geometric quantization
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In
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given
classical theory Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. 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. For example, the similarity between the Heisenberg equation in the Heisenberg picture of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and the Hamilton equation in classical physics should be built in.


Origins

One of the earliest attempts at a natural quantization was
Weyl quantization Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, proposed by
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
) with a real-valued function on classical
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
. The position and momentum in this phase space are mapped to the generators of the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
, and the Hilbert space appears as a
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
of the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
. In 1946, H. J. Groenewold considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions. The modern theory of geometric quantization was developed by
Bertram Kostant Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he gradua ...
and Jean-Marie Souriau in the 1970s. One of the motivations of the theory was to understand and generalize Kirillov's orbit method in representation theory.


Deformation quantization

More generally, this technique leads to
deformation quantization Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * De ...
, where the ★-product is taken to be a deformation of the algebra of functions on a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sym ...
or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory. For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2. (This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom.) As a mere representation change, however, Weyl's map underlies the alternate
phase-space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mo ...
of conventional quantum mechanics.


Geometric quantization

The geometric quantization procedure falls into the following three steps: prequantization, polarization, and metaplectic correction. Prequantization produces a natural Hilbert space together with a quantization procedure for observables that exactly transforms Poisson brackets on the classical side into commutators on the quantum side. Nevertheless, the prequantum Hilbert space is generally understood to be "too big". The idea is that one should then select a Poisson-commuting set of ''n'' variables on the 2''n''-dimensional phase space and consider functions (or, more properly, sections) that depend only on these ''n'' variables. The ''n'' variables can be either real-valued, resulting in a position-style Hilbert space, or complex analytic, producing something like the
Segal–Bargmann space In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integr ...
. A polarization is a coordinate-independent description of such a choice of ''n'' Poisson-commuting functions. The metaplectic correction (also known as the half-form correction) is a technical modification of the above procedure that is necessary in the case of real polarizations and often convenient for complex polarizations.


Prequantization

Suppose (M,\omega) is a symplectic manifold with symplectic form \omega. Suppose at first that \omega is exact, meaning that there is a globally defined ''symplectic potential'' \theta with d\theta=\omega. We can consider the "prequantum Hilbert space" of square-integrable functions on M (with respect to the Liouville volume measure). For each smooth function f on M, we can define the Kostant–Souriau prequantum operator :Q(f):= - i\hbar\left( X_f +\frac\theta(X_f)\right) +f. where X_f is the Hamiltonian vector field associated to f. More generally, suppose (M,\omega) has the property that the integral of \omega/(2\pi\hbar) over any closed surface is an integer. Then we can construct a line bundle L with connection whose curvature 2-form is \omega/\hbar. In that case, the prequantum Hilbert space is the space of square-integrable sections of L, and we replace the formula for Q(f) above with :Q(f)= - i\hbar\nabla_+f, with \nabla the connection. The prequantum operators satisfy : (f),Q(g)i\hbar Q(\ ) for all smooth functions f and g. The construction of the preceding Hilbert space and the operators Q(f) is known as ''prequantization''.


Polarization

The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in M a Lagrangian subspace of the complexified tangent space of M. The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the subspace at each point. The ''quantum'' (as opposed to prequantum) Hilbert space is the space of sections of L that are covariantly constant in the direction of the polarization. The idea is that in the quantum Hilbert space, the sections should be functions of only n variables on the 2n-dimensional classical phase space. If f is a function for which the associated Hamiltonian flow preserves the polarization, then Q(f) will preserve the quantum Hilbert space. The assumption that the flow of f preserve the polarization is a strong one. Typically not very many functions will satisfy this assumption.


Half-form correction

The half-form correction—also known as the metaplectic correction—is a technical modification to the above procedure that is necessary in the case of real polarizations to obtain a nonzero quantum Hilbert space; it is also often useful in the complex case. The line bundle L is replaced by the tensor product of L with the square root of the canonical bundle of L. In the case of the vertical polarization, for example, instead of considering functions f(x) of x that are independent of p, one considers objects of the form f(x)\sqrt. The formula for Q(f) must then be supplemented by an additional Lie derivative term. In the case of a complex polarization on the plane, for example, the half-form correction allows the quantization of the harmonic oscillator to reproduce the standard quantum mechanical formula for the energies, (n+1/2)\hbar\omega, with the "+1/2" coming courtesy of the half-forms.


Poisson manifolds

Geometric quantization of Poisson manifolds and symplectic foliations also is developed. For instance, this is the case of partially integrable and superintegrable Hamiltonian systems and non-autonomous mechanics.


Example

In the case that the symplectic manifold is the 2-sphere, it can be realized as a
coadjoint orbit In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadjoint ...
in \mathfrak(2)^*. Assuming that the area of the sphere is an integer multiple of 2\pi\hbar, we can perform geometric quantization and the resulting Hilbert space carries an irreducible representation of SU(2). In the case that the area of the sphere is 2\pi\hbar, we obtain the two-dimensional spin-½ representation.


See also

* Half-form * Lagrangian foliation * Kirillov orbit method * Quantization commutes with reduction


Notes


Citations


Sources

* * * * * * * * *


External links

* William Ritter's review of Geometric Quantization presents a general framework for all problems in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and fits geometric quantization into this framework
John Baez's review of Geometric Quantization
by
John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appli ...
is short and pedagogical
Matthias Blau's primer on Geometric Quantization
one of the very few good primers (ps format only) * A. Echeverria-Enriquez, M. Munoz-Lecanda, N. Roman-Roy, Mathematical foundations of geometric quantization, . * G. Sardanashvily, Geometric quantization of symplectic foliations, {{arXiv, math/0110196. Functional analysis Mathematical quantization