-perturbative strategy for constructing models. The speaker highlights the evolution of these ideas and their significance in both perturbative and non-perturbative quantum field theory.

The speaker elaborates on Symanzik's Euclidean strategy for constructing quantum field theory models, which gained prominence in both the renormalization group methods by Wilson and constructive field theory. They explain how Euclidean field theories serve as a starting point for constructing quantum field theory models by analytically continuing back to the original theory. This approach requires careful encoding of physical structure in Euclidean models, a process illustrated by the Osterwalder-Schrader reconstruction theorem.

The speaker delves into the Whiteman axioms for quantum field theory, emphasizing their role in ensuring the analyticity of vacuum expectation values (Whiteman functions) and their ability to be analytically continued to Schwinger functions. They outline the mathematical properties required for this continuation and discuss how these properties, rooted in physical principles like covariance and locality, facilitate the transformation of Whiteman functions across Euclidean domains.

The speaker explains the conditions under which Whiteman functions can be analytically continued, focusing on the assumptions of Lorentz covariance, local commutativity, and the spectral condition. These assumptions guarantee the analyticity of Whiteman functions in the relevant domain, thus legitimizing the use of analytic continuation. They draw parallels between this explanation and Dyson's justification for variable changes in QED, emphasizing the importance of avoiding singularities.

The speaker transitions to discussing the reconstruction of quantum field theory models from Euclidean field theories, guided by the Osterwalder-Schrader theorem. They outline the steps involved in this reconstruction, including defining Euclidean fields and vacuum states, and ensuring these fields satisfy the appropriate properties to construct a corresponding quantum field theory. This process highlights the need for encoding physical structure in Euclidean models to ensure the successful application of analytic continuation.

The speaker provides a detailed overview of the reconstruction process, explaining how Euclidean fields are used to construct a Hilbert space and how these fields are identified with quantum field theory operators. They emphasize the importance of ensuring that the Euclidean theory encodes the necessary physical structure to reconstruct a quantum field theory that meets the Whiteman axioms. This section underscores the technical complexities and theoretical considerations involved in this process.

The speaker contrasts the two types of explanations for analytic continuation: mathematical tricks and physical symmetries. They argue that while physical conditions motivate the required analyticity properties, the explanation ultimately relies on the mathematical tractability provided by these conditions. The speaker acknowledges that constructing Euclidean field theories with the correct physical structure is essential for certain projects, but maintains that this does not fundamentally alter the nature of analytic continuation as a mathematical technique.

The speaker revisits the concept of structural equivalence between Euclidean and quantum field theories, clarifying that the Euclidean models must minimally encode the quantum field theory's structure to be useful. They address concerns about the robustness of this encoding and emphasize that while additional Euclidean structures may be present, the primary goal is to ensure that essential quantum field theory properties are preserved. This section reinforces the importance of careful model construction in theoretical physics.

The speaker explores the conditions under which analytic continuation can be reversed and discusses the practical implications of this process. They note that successful reversal depends on the details of the models and the context in which they are applied. The speaker acknowledges the complexity of these considerations and suggests that different goals in physics, such as quantum gravity, might necessitate more robust approaches to analytic continuation.

The speaker emphasizes the importance of context when applying analytic continuation techniques. They highlight that the specific goals and details of the models in question can influence whether a mathematical trick or a deeper physical explanation is appropriate. The speaker suggests that certain applications, such as calculating specific quantities or addressing singularities, might favor the mathematical trick explanation, while more fundamental questions in physics might require a more comprehensive approach.

The speaker briefly touches on the broader implications of analytic continuation across different areas of physics, including classical statistical mechanics and quantum theories. They acknowledge the utility of this technique in various contexts and suggest that understanding its role in classical physics might provide insights into its application in quantum field theory. This section serves as a bridge to broader discussions about the nature of relativistic and non-relativistic models.

The speaker reflects on the role of Euclidean symmetries in analytic continuation, addressing concerns about the spatial and temporal interpretations of these transformations. They suggest that while Euclidean symmetries are mathematically convenient, they do not necessarily imply a direct physical correspondence. The speaker concludes by reiterating the importance of distinguishing between mathematical techniques and physical interpretations in theoretical physics.

The speaker wraps up the discussion by reiterating the importance of constructing robust theoretical models that accurately encode the physical structures of quantum field theories. They emphasize that while analytic continuation is a powerful mathematical tool, its application must be carefully considered in the context of specific physical theories and goals. The speaker highlights the need for ongoing research and critical examination of these techniques in theoretical physics.