[Paper Review] Group Action in Topos Quantum Physics

Cecilia Flori|arXiv (Cornell University)|Oct 7, 2011

Quantum Mechanics and Applications21 references4 citations

TL;DR

This paper introduces a group action formalism in topos quantum theory by replacing the standard presheaf topos with a sheaf topos over a product category involving contexts and a unit interval, enabling continuous group actions and eliminating the need for twisted presheaves. The key contribution is a geometrically consistent, observer-independent formulation of unitary symmetries in quantum theory via sheaf-theoretic structures that support continuous transformations and internal truth values for group-related propositions.

ABSTRACT

Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Döring, as an alternative mathematical structure within which to formulate physical theories. In particular it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include the notion of a group and group transformation in such a way that we overcome the problem of twisted presheaves. In order to implement this we need to change the type of topos involved, so as to render the notion of continuity of the group action meaningful.

Motivation & Objective

• To overcome the limitations of twisted presheaves in topos quantum theory when formulating group actions.
• To provide a geometrically consistent and continuous representation of unitary group transformations in quantum theory.
• To extend the topos framework to include group symmetries without relying on external notions of measurement or observer.
• To generalize the topos formulation to accommodate both pure and mixed states under group actions via sheaf-theoretic structures.
• To establish a foundation for quantization procedures related by unitary transformations within the topos framework.

Proposed method

• Replace the standard presheaf topos $\mathbf{Sets}^{\mathcal{V}(\mathcal{H})^{\mathrm{op}}}$ with the sheaf topos $\mathrm{Sh}(\mathcal{V}(\mathcal{H}) \times (0,1)_L)$ to incorporate topological continuity.
• Define group actions via morphisms in the sheaf topos that act on spectra of transformed algebras, ensuring continuity through the sheaf structure.
• Utilize the etalé bundle representation of sheaves to model the pullback of sections under group-induced maps, ensuring local homeomorphism compatibility.
• Construct the tensor product $A(-) \otimes_X \mathrm{Hom}_Y(y, f(-))$ to represent the pullback of sheaves under group-induced maps, with equivalence classes encoding fiber data.
• Apply the notion of $f$-induced sheaf morphisms to ensure compatibility across overlapping open sets, preserving sheaf axioms under group transformations.
• Use the internal logic of the sheaf topos to define truth values of propositions under group actions, ensuring they transform covariantly and continuously.

Experimental results

Research questions

• RQ1 How can group actions be consistently formulated in topos quantum theory without resorting to twisted presheaves?
• RQ2 What topological structure is required to render group actions continuous within the topos framework?
• RQ3 How can unitary symmetries be represented geometrically in a topos-theoretic setting that preserves internal logic and truth values?
• RQ4 Can the sheaf topos formulation support both pure and mixed quantum states under group actions, with probabilities derived internally?
• RQ5 What is the role of the Lie group's geometry and its orbits on the context category $\mathcal{V}(\mathcal{H})$ in the topos formulation?

Key findings

• The sheaf topos $\mathrm{Sh}(\mathcal{V}(\mathcal{H}) \times (0,1)_L)$ supports continuous group actions by providing a topological structure absent in presheaves.
• Twisted presheaves are eliminated by replacing the context category with a product category that includes a unit interval, enabling continuous transformation of spectra.
• Group actions on propositions are now represented by continuous morphisms in the topos, ensuring covariant transformation of truth values.
• The pullback of sheaves via group-induced maps is isomorphic to the etalé bundle $f \circ p_A$, ensuring geometric consistency and local triviality.
• The internal logic of the sheaf topos allows truth values of propositions to transform continuously under group actions, supporting a propensity-based interpretation of probabilities.
• The formulation generalizes to all unitary symmetries and provides a geometrically coherent framework for quantization procedures related by unitary transformations.