Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

Quantum physics; Mathematical physics; Matrix theory; Algebr

Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories

Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamiolkowski isomorphism, one and the same --- namely the homogeneous self-dual cones of positive semi-definite linear endomorphisms on finite dimensional complex Hilbert spaces. From the perspective of category theory, these cones are the sets of morphisms in finite dimensional quantum theory as a dagger compact closed category. Understanding the intricate geometry of these cones and charting the wider landscape for their host category is imperative for foundational physics. In Part I of this thesis, we study the shape of finite dimensional quantum theory in terms of quantum information. We introduce novel geometric structures inscribed within quantum cones: conical t-designs. Conical t-designs are a natural, strictly inclusive generalization of complex projective t-designs. We prove that symmetric informationally complete measurements of arbitrary rank (SIMs), and full sets of mutually unbiased measurements of arbitrary rank (MUMs) are conical 2-designs. SIMs and MUMs correspond to highly symmetric polytopes within the Bloch body. The same holds for the entire class of homogeneous conical 2-designs; moreover, we establish necessary and sufficient conditions for a Bloch polytope to represent a homogeneous conical 2-design. Furthermore, we show that infinite families of such designs exist in all finite dimensions. It turns out that conical 2-designs are naturally adapted to a geometric description of bipartite entanglement. We prove that a quantum measurement is a conical 2-design if and only if there exists a (regular) entanglement monotone whose restriction to pure states is a function of the norm of the probability vector over the outcomes of the bipartite measurement formed from its tensor products. In that case the concurrence is such a monotone. In addition to monotones, we formulate entanglement witnesses in terms of geometric conditions on the aforementioned conical 2-design probabilities. In Part II of this thesis, we move beyond quantum theory within the vein of Euclidean Jordan algebras (EJAs). In light of the Koecher-Vinberg theorem, the positive cones of EJAs are the only homogeneous self-dual cones to be found in a finite dimensional setting. We consider physical theories based on EJAs subject to nonsignaling axioms regarding their compositional structure. We prove that any such Jordanic composite is a Jordan ideal of Hanche-Olsen's universal tensor product. Consequently, no Jordanic composite exists having the exceptional Jordan algebra as a direct summand, nor does any such composite exist if either factor is exceptional. So we focus on special EJAs of self-adjoint matrices over the real, complex, and quaternionic division rings. We demonstrate that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded EJAs, having fewer objects but more morphisms, that is dagger compact closed. This category unifies finite dimensional real, complex and quaternionic quantum theories, except that the composite of two complex quantum systems comes with an extra classical bit. Our notion of composite requires neither tomographic locality, nor preservation of purity under monoidal products. The categories we construct include examples in which both of these conditions fail. Our unification cannot be extended to include any finite dimensional spin factors (save the rebit, qubit, and quabit) without destroying compact closure

Foundations of Mechanics, Second Edition

Preface to the Second Edition. Since the first edition of this book appeared in 1967, there has been a great deal of activity in the field of symplectic geometry and Hamiltonian systems. In addition to the recent textbooks of Arnold, Arnold-Avez, Godbillon, Guillemin-Sternberg, Siegel-Moser, and Souriau, there have been many research articles published. Two good collections are "Symposia Mathematica," vol. XIV, and "Géométrie Symplectique el Physique Mathématique," CNRS, Colloque Internationaux, no. 237. There are also important survey articles, such as Weinstein [1977b]. The text and bibliography contain many of the important new references we are aware of. We have continued to find the classic works, especially Whittaker [1959], invaluable. The basic audience for the book remains the same: mathematicians, physicists, and engineers interested in geometrical methods in mechanics, assuming a background in calculus, linear algebra, some classical analysis, and point set topology. We include most of the basic results in manifold theory, as well as some key facts from point set topology and Lie group theory. Other things used without proof are clearly noted. We have updated the material on symmetry groups and qualitative theory, added new sections on the rigid body, topology and mechanics, and quantization, and other topics, and have made numerous corrections and additions. In fact, some of the results in this edition are new. We have made two major changes in notation: we now use f^* for pull-back (the first edition used f[sub]*), in accordance with standard usage, and have adopted the "Bourbaki" convention for wedge product. The latter eliminates many annoying factors of 2. A. N. Kolmogorov's address at the 1954 International Congress of Mathematicians marked an important historical point in the development of the theory, and is reproduced as an appendix. The work of Kolmogorov, Arnold, and Moser and its application to Laplace's question of stability of the solar system remains one of the goals of the exposition. For complete details of all tbe theorems needed in this direction, outside references will have to be consulted, such as Siegel-Moser [1971] and Moser [1973a]. We are pleased to acknowledge valuable assistance from Paul Chernoff, Wlodek Tulczyjew, Morris Hirsh, Alan Weinstein, and our invaluable assistant authors, Richard Cushman and Tudor Ratiu, who all contributed some of their original material for incorporation into the text. Also, we are grateful to Ethan Akin, Kentaro Mikami, Judy Arms, Harold Naparst, Michael Buchner, Ed Nelson, Robert Cahn, Sheldon Newhouse, Emil Chorosoff, George Oster, André Deprit, Jean-Paul Penot, Bob Devaney, Joel Robbin, Hans Duistermaat, Clark Robinson, John Guckenheimer, David Rod, Martin Gutzwiller, William Satzer, Richard Hansen, Dieter Schmidt, Morris Kirsch, Mike Shub, Michael Hoffman, Steve Smale, Andrei Iacob, Rich Spencer, Robert Jantzen, Mike Spivak, Therese Langer, Dan Sunday, Ken Meyer, Floris Takens, [and] Randy Wohl for contributions, remarks, and corrections which we have included in this edition. Further, we express our gratitude to Chris Shaw, who made exceptional efforts to transfom our sketches into the graphics which illustrate the text, to Peter Coha for his assistance in organizing the Museum and Bibliography, and to Ruthie Cephas, Jody Hilbun, Marnie McElhiney, Ruth (Bionic Fingers) Suzuki, and Ikuko Workman for their superb typing job. Theoretical mechanics is an ever-expanding subject. We will appreciate comments from readers regarding new results and shortcomings in this edition. RALPH ABRAHAM, JERROLD E. MARSDEN</p