David Baker: Broken Symmetry and Spacetime

The speaker, John Baker from the University of Michigan's philosophy department, introduces a discussion on symmetry breaking in quantum field theory. He outlines an apparent inconsistency involving the physical equivalence of states under unitarily inequivalent representations and the implications for global spacetime symmetries. The talk aims to explore the algebraic approach to quantum field theory, where spontaneous symmetry breaking leads to different ground states within unitarily equivalent representations, challenging the view that global spacetime symmetries merely redescribe the same physical possibility.

This paragraph delves into the concept of global spacetime symmetries, such as translations and rotations, and their philosophical implications. The speaker argues that these symmetries are often viewed as merely coordinate changes without distinct physical possibilities. However, when spacetime symmetry is spontaneously broken, these global symmetries unexpectedly relate states within unitarily inequivalent representations, which contradicts the initial assumption of their irrelevance to physical distinctions. The speaker also discusses the notion of spontaneous symmetry breaking, using the analogy of a ball on a sombrero to illustrate the concept.

The speaker addresses the challenge of translating between quantum states in the context of symmetry breaking. He discusses the principle of equivalence, which states that spacetime symmetry transformations do not yield new physical possibilities. The speaker explores the implications of this principle in quantum theory, particularly when dealing with systems like the 'ball and sombrero' example, where ground states related by global rotations are considered. The difficulty arises in maintaining a translation scheme that preserves the physical equivalence of these states across different representations.

In this paragraph, the speaker examines the role of observables in quantum representations, focusing on the distinction between conservative and liberal views about which observables are physically meaningful. The conservative view limits physical quantities to those invariant across representations, while the liberal view acknowledges that some physical quantities, like the stress-energy tensor, may be representation-dependent. The speaker raises the question of how to interpret the existence of multiple, unitarily inequivalent representations of the canonical commutation relations, given that they satisfy the same commutation relations.

The speaker discusses spontaneous symmetry breaking in quantum systems, using the example of a magnet to illustrate how a physical system can exhibit a preferred direction after symmetry breaking, despite the underlying physics being rotationally symmetric. This leads to a situation where the ground states of the system, which are related by symmetry transformations, reside in unitarily inequivalent representations. The speaker questions the physical interpretation of these inequivalent representations and how they relate to our understanding of observables.

In this paragraph, the speaker explores the paradox of physical equivalence in quantum representations, particularly in the context of an infinite magnet. The magnet, when rotated, appears to generate new physical possibilities due to the unitarily inequivalent representations of its ground states. This challenges the principle of liberalism about observables, which assumes that physically equivalent states should be intertranslatable. The speaker suggests that there may be a need to reconsider the assumptions underlying the translation scheme between different representations.

The speaker questions the H and Clifton translation scheme, which posits that physically equivalent representations must be unitarily equivalent. The speaker argues that this requirement may be too stringent, as it does not allow for changes in the numerical expectation values of the vile operators. The speaker proposes that different numerical values could represent the same physical property, challenging the H and Clifton assumption that translation schemes must preserve these values exactly.

In this paragraph, the speaker explores the nature of physical equivalence, considering the implications of unitary and anti-unitary transformations. The speaker discusses the role of inner products in determining transition probabilities and the potential bizarre consequences of these probabilities being affected by symmetry transformations. The speaker also addresses the question of what constitutes physical equivalence and how it relates to the conservation of observables.

The speaker discusses the challenge of rotational symmetry in quantum systems, particularly in the context of an infinite paramagnet. The speaker questions the physical differences between representations and the implications of unitary inequivalence. The discussion touches on the possibility of a preferred direction in space, even in the absence of apparent asymmetry, and the potential need for a new understanding of the underlying principles of quantum mechanics.

In this paragraph, the speaker addresses the issue of representation and observables in quantum theory, focusing on the challenges of extending models to infinite systems. The speaker discusses the need to find a way to represent poly-spin systems in a separable Hilbert space and the implications of different representations for the understanding of global magnetization. The speaker also questions the assumptions underlying the use of spatial translations and the potential impact on the interpretation of observables.

The speaker concludes by suggesting that the future direction of quantum representation theory may involve a deeper exploration of the physical differences between representations and the development of a more intuitive understanding of unitary equivalence and inequivalence. The speaker also hints at the possibility of a panel discussion to further explore these complex issues.