Semiclassical limit of geometric quantization book

Semiclassical analysis is the name reserved for areas where some asymptotic approximation of a quantum mechanical object is employed. Classical limit and quantization of hamiltonian systems. The book by brack and bhaduri 1 is an excellent introduction to the semiclassical methods. Geometric quantization, semiclassical limits, and formal. Greenberg, abraham klein, and ivaylo zlatev department of physics, university of pennsylvania, philadelphia, pennsylvania 191046396 chingteh li department of physics, national taiwan university, tapai, taiwan 10764, republic of china. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in pde. Sternberg spectral asymptotics in the semiclassical limit by m. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear pde.

Semiclassical and large quantum number limits of the. Scoppola, new approach to the semiclassical limit of quantum mechanics, imultiple tunnelings in one dimension, preprint 180, feb. Milonni air force weapons laboratory, kirtlandafr force base, new mexico 87117, u. The semiclassical contribution to the density of states 35. Gianfausto dellantonio lectures on the mathematics of quantum mechanics february 12, 2015 mathematical department, universita sapienza rome.

Semiclassical physics, or simply semiclassical refers to a theory in which one part of a system is described quantummechanically whereas the other is treated classically. Download pdf semiclassical analysis graduate studies in. The basic objects, concepts, and results are given. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic estimates of pairings of such eigenvectors. We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum. Deterministic chaos in classical and quantum mechanics 15 iv. It still remains a useful and classic introduction to the subject. Gutzwillers monograph 2 is an advanced introduction focusing on chaotic dynamics both in classical hamiltonian settings and in the semiclassical quantization. For example, external fields will be constant, or when changing will be classically described. An important example is the photoelectric effect, which is not proving em.

Lets say we have a symplectic manifold whose symplectic form is integral. The concept of phase space plays a decisive role in the study of the transition from classical to quantum physics. Momentum space quantization for compact lie groups. Maciej zworski, university of california, berkeley, berkeley, ca. The book is intended to be a graduate level text introducing readers. This is guilleminsternberg conjecture proved by meinrenkensjamaar 1999 for g compact and any in the symplectic case. Graduate studies in mathematics publication year 2012. You can also consider the opposite approximation, i. The physics of phase space nonlinear dynamics and chaos.

In general, it incorporates a development in powers of plancks constant, resulting in the classical physics of power 0, and the. We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of. Lectures on the geometry of quantization berkeley math. Semiclassical properties of geometric quantization with metaplectic correction article in communications in mathematical physics 2702 february 2006 with 22 reads how we measure reads.

The geometric approach to quantization was introduced by konstant and souriau more than 20 years ago. To date, this program has succeeded in unifying various older methods of quantizing. Semiclassical analysis american mathematical society. The semiclassical asymptotic in representation theory is the limit when all. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. Semiclassical quantization and spectral limits of h. Jain on the occasion of his 70th birthday abstract. Semiclassical approximation for the radial wave equation 12 iii. Semiclassical properties of geometric quantization with. Quantization and semiclassical analysis information. Geometric quantization and semiclassical limits of. Quantitative limiting absorption principle in the semiclassical limit. In a natural semiclassical limit of this model, the wavefunction localizes on the submanifold of vanishing loops. My main research interests are berezintoeplitz operators, that one encounters one studying the semiclassical limit of geometric quantization, and classical and quantum integrable systems, in particular semitoric systems.

Four examples of a semiclassical approximation include. Thus, by applying semiclassical considerations one hopes to get an insight into some of the structures of the full, elusive, theory. Semiclassical quantization rules near separatrices 625 it is to be noted that near u0 the energy levels are more dense than on the average. Here we will recall the basic textbook derivation of maslov. Second quantization further quantizes this electromagnetic field, so. Following earlier work, we view twodimensional nonlinear sigma model as single particle quantum mechanics in the free loop space of the target space.

This concerns the correct semiclassical limit behaviour of the operator quantization, the unique berezintoeplitz deformation quantization star product, covariant and contravariant berezin symbols. Y n in line with the principles of the kirillovkostant orbit method. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety v consists of the. Publications professor sir michael victor berry, frs. From heisenberg matrix mechanics to semiclassical quantization.

It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. A bynow standard example is geometric quantization is the construction of the segalbargmann space for cn using geometric quantization with a. We prove that its semiclassical limit does not depend on the choice of the complex structure. Published in chapter of the book physics, geometry, and topology, edited by h. Theorem cannas da silvakarshontolman 2000 regular value of. Laser resonators, the quantum theory of coherence and the quantization of electromagnetic fields are thoroughly covered and with some rigor. First, the classical and semiclassical adjectives are not quite synonyma. Here the crucial observation is that whenever an eigenvalue of. Semiclassical and adiabatic approximation in quantum. The large n limit of gauge theories, which is of importance in collective field theory and in the study of relation between gauge and string theories is formally very similar to semiclassical expansion, where the role of planck constant is played by 1 n 2 1n2. More recently another approach to the semiclassical quantization problem was presented by bogomolny 27. The book 29 treats further topics in symplectic geometry. The book 29 treats further topics in symplectic geometry and mechanics, with special attention to the role of symmetry groups, a topic pretty much ignored in the present notes.

In physics, canonical quantization is a procedure for quantizing a classical theory, while. In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. One would expect that the semiclassical expansion should be related to the tubular expansion of the theory around the submanifold. A potential well with a sufficiently strong attractive inversesquare tail supports an infinite dipole series of bound states, but the limit of infinite. On a semiclassical limit of loop space quantum mechanics. For example, the similarity between the heisenberg. This wkb method makes sense for a more general class of wave equations. For the formalization of this method in symplectic geometrygeometric quantization see at semiclassical state. For more extensive treatment of the pde aspects of the subject. Semiclassical means a treatment of a quantum system whose part is described classically, and another part quantum mechanically. As shown by perelomov and popov, traces of powers of a n and b n are scalar operators in a n this is the quantum analogue of. Charles, semiclassical properties of geometric quantization with metaplectic.

Consider, for instance, the case when x 0 is the absolute maximum of the potential ux such that ux semiclassical limits, and formal quantization. Diracs book details his popular rule of supplanting poisson brackets by. Woodhouse gives a method of building quantum hilbert spaces from classical systems. For instance in wave optics this yields the short wavelength limit of the geometrical optics approximation. A fairly comprehensive textbook with modern developments is. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety v consists. The orbit is periodic in the full phase space only if the initial and the. Geometric quantization 1 introduction the aim of the geometric quantization program is to describe a quantization procedure in terms of natural geometric structures. Semiclassical ideas have been very important in the history of quantum theory.

However, quantization of energy requires there to be a minimum volume of phase space surrounding each of the finite number of states in a specified energy range. Spectral asymptotics in the semiclassical limit by m. To overcome this difficulty one has to glue together various local solutions of the form. Fields may be classical, particle positions inside the fields quantum mechanical. This prompts replacement of the differential geometric concept of symplectic leaves with the algebraic concept of. In geometric quantization, this is split into two steps.

This prompts replacement of the differentialgeometric concept of symplectic leaves with the algebraic. Physics reports section c of physics letters 25, no. Kahler structure in the commutative limit of matrix geometry. Semiclassical quantisation rules for the dirac and pauli. Nambu mechanics, geometric quantization, kahler quantization, projective em bedding. Quantization of two types of multisymplectic manifolds scholarship.

In the context of geometric quantization the classical limit corresponds to taking high tensor. This is particularly the case in areas such as nonlinear dynamics and chaos, geometric quantization and the study of the various semiclassical theories, which are the setting of. Second, we will quantize semiclassical states in these symplectic manifolds. B n may be viewed as a natural quantization of x n.