James Fraser: Rethinking Perturbation Theory (in the 1950s)

Rotman Institute of Philosophy

9 Jul 201971:27

EducationalLearning

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TLDRThis lecture delves into the causal perturbation theory approach in quantum field theory, discussing its historical development and philosophical implications. It explores the work of Stueckelberg, Berg, and Bogoliubov, highlighting their efforts to address foundational issues like the relationship between perturbative and non-perturbative QFT, and the problem of ultraviolet divergences. The speaker also touches on the emergence of the renormalization group and the rigorous mathematical framework that has evolved within mathematical physics.

Takeaways

📚 The speaker discusses the history and development of causal perturbation theory in quantum field theory, highlighting its resurgence in the literature over the past decade.
🤝 The collaboration between the speaker, Michael Miller, and Alexander Bloom is focused on integrating the historical context with the philosophical implications of causal perturbation theory.
🔬 The causal perturbation theory framework emerged in the 1950s in the work of Stückelberg, Berg, and Buggalo Buff, aiming to address foundational issues in quantum field theory.
🕒 The conventional approach to perturbative quantum field theory (QFT) starts with the S-matrix and involves an interaction picture time evolution equation, which has been criticized for its formal character and shaky foundations.
🔍 The causal perturbation theory offers an alternative approach to setting up the perturbation series based on a causality condition, potentially resolving issues with ultraviolet divergences and renormalization.
🌐 The renormalization group, which has its roots in the early work on causal perturbation theory, is connected to the scaling properties of QFT and the handling of ultraviolet divergences.
📉 The problem of ultraviolet divergences in the perturbation series is reinterpreted in the causal approach as an issue of ambiguity at coincident points, rather than as divergent integrals that need renormalization.
📈 The use of distribution theory in the causal perturbation theory approach allows for a more rigorous treatment of products of distributions, which are at the heart of the perturbation series.
🛠️ The causal perturbation theory approach is seen as a conceptually cleaner way to set up perturbative QFT, as it does not rely on the interaction picture time evolution equation, which is at odds with the Haag's theorem.
🔬 The speaker also touches on the broader implications of the causal perturbation theory for the understanding of the relationship between perturbative and non-perturbative QFT, and the conceptual foundations of quantum field theory as a whole.

Q & A

What is the main topic of the conference where the speaker is presenting?
-The main topic of the conference is the foundations of quantum field theory, with a focus on causal perturbation theory and its historical development.
Who are the key figures mentioned in the development of causal perturbation theory?
-Key figures mentioned include Michael Miller, Alexander Bloom, and historical figures such as Stoker Berg, and Bugalo Buff.
What is the historical significance of the causal perturbation theory approach in quantum field theory?
-The causal perturbation theory approach is significant because it offers an alternative framework to the conventional perturbative quantum field theory, addressing foundational issues like the relationship between perturbative and non-perturbative theories and the problem of ultraviolet divergences.
What are the two foundational issues illuminated by the work on causal perturbation theory in the 1950s?
-The two foundational issues are understanding the relationship between perturbative and non-perturbative quantum field theory, and understanding the problem of ultraviolet divergences in the perturbation series and the renormalization procedure.
What is the conventional approach to setting up perturbative quantum field theory?
-The conventional approach involves splitting the Hamiltonian into free and interacting parts, evolving states under the interaction Hamiltonian, and using the interaction picture time evolution operator to derive a series expansion for the S-matrix.
What are the problems associated with the conventional approach to perturbative quantum field theory?
-The problems include the issue of ultraviolet divergences and renormalization, which requires a procedure to handle the divergences that arise in the perturbation series, and the shaky foundations of the interaction picture time evolution equation, which may not be consistent with the principles of quantum field theory.
What is the alternative approach to setting up the perturbation series proposed by causal perturbation theory?
-The alternative approach involves using a causality condition to restrict the form of the perturbation series, starting from a causal form of the propagator and iteratively constructing higher terms without employing a time evolution equation.
How does the causal perturbation theory approach address the problem of ultraviolet divergences?
-The causal perturbation theory approach reinterprets the problem of ultraviolet divergences as an issue of ambiguity at coincident points, suggesting that renormalization can be seen as fixing these ambiguities rather than subtracting divergences.
What is the role of distribution theory in the causal perturbation theory approach?
-Distribution theory plays a crucial role in providing a rigorous mathematical framework for understanding the products of singular distributions that occur in the perturbation series, and for defining these products in a way that respects the causality condition.
How does the causal perturbation theory approach relate to the renormalization group?
-The causal perturbation theory approach laid the groundwork for understanding the renormalization group by emphasizing the ambiguities in the perturbation series. It led to the introduction of the renormalization group as a way to describe transformations between different ways of fixing these ambiguities.

Outlines

00:00

📚 Introduction to Causal Perturbation Theory

The speaker begins by acknowledging the ten-year gap since the last similar conference on the topic and notes the confusion that has arisen in the literature due to the complexity of the subject. They express gratitude to the organizers and introduce the theme of their talk, which revolves around the causal perturbation theory approach in quantum field theory. They mention a collaborative project with Michael Miller and Alexander Bloom, focusing on the historical and philosophical aspects of this approach, particularly its emergence in the 1950s and its treatment of ultraviolet divergences and the relationship between perturbative and non-perturbative quantum field theory.

05:01

🔬 Historical Context of Causal Perturbation Theory

The speaker discusses the historical background of causal perturbation theory, highlighting the work of Stoker, Berg, and Bugalo Buff in the 1950s. They delve into the conventional approach to quantum field theory, which involves setting up perturbative expansions using the S-matrix and the interaction picture time evolution equation. The speaker identifies two foundational issues: the relationship between perturbative and non-perturbative quantum field theory and the problem of ultraviolet divergences, which they plan to address through the lens of causal perturbation theory.

10:01

💡 The Emergence of Causal Perturbation Theory

The speaker explores the origins of causal perturbation theory, starting with the work of Stueckelberg and his observation of acausal components in the perturbation series. They mention the development of the theory by Gelfand and栗夫, who introduced a causality condition to address the issue of acausality and to derive the perturbation series without relying on the interaction picture time evolution equation. The speaker emphasizes the conceptual shift from viewing ultraviolet divergences as a problem of renormalization to one of ambiguity in the definition of products of distributions.

15:03

📉 Reinterpreting Ultraviolet Divergences

The speaker discusses the reinterpretation of ultraviolet divergences within the causal perturbation theory framework. They explain that Stueckelberg and Peterman viewed these divergences as ambiguities in the definition of products of distributions at coincident points, rather than as actual infinities that need to be renormalized. The speaker outlines the distribution theory approach to addressing these ambiguities and the idea of extending the space of test functions to provide a proper definition for these products.

20:04

🔧 The Role of Renormalization Group in Causal Perturbation Theory

The speaker connects the reinterpretation of ultraviolet divergences to the early development of the renormalization group. They mention the work of Gelman and Low, who introduced the renormalization group beta function to study the ultraviolet asymptotic behavior of QED. The speaker suggests that the causal perturbation theory's emphasis on ambiguities in the perturbation series played a role in the emergence of the renormalization group concept, although the exact relationship between the early normalization group and Wilson's renormalization group remains a topic for further exploration.

25:05

🤔 Open Questions and Future Directions

In the final part of the talk, the speaker acknowledges the open questions and potential future directions for research in causal perturbation theory. They express a desire to further investigate the relationship between different interpretations of the renormalization group and to clarify the conceptual differences between them. The speaker also hints at the possibility of extending the causal perturbation theory approach to non-renormalizable theories and gauge theories, suggesting that the framework may offer a more coherent understanding of quantum field theory.

Mindmap

Keywords

💡Quantum Field Theory

Quantum Field Theory (QFT) is a fundamental framework in physics that combines quantum mechanics with special relativity, providing a description of the quantum mechanical nature of fields. In the video, the theme revolves around the foundations of QFT, particularly the causal perturbation theory approach, which seeks to address foundational issues like the relationship between perturbative and non-perturbative QFT.

💡Causal Perturbation Theory

Causal Perturbation Theory is an approach within QFT that emphasizes the causality condition in the perturbative expansion of the S-matrix. The speaker discusses the historical development of this approach in the 1950s and its philosophical implications. It is presented as an alternative to the conventional approach, aiming to resolve issues related to ultraviolet divergences and the renormalization procedure.

💡Renormalization Group

The Renormalization Group is a set of techniques used in QFT to handle infinities that arise in calculations. It involves a change of scale in the theory, allowing for the systematic study of the behavior of physical systems at different scales. The script mentions the early development of the renormalization group in the context of causal perturbation theory and its connection to the scaling properties of QFT.

💡Ultraviolet Divergences

Ultraviolet Divergences refer to the mathematical infinities that appear in calculations of quantum field theories at very high energies or short distances. In the video, the causal perturbation theory approach offers a different perspective on these divergences, interpreting them as ambiguities in the definition of products of distributions rather than actual infinities that need to be subtracted out.

💡S-Matrix

The S-Matrix is a fundamental concept in QFT that describes the scattering process, relating the states of a system before and after an interaction. The script discusses the perturbative expansion of the S-Matrix and how the causal perturbation theory offers an alternative derivation of this expansion based on causality conditions.

💡Perturbative Quantum Field Theory

Perturbative Quantum Field Theory is a method of approximating solutions to QFT by expanding around a known solution, typically the free field theory. The script raises questions about the relationship between perturbative and non-perturbative approaches and how the causal perturbation theory might illuminate this relationship.

💡Non-Perturbative Quantum Field Theory

Non-Perturbative Quantum Field Theory refers to methods that go beyond the perturbative expansion to describe the full dynamics of a quantum field theory. The video discusses the causal perturbation theory's potential to provide insights into the non-perturbative aspects of QFT, particularly in the context of the renormalization group.

💡Renormalization

Renormalization is a procedure in QFT that deals with the infinities that arise in perturbative calculations by redefining parameters of the theory. The script discusses the causal perturbation theory's reinterpretation of renormalization as fixing ambiguities in the perturbation series, rather than as a process of subtracting infinities.

💡Causality Condition

The Causality Condition is a fundamental principle in physics stating that the effect cannot precede the cause. In the context of the video, the causality condition is used to derive the perturbation series for the S-matrix in a way that respects the temporal ordering of events, leading to a different perspective on the structure of QFT.

💡Distributions

In the script, distributions refer to mathematical objects that generalize the concept of functions, allowing for the inclusion of singular functions like the Dirac delta function. The causal perturbation theory approach uses distribution theory to provide a rigorous treatment of the products of distributions that appear in the perturbation series of QFT.

💡Epstein-Glaser Renormalization

Epstein-Glaser Renormalization is a method in mathematical physics that provides a rigorous treatment of renormalization in QFT by using distribution theory. The script mentions this method as a development that brought the distribution theoretic analysis of renormalization to fruition, building on the earlier ideas of causal perturbation theory.

Highlights

It has been ten years since a similar conference on the philosophy of quantum field theory was held, and the literature has become increasingly confusing.

The speaker thanks the organizers and mentions the good progress of the conference so far.

Michael Miller and the speaker are both presenting on causal perturbation theory, a project they are working on with Alexander Bloom.

The project originated from work at the Max Planck Institute, exploring early papers on renormalization group theory.

The speaker will focus on the historical emergence of causal perturbation theory in the 1950s.

Two foundational issues in quantum field theory will be discussed: the relationship between perturbative and non-perturbative QFT, and the problem of ultraviolet divergences.

The conventional approach to perturbative quantum field theory starts with a time evolution equation in the interaction picture.

The S-matrix is derived from the interaction picture time evolution operator, leading to the Dyson series.

Renormalization is introduced as a procedure to handle divergences in the perturbation series.

Causal perturbation theory offers an alternative approach to setting up the perturbation series based on causality conditions.

The causal approach aims to resolve conceptual puzzles related to ultraviolet divergences and renormalization.

The speaker discusses the work of Stueckelberg and Bogoliubov in developing causal perturbation theory.

Bogoliubov's version of causality condition leads to a clearer derivation of the Dyson series without a time evolution equation.

Causal perturbation theory provides a conceptually cleaner way to set up perturbative quantum field theory.

The approach reinterprets ultraviolet divergences as ambiguities at coincident points, leading to a different view on renormalization.

Distribution theory is introduced to provide a rigorous mathematical framework for handling products of distributions in the perturbation series.

The early ideas on causal perturbation theory laid the groundwork for modern distribution-theoretic analysis in quantum field theory.

The speaker concludes by discussing the connection between causal perturbation theory and the renormalization group.

Transcripts

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Related Tags

Quantum Field Causal Perturbation Renormalization Philosophy of Physics Historical Analysis Theoretical Physics Conference Talk Scientific Debate Mathematical Physics Conceptual Clarity