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Open Access Publications from the University of California

Scholarly Works (25 results)

  • Article
  • Peer Reviewed
UC Riverside Previously Published Works (2012)
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the 'three-fold way'. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly 'complex' representations), those that are self-dual thanks to a symmetric bilinear pairing (which are 'real', in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are 'quaternionic', in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds-real, complex and quaternionic-can be seen as Hilbert spaces of the other kinds, equipped with extra structure. © 2011 Springer Science+Business Media, LLC.
    • Article
    • Peer Reviewed
    UC Riverside Previously Published Works (2014)
    Maldacena and Susskind have proposed a correspondence between wormholes and entanglement, dubbed ER=EPR. We study this in the context of three-dimensional topological quantum field theory (TQFT), where we show that the formation of a wormhole is the same process as creating a particle-antiparticle pair. A key feature of the ER=EPR proposal is that certain apparently entangled degrees of freedom turn out to be the same. We name this phenomenon 'fake entanglement', and show how it arises in our TQFT model.
      • Article
      • Peer Reviewed
      UC Riverside Previously Published Works (2014)
      Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2.Its Lie algebra g2acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2:it acts as the symmetries of a ‘spinorial ball rolling on a projective plane’, again when the ratio of radii is 1:3. We explain this ratio in simple terms, use the dot product and cross product of split octonions to describe the G2incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.
        Cover page: G 2 \mathrm {G}_2 and the rolling ball
        • Article
        • Peer Reviewed
        UC Riverside Previously Published Works (2013)
        Noether's theorem links the symmetries of a quantum system with its conserved quantities, and is a cornerstone of quantum mechanics. Here we prove a version of Noether's theorem for Markov processes. In quantum mechanics, an observable commutes with the Hamiltonian if and only if its expected value remains constant in time for every state. For Markov processes that no longer hold, but an observable commutes with the Hamiltonian if and only if both its expected value and standard deviation are constant in time for every state. © 2013 American Institute of Physics.
          • Article
          • Peer Reviewed
          UC Riverside Previously Published Works (2014)
          We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence. We use a number of interesting categories related to probability theory. In particular, we consider a category FinStat where an object is a Finite set equipped with a probability distribution, while a morphism is a measure-preserving function f: X → Y together with a stochastic right inverse s: Y → X. The function f can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement. Given this data we can define the entropy of the probability distribution on X relative to the prior' given by pushing the probability distribution on Y forwards along s. We say that s is optimal' if these distributions agree. We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid [0;∞] which vanishes when s is optimal must be a scalar multiple of this relative entropy. Our proof is independent of all earlier characterizations, but inspired by the work of Petz. © John C. Baez and Tobias Fritz, 2014.
            Cover page: A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY
            • Thesis
            • Peer Reviewed
            UC Riverside Electronic Theses and Dissertations (2021)

            This thesis aims to develop a compositional theory for the operational semantics of networks. The networks considered are described by either internal or enriched graphs. In the internal case we focus on $\mathsf{Q}$-nets, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$. $\mathsf{Q}$-nets include many known variants of Petri nets including pre-nets, integer nets, elementary net systems, and bounded nets. In the enriched case we focus on graphs enriched in a quantale $R$ regarded as matrices with entries in $R$. These $R$-matrices represent distance networks, Markov processes, capacity networks, non-deterministic finite automata, simple graphs, and more. The operational semantics of $\mathsf{Q}$-nets is constructed as an adjunction between $\mathsf{Q}$-nets and categories internal to the category of models of $\mathsf{Q}$. The left adjoint of this adjunction sends a $\mathsf{Q}$-net $P$ to an internal category $F_{\mathsf{Q}}(P)$ whose morphisms represent all possible firing sequences in $P$. Similarly, the operational semantics of $R$-matrices is constructed as an adjunction between $R$-matrices and categories enriched in $R$. The left adjoint of this adjunction sends an $R$-matrix $M$ to the $R$-category $F_R(M)$ whose hom-objects are solutions of the algebraic path problem: a generalization of the shortest path problem to graphs weighted in $R$. For both $\mathsf{Q}$-nets and $R$-matrices we use the theory of structured cospans to study the compositionality of the above operational semantics. For each type of network we construct a double category whose morphisms are ``open networks", i.e.\ networks with certain vertices designated as input or output. The operational semantics gives a double functor from a double category of open networks to a double category of open enriched or internal categories. These double functors give a compositional framework for computing the operational semantics of $\mathsf{Q}$-nets and $R$-matrices: their functoriality and coherence give relationships between the operational semantics of a network and the operational semantics of the smaller networks from which it is composed. We introduce the black-boxing of an open network, a profunctor describing the externally observable behavior of an open network. We introduce a class of open networks called ``functional open networks" for which black-boxing preserves composition.

              Cover page: Composing Behaviors of NetworksCreative Commons 'BY-NC' version 4.0 license
              • Article
              • Peer Reviewed
              UC Riverside Previously Published Works (2015)
              Control theory uses 'signal-flow diagrams' to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the symmetric monoidal category FinVectk of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVectk in terms of the generators used in signal-flow diagrams. A broader class of signal-flow diagrams also includes `caps' and `cups' to model feedback. We show these diagrams can be seen as string diagrams for the symmetric monoidal category FinRelk, where objects are still finite-dimensional vector spaces but the morphisms are linear relations. We also give a presentation for FinRelk. The relations say, among other things, that the 1-dimensional vector space k has two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid. This sort of structure, but with tensor product replacing direct sum, is familiar from the `ZX-calculus' obeyed by a finite-dimensional Hilbert space with two mutually unbiased bases.
                Cover page: CATEGORIES IN CONTROL
                • Thesis
                • Peer Reviewed
                UC Riverside Electronic Theses and Dissertations (2010)

                This thesis develops the foundations of the program of groupoidification and presents an application of this program --- the Fundamental Theorem of Hecke Operators. In stating this theorem, we develop a theory of enriched bicategories and construct the Hecke bicategory --- a categorification of the intertwining operators between permutation representations of a finite group. As an immediate corollary, we obtain a categorification of the Iwahori-Hecke algebra, which leads to solutions of the Zamolodchikov tetrahedron equation. Such solutions are a positive step towards invariants of 2-tangles in 4-dimensional space and constructions of higher-categories with braided structures.

                  Cover page: Foundations of Categorified Representation Theory
                  • Article
                  • Peer Reviewed
                  UC Riverside Previously Published Works (2011)
                  In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group', and the Lie 3-superalgebra that governs 11-dimensional supergravity. © 2010 The Author(s).
                    • Article
                    • Peer Reviewed
                    UC Riverside Previously Published Works (2011)
                    Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincaré Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincaré Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an (n + 1)-cocycle on a Lie superalgebra yields a "Lie n-superalgebra": that is, roughly speaking, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincaré superalgebra in dimensions 3, 4, 6 and 10, and Lie 3-superalgebras extending the Poincaré superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati, Schreiber and Stasheff's work on higher gauge theory, Lie 2-superalgebra connections describe the parallel transport of strings, while Lie 3-superalgebra connections describe the parallel transport of 2-branes. Moreover, in the octonionic case, these connections concisely summarize the fields appearing in 10- and 11-dimensional supergravity. © 2011 International Press.
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