Hilary Greaves: Towards a Geometrical Understanding of the CPT Theorem

The speaker begins by apologizing for the anti-store nature of the talk and for potentially confusing the audience with complex classical field theory references. They express eagerness to receive feedback and clarify their intentions for giving the talk, which is to explore the relationship between quantum field theory and its invariance under the restricted Lorentz group. The speaker also touches on the difficulty of understanding mathematical talks without a strong background and their desire to provide enough technical content for a knowledgeable audience.

The speaker outlines a strategy of asking 'stupid questions' to simplify complex concepts. They pose three fundamental questions about the CPT theorem's connection to symmetries, the intimate relationship between space and time, and the quantum nature of the theorem. The speaker aims to dissolve these puzzles by considering the symmetries of Lorentz invariance and the implications of charge conjugation, parity, and time reversal (CPT). They also discuss the possibility of a classical PT theorem and the confusion surrounding the naming conventions in physics.

The speaker delves into the classical PT theorem, discussing its implications for quantum field theory and the potential for a more elegant argument. They explore the relationship between physicists' understanding and the mathematical formalism of the theorem, highlighting the unsatisfactory state of affairs in the field. The speaker also addresses the question of whether the PT theorem is specific to quantum field theory or if it can be applied to classical field theory, suggesting that the theorem may be more broadly applicable.

The speaker introduces tensors as fundamental objects in classical field theory, explaining their mathematical properties and physical examples. They discuss the transformation of tensors under the Lorentz group and the implications for the space-time structure. The speaker also touches on the construction of quantum field theory from the space of positive frequency solutions, emphasizing the importance of understanding the mathematical framework behind physical theories.

The speaker presents a classical PT theorem for tensor fields, discussing the conditions under which the theorem holds and its implications for the invariance of physical laws. They explore the idea of polynomial tensor fields and their transformation properties, leading to a deeper understanding of the PT theorem's role in classical field theory. The speaker also addresses the limitations of the current understanding of the PT theorem and the need for a more rigorous mathematical approach.

The speaker discusses the complexities of PT symmetry in quantum field theory, particularly in the context of complex tensor fields. They explore the transformation properties of these fields and the challenges in proving the PT theorem in a quantum context. The speaker also addresses the question of charge reversal under PT operations and the differences between classical and quantum field theories in this regard.

The speaker acknowledges the difficulty in proving the PT theorem for spin fields, given their complex nature and the challenges in applying the theorem to classical field theory. They discuss the need for a more general approach to the PT theorem that can account for the complexities of spin fields and their transformation properties. The speaker also considers alternative strategies for proving the theorem, including the use of polynomial systems and the role of complex conjugation.

The speaker expresses dissatisfaction with the current state of the PT theorem in classical field theory, particularly in the context of spin fields. They suggest that a more general theorem may be possible if certain assumptions about the nature of field equations can be relaxed. The speaker also discusses the potential for a more elegant and simple argument for the PT theorem, one that could better capture the fundamental principles of quantum field theory.

The speaker reflects on the fundamental nature of the PT theorem and its implications for our understanding of quantum field theory. They consider the possibility that the theorem may be more deeply rooted in the structure of physical laws than currently appreciated. The speaker also discusses the potential for further research into the PT theorem, including exploring its relationship with other fundamental symmetries and conservation laws.

The speaker engages in a discussion about the paradoxical behavior of charge reversal under PT operations in classical and quantum field theories. They explore the implications of this phenomenon for the understanding of physical symmetries and the conservation laws. The speaker also addresses the audience's questions and concerns, highlighting the need for further exploration and clarification of these complex issues.

The speaker delves into the relationship between PT symmetry and charge conservation, discussing the implications of these symmetries for the behavior of physical systems. They explore the idea that PT symmetry may not necessarily imply charge conservation, particularly in the context of certain quantum field theories. The speaker also addresses the audience's questions, providing insights into the complex interplay between symmetries and conservation laws in physics.

The speaker investigates the phenomenon of parity violation, particularly in the context of quantum field theory. They discuss the implications of parity violation for our understanding of fundamental physical principles and the behavior of elementary particles. The speaker also explores the relationship between parity violation and time reversal non-invariance, highlighting the complex interplay between these symmetries in the physical world.

The speaker discusses the consequences of time reversal non-invariance in quantum field theory, particularly in the context of the PT theorem. They explore the idea that the violation of time reversal symmetry may have profound implications for our understanding of physical laws and the behavior of the universe. The speaker also addresses the audience's questions, providing insights into the complex relationship between time reversal symmetry and other fundamental symmetries.