`What is a Thing?': Topos Theory in the Foundations of Physics (0803.0417v1)

Abstract: The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question `What is a thing?''. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow $\breve{A}_\phi:\Si_\phi\map\R_\phi$ where $\Si_\phi$ and $\R_\phi$ are two special objects (thestate-object' and quantity-value object') in the appropriate topos, $\tau_\phi$. We discuss two different types of language that can be attached to a system, $S$. The first, $\PL{S}$, is a propositional language; the second, $\L{S}$, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of $\PL{S}$ we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we termdaseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf $\Sig$--the topos quantum analogue of a classical state space. The topos concerned is $\SetH{}$: the category of contravariant set-valued functors on the category (partially ordered set) $\V{}$ of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space $\Hi$.

• The paper explores using topos theory as a novel mathematical framework to reformulate foundational aspects of physics, particularly quantum theory.
• It proposes representing physical quantities and propositions via spectral presheaves and 'daseinisation' within a topos, offering an alternative to traditional set-theoretic approaches.
• This framework aims to address conceptual challenges, move away from relying on real numbers, and provide a basis for a coherent neo-realist interpretation of quantum mechanics.

Topos Theory in the Foundations of Physics

The paper authored by Andreas Döring and Chris Isham presents an exploratory framework in the foundations of physics. This theoretical investigation attempts to reformulate the conventional structure of physical theories, particularly quantum physics, using the rich mathematical apparatus of topos theory. The authors advocate for a fundamental revision of how we construct physical theories, aiming to enhance the representation of quantum propositions and to provide a novel perspective on realizing a form of realism within quantum theory.

Core Motivations and Approach

The authors start by acknowledging long-standing conceptual challenges in quantum gravity and quantum theory, focusing on issues that arise from the standard formalism's reliance on real and complex numbers. They argue that conventional quantum mechanics, when applied to quantum gravity, might impose unwarranted spatio-temporal assumptions. The paper suggests shifting from the use of real numbers as default values for physical quantities to a framework where space and time may have different and more abstract representations.

Topos Theory as a New Mathematical Framework

Central to this framework is the use of topos theory, a branch of category theory, which generalizes set theory and can be tailored to incorporate intuitionistic logic instead of classical Boolean logic. The paper proposes that physical systems can be associated with particular topoi, each providing its own internal logic and mathematical structure. This aligns with the contention that different systems should be studied within potentially diverse categorical contexts, offering mathematical flexibility and intuitively "quantum" domains.

Representation of Physical Quantities and Propositions

Physical quantities in this topos-theoretic framework are represented as arrows between objects in a topos, moving away from traditional set-theoretic perspectives. Two primary objects within the topos are the state object (Σ) and the quantity-value object (R). The paper particularly leverages the concept of spectral presheaves to represent self-adjoint operators in quantum mechanics. The notion of 'daseinisation’ is introduced to derive these representations, which approximates projections and provides a basis for assigning generalized truth values to quantum propositions.

Handling Propositions and Realism

Döring and Isham confront the question of realism in quantum theory through topos theory. The spectral presheaf, with its structure that does not support global elements (reflecting the Kochen-Specker theorem's objections to traditional valuations), plays a critical role. By representing quantum propositions with sub-objects of the spectral presheaf, they propose capturing a form of realism within this more flexible logical structure, which departs from the probabilistic interpretations characteristic of the Copenhagen interpretation.

Numerical and Categorical Details

The authors explore various mathematical constructs, such as the outer and inner daseinisations of operators, and a range of novel presheaves like de Groote presheaves. They demonstrate how, by employing such structures, the quantum theory can be framed within a new logic where truth objects in the topos context indirectly reflect states, but with more nuanced truth values reflective of the logical structure imposed by the topos.

Implications and Future Outlook

The paper proposes that this topos approach not only provides an alternative to the A-Categorical quantum framework but also lays a foundation for potentially reconciling quantum mechanics with quantum gravity's demanding theoretical landscape. By averting the intrinsic use of real and complex numbers, this work aims to develop a coherent neo-realist interpretation of quantum mechanics, offering a systematic method to revisit long-standing philosophical questions about measurement, observation, and the reality of quantum states.

Conclusion

While the paper remains high-level and theoretical, with a deep reliance on advanced mathematics, it sheds light on profound conceptual issues challenging the intersection of mathematics and physics. This exploratory work anticipates future development and practical insights that could arise from pursuing this line of foundational research—an endeavor that could bridge gaps in our understanding of physics at its most fundamental levels.