Reflections on Quanta, Symmetries, and Supersymmetries

verfasst von: V.S. Varadarajan

Verlag: Springer New York

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This is a collection of essays based on lectures that author has given on various occasions on foundation of quantum theory, symmetries and representation theory, and the quantum theory of the superworld created by physicists. The lectures are linked by a unifying theme: how the quantum world and superworld appear under the lens of symmetry and supersymmetry. In the world of ultra-small times and distances such as the Planck length and Planck time, physicists believe no measurements are possible and so the structure of spacetime itself is an unknown that has to be first understood. There have been suggestions (Volovich hypothesis) that world geometry at such energy regimes is non-archimedian and some of the lectures explore the consequences of such a hypothesis. Ultimately, symmetries and supersymmetries are described by the representation of groups and supergroups. The author's interest in representation is a lifelong one and evolved slowly, and owes a great deal to conversations and discussions he had with George Mackey and Harish-Chandra. The book concludes with a retrospective look at these conversations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Prologue

Abstract

Some thoughts on reality and its description, as well as some personal recollections.

V. S. Varadarajan

Chapter 2. Quantum Algebra

Abstract

Quantum algebra was created by Dirac. Its evolution also bears the imprint of the genius of many great mathematicians and physicists such as Weyl, von Neumann, Schwinger, Moyal, Flato, and others. It has inspired developments in deformation theory, representation theory, quantum groups, and many other mathematical themes.

V. S. Varadarajan

Chapter 3. Probability in the quantum world

Abstract

Probability has been a part of quantum theory from its very beginnings. The original probabilistic interpretation of quantum mechanics was put forward by Max Born. However the idea that the understanding of Nature had to be statistical was unacceptable to a lot of people including Einstein, and there was substantial criticism of this aspect of quantum theory. Eventually, the ideas of von Neumann, Bell, Feynman, Mackey, Glea- son, and many others on the probabilistic aspects of quantum theory clarified the situation and answered the criticisms. These contributions have made the role of probability in the quantum world both far-reaching and profound.

V. S. Varadarajan

Chapter 4. Super geometry

Abstract

Supersymmetry was discovered by physicists around 1970. At first it was just a device for treating the bosonic and fermionc aspects of quantum statistical mechanics and quantum field theory on an equal footing. Eventually, due to the remarkable ideas and contributions of many physicists and mathematicians, supersymmetry came to be understood as the symmetry of a new kind of geometrical object, namely, a super manifold. The super manifolds are objects in super geometry, which is a deep generalization of conventional differential and algebraic geometry. The symmetries of super manifolds form super Lie groups.

V. S. Varadarajan

Chapter 5. Unitary representations of super Lie groups

Abstract

The concept of a unitary representation(UR) of a super Lie group is formulated via super Harish-Chandra pairs. For super semidirectproducts the classical Wigner-Mackey theory of little groups works perfectly in the supersymmetric setting, and leads to a description of all of their unitary irreducible representations (UIR). The Clifford structure of the representations and the concept of super multiplets all make sense in the general context of super semidirect products, which includes all cases studied by the physicists, and leads to many of their major predictions: multiplet structure (both for minimal and extended supersymmetry), and the famous susy partners.

V. S. Varadarajan

Chapter 6. Nonarchimedean physics

Abstract

There is a natural time scale, the Planck scale, that emerges when general relativity and quantum mechanics are both significant. No measurements are possible in regions of smaller sizes than the Planck units. In particular we cannot compare distances and times in sub-Planckian domains. In 1987 Igor Volovich proposed the bold hypothesis that in such domains the geometry of spacetime is nonarchimedean. It is then natural to ask what are the consequences of this hypothesis. We address this question in the Dirac mode.

V. S. Varadarajan

Chapter 7. Differential equations with irregular singularities

Abstract

The first global studies of differential equations with rational coefficients are those of Riemann on the hypergeometric equations. These are special cases of Fuchsian equations, or, equations with regular singularities. Their theory is essentially controlled by the monodromy action. The equations with irregular singularities tell a completely different story. Here the central fact is that formal solutions do not always converge. Their theory goes back to Fabry in 1885 who discovered the phenomenon of ramification, and the decisive developments came from Hukuhara, Levelt, Turrittin, and others. In more recent times, the ideas of Balser, Deligne, Malgrange, Ramis, Subiya, Babbitt and myself, and a host of others, have created a more modern view of irregular linear differential equations that relates them to themes in commutative algebra, linear algebraic groups, and algebraic geometry.

V. S. Varadarajan

Chapter 8. Mackey, Harish-Chandra, and representation theory

Abstract

Some personal reminiscences of Mackey and Harish-Chandra together with brief comments on their work.

V. S. Varadarajan

Titel: Reflections on Quanta, Symmetries, and Supersymmetries
verfasst von: V.S. Varadarajan
Verlag: Springer New York
Electronic ISBN: 978-1-4419-0667-0
Print ISBN: 978-1-4419-0666-3
DOI: https://doi.org/10.1007/978-1-4419-0667-0