Smooth Manifolds And Observables

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Peter Walters. Steven Roman. Joseph H.

Manifolds 3.1 : Partitions of Unity on Smooth Manifolds

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Mathematisch-Naturwissenschaftliche Fakultät - Jahrgang 2018

Bestselling Series. Harry Potter. Popular Features. New Releases. Smooth Manifolds and Observables. Description This book gives an introduction to fiber spaces and differential operators on smooth manifolds. Over the last 20 years, the authors developed an algebraic approach to the subject and they explain in this book why differential calculus on manifolds can be considered as an aspect of commutative algebra.

This new approach is based on the fundamental notion of observable which is used by physicists and will further the understanding of the mathematics underlying quantum field theory. Product details Format Paperback pages Dimensions x x Other books in this series. Introduction to Smooth Manifolds John M. Add to basket. Algebraic Geometry Robin Hartshorne. Mathematical Methods of Classical Mechanics V. Algebra Thomas W.

Commutative Algebra David Eisenbud. Representation Theory William Fulton. Introduction to Riemannian Manifolds John M. Graph Theory Adrian Bondy. Differential Geometry Loring W. Introduction to Topological Manifolds John M. Advanced Linear Algebra Steven Roman. Review Text From the reviews: "Main themes of the book are manifolds, fibre bundles and differential operators acting on sections of vector bundles.

A classical treatment of these topics starts with a coordinate description of a manifold M Thus, if a vector field is both Hamiltonian and Killing, then it preserves both parts of the inner product, and is thus implemented by a set of unitary transformations. Because of the Killing condition, these transformations form a one-parameter group7 , 7 This fact, as well as a detailed discussion of Killing vector fields is given by Wald , Appendix C.

The converse may be argued similarly. Therefore, what identifies quan- tum observables corresponding to self-adjoint operators from among the many functions on phase space is that they generate a Hamiltonian vector field that is Killing. Let me summarize the observations of this section that are most essential for the next. The manifold P is connected and geodesically complete with respect to the metric gab. Viewing quantum theory as a special class of dynamical systems, this corresponds to the following elementary fact. Lemma 3. If F a is a Killing field on a connected and geodesically complete Riemannian manifold, then F a is complete.

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But the Hopf-Rinow theorem guarantees that every Cauchy sequence on a geodesically complete Riemannian manifold converges Lee , Theorem 6. By Lemma 1, P is connected and geodesically complete with respect to gab. Thus, restricting a dynamical system to the observables of orthodox quantum theory precludes the possibility of a time observable.

Time observables in extensions of quantum theory A half-bounded Hamiltonian precludes timely observables among the im- poverished class of functions characterizing quantum observables. But we have seen that timely functions can still appear in more general dynamical systems. This suggests that the lack of time observables is a somewhat fragile aspect of quantum theory, which can be regained in modest extensions of the theory.

This assures that A has a real spectrum. A symmetric operator that does not extend to any self-adjoint operator is called maximal symmetric.

However, Cooper , showed that maximal symmetric operators lack this assurance9 , instead allowing for the analogue of an incomplete vector field. In particular, call an operator U an isometry if it is unitary on a closed 8 In finite-dimensional Hilbert spaces, an operator is symmetric if and only if it is self-adjoint in which case it is often called Hermitian. But this equivalence fails for infinite-dimensional Hilbert spaces.

Smooth Manifolds and Observables - Jet Nestruev - Google книги

This feature of symmetric operators lifts the requirement of a complete vector field, thereby opening the door for timely observables. The Hamilto- nian flow it generates corresponds to a set of operators that preserve the Hilbert space inner product, and therefore preserve the metric. But since these isome- tries do not form a group defined for all real parameter values, the corresponding Hamiltonian vector field is not Killing An example is provided by the positive-momentum free particle discussed in Section 3. Similar examples have been studied by Busch et al.

We now have a more general perspective on why such time observables are possible. Weinberg functions A second, more general route to time observables begins with the geomet- ric perspective. Suppose we are a even more inclusive with our observables, by dropping not only the requirement that an observable function generate a Killing field as with self-adjoint operators , but also that it even generate a set of isometries as with symmetric operators. Suppose we allow any smooth function that generates a vector fields that covers the entire phase space.

A , who show that they characterize a class of extensions of quantum theory proposed by Weinberg For this reason, they refer to these functions as Weinberg functions. Unlike orthodox quantum observables, the Weinberg functions can quite easily be timely. To illustrate, consider the following example, which is due to John D. Cooper showed that this set forms a strongly continuous semigroup for which inverses are not assured of isometries for either the positive or the negative values of s.

However, it is smooth and defined on the entire manifold, and therefore counts as a Weinberg function according to the definition of Ashtekar and Schilling. Once we have specified what it means to be a timely observable, the language of dynamical systems provides a general perspective from which to discuss their existence.

We began by observing a local sense in which timely ob- servables are guaranteed to exist in all dynamical systems. We then showed that a global timely observable can only exist if it generates an incomplete Hamil- tonian vector field. This property is not possible among the observables among the observables of quantum theory, when the Hamiltonian is half-bounded. From this perspective, the quantum prohibition on time observables does not appear to be such a permanent feature of the description of the physical world.

Nor- ton, Nicholas Teh, and Jos Uffink for comments that led to improvements, and especially to Thomas Pashby for many invaluable discussions. Dynamical time versus system time in quantum mechanics, Chinese Physics B 21 7 : Ashtekar, A. Geometrical formulation of quantum mechanics, in A.

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Smooth Manifolds and Observables

Harvey ed. Ballmann, W. Blank, J. Busch, P. Time observables in quantum theory, Physics Letters A : — Butterfield, J. On Time in Quantum Physics, in H. Dyke and A. Cooper, J. One-parameter semigroups of isometric operators in hilbert space, Annals of Mathematics 48 4 : — Lee, J. Marsden, J.