Categories for the practising physicist (0905.3010v2)

Abstract: In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are particularly relevant for quantum foundations and for quantum informatics. Special attention is given to the category which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, and the tensor product as its monoidal structure (FdHilb). We also provide a detailed discussion of the category which has sets as objects, relations as morphisms, and the cartesian product as its monoidal structure (Rel), and thirdly, categories with manifolds as objects and cobordisms between these as morphisms (2Cob). While sets, Hilbert spaces and manifolds do not share any non-trivial common structure, these three categories are in fact structurally very similar. Shared features are diagrammatic calculus, compact closed structure and particular kinds of internal comonoids which play an important role in each of them. The categories FdHilb and Rel moreover admit a categorical matrix calculus. Together these features guide us towards topological quantum field theories. We also discuss posetal categories, how group representations are in fact categorical constructs, and what strictification and coherence of monoidal categories is all about. In our attempt to complement the existing literature we omitted some very basic topics. For these we refer the reader to other available sources.

• The paper presents category theory’s application through symmetric monoidal and compact closed categories to model quantum processes and entanglement.
• It details how the FdHilb and Rel categories formalize quantum protocols and facilitate a new perspective on parallel computation in physics.
• It demonstrates that diagrammatic reasoning improves the analysis of complex quantum interactions, advancing practical methodologies in quantum computation.

An Expert Overview of "Categories for the Practising Physicist" by Bob Coecke and Eric Oliver Paquette

The document titled "Categories for the Practising Physicist" by Bob Coecke and Eric Oliver Paquette is an exploration of category theory, specifically its application and interpretation in the physical sciences. Category theory has traditionally found its role in mathematics, shedding light on structures and revealing connections between seemingly disparate ideas. This paper uniquely tailors this mathematical structure to a physical context, particularly beneficial for quantum mechanics and field theories.

The authors focus heavily on monoidal categories, particularly symmetric monoidal categories, which are foundational in understanding complex interactions in physics, such as quantum entanglement and computation. A monoidal category is equipped with a tensor operation symbolic of parallel composition, while the symmetric nature implies an inherent commutativity, crucial for describing systems in which the order of operations doesn't affect the outcome. The paper postulates a physical interpretation of these mathematical constructs, emphasizing how the formalism not only captures traditional mathematical elegance but also aligns with physical intuitions.

Numerical and Theoretical Results

The document describes FdHilb, the category of finite-dimensional Hilbert spaces with linear maps as morphisms and the tensor product as the structural equation, asserting its fundamental role in formalizing quantum mechanics. This framework aligns with quantum protocols such as teleportation, where Hilbert spaces provide a rigorous language to describe quantum states and their transformations. Additionally, the category Rel, constituting sets as objects and relations as morphisms, provides a perspective that diverges from classical functional assignment, offering insights into parallel computation and information flow in physical systems.

The authors meticulously dissect monoidal structures to outline compact closed categories, pivotal when expressing processes such as quantum teleportation within a category of finite-dimensional vector spaces. Compactness herein allows for the representation of phenomena like self-duality and entanglement, underpinning a significant part of quantum computational theory.

Implications and Future Prospects

The exploration of category theory in physics implicates future advancements in quantum computation, where the ability to conceptualize transformations and entanglements abstractly but precisely could yield new computational paradigms. The graphical calculus introduced, which leverages intuitive diagrammatic reasoning, might vastly improve the accessibility of complex quantum manipulations, empowering physicists with a tool both rigorous and visually comprehensive.

The prospects extend to topological quantum field theories (TQFTs), as expressed in the section discussing cobordism categories; these ideas promise progress in string theory and quantum gravity, where the intersection of geometry and physics becomes unavoidable.

Conclusion

This paper is a monumental step towards marrying the abstraction of category theory with the intricate fabric of physical processes. By framing these mathematical concepts in a language accessible to physicists, Coecke and Paquette provide a bridge that unveils the conceptual lineages between mathematics and physics. These advances not only promise to deepen understanding in theoretical physics but also open frontiers in computational capabilities, effectively harnessing the structure of nature's most foundational laws.