Geometric Quantization Theory Research Articles

Prequantization, Geometric Quantization, Corrected Geometric Quantization

A comparison on some facts concerning the geometric quantization of symplectic manifolds is presented here. Criticism and improvements on the sophisticated theory of geometric quantization are presented touching briefly, all the “salient points of the theory”. The unfamiliar reader can consider this as a “soft” introduction to the topic.

Conversion of second-class constraints and resolving the zero-curvature conditions in the geometric quantization theory

In the approach to the geometric quantization, based on the conversion of second-class constraints, we resolve the respective non-linear zero curvature conditions for the extended symplectic potential. From the zero curvature conditions, we deduce new, linear, equations for the extended symplectic potential. Then we show that being the linear equations satisfied, their solution does certainly satisfy the non-linear zero curvature condition, as well. Finally, we give the functional resolution to the new linear equations, and then deduce the respective path integral representation. We do our consideration as to the general case of a phase superspace where both Boson and Fermion coordinates are present on equal footing.

Entangled and Sequential Quantum Protocols with Dephasing

Sequences of commuting quantum operators can be parallelized using entanglement. This transformation is behind some optimal quantum metrology protocols and recent results on quantum circuit complexity. We show that dephasing quantum maps in arbitrary dimension can also be parallelized. This implies that for general dephasing noise the protocol with entanglement is not more fragile than the corresponding sequential protocol and, conversely, the sequential protocol is not less effective than the entangled one. We derive this result using tensor networks. Furthermore, we only use transformations strictly valid within string diagrams in dagger compact closed categories. Therefore, they apply verbatim to other theories, such as geometric quantization and topological quantum field theory. This clarifies and characterizes to some extent the role of entanglement in general quantum theories.

Lagrangian tori in the projective plane

We extend the discussion of the homological mirror symmetry for toric manifolds to the more general case of monotonic symplectic manifolds with real polarizations. We claim that the Hori-Vafa conjecture, proved for toric Fano varieties, can be verified in a much wider context. Then the Bohr-Sommerfeld notion regarding the canonical class Lagrangian submanifold appears and plays an important role. A bridge is thus manifested between the geometric quantization and homological mirror symmetry programs for the projective plane in terms of its Lagrangian geometry. This allows using standard facts from the theory of geometric quantization to obtain some results in the framework of the theory of homological mirror symmetry.

Signature Quantization

We associate to the action of a compact Lie group G on a line bundle over a compact oriented even-dimensional manifold a virtual representation of G using a twisted version of the signature operator. We obtain analogues of various theorems in the more standard theory of geometric quantization. Some of these results were announced in Guillemin, Sternberg and Weitsman, 2003.

Signature quantization.

We associate to the action of a compact Lie group G on a line bundle over a compact oriented even-dimensional manifold a virtual representation of G using a twisted version of the signature operator. We obtain analogues of various theorems in the more standard theory of geometric quantization.

N -symplectic algebra of observables in covariant Lagrangian field theory

n -symplectic geometry on the adapted frame bundle λ:LπE→E of an n=(m+k)-dimensional fiber bundle π:E→M is used to set up an algebra of observables for covariant Lagrangian field theories. Using the principle bundle ρ:LπE→J1π we lift a Lagrangian L:J1π→R to a Lagrangian L≔ρ*(L):LπE→R, and then use L to define a “modified n-symplectic potential” θ̂L on LπE, the Cartan–Hamilton–Poincaré (CHP) Rn-valued 1-form. If the lifted Lagrangian is nonzero, then (LπE,dθ̂L) is an n-symplectic manifold. To characterize the observables we define a lifted Legendre transformation φL from LπE into LE. The image QL≔φL(LπE) is a submanifold of LE, and (QL,d(θ̂|QL)) is shown to be an n-symplectic manifold. We prove the theorem that θ̂L=φL*(θ|QL), and pull back the reduced canonical n-symplectic geometry on QL to LπE to define the algebras of observables on the n-symplectic manifold (LπE,dθ̂L). To find the reduced n-symplectic algebra on QL we set up the equations of n-symplectic reduction, and apply the general theory to the model of a k-tuple of massless scalar fields on Minkowski space–time. The formalism set forth in this paper lays the ground work for a geometric quantization theory of fields.

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let Π be the fundamental group of a closed orientable surface of genus g ≥ 1, and let R(Π, G)/G be the space of conjugacy classes of representations of Π into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map φ¯ on R(Π, G)/G and investigate whether the fibres of φ¯ are isotropic with respect to the natural symplectic structure on R(Π, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of φ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of φ¯ are also isotropic, but we give an example to show that in general they are not.

Symplectic Dirac Operators on Kähler Manifolds

The notion of symplectic spinors over a symplectic manifold was already introduced by B. Kostant in 1974 as a tool in his theory of geometric quantization. We observed that it is possible to define canonical symplectic Dirac operators acting on symplectic spinor fields in a similar way as one knows it from the Dirac operator on Riemannian manifolds. In this paper we study symplectic Dirac operators provided that the underlying manifold is a Kähler manifold with the Kähler form as symplectic structure.

Extensions of flat partial connections

It is proved that any flat partial connection ∇ F on a line bundle L over a manifold M, where F is a complex integrable subbundle of (TM) C , admits an extension to a connection ∇ on L. The kernel of the curvature form ω of ∇ is investigated. These results are applied to description of the Bohr–Sommerfeld subset in geometric quantization theory and quantization of the motion in the potential λ 2x 2+μ 2/x 2 .

Symplectic geometry on T ∗M derived from n-symplectic geometry on LM

We establish the relationship between n-symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M and canonical symplectic geometry on the cotangent bundle T ∗M . We show that all basic features of the canonical symplectic geometry of polynomial observables on T ∗ M are induced from the n-symplectic geometry on LM. Moreover, the C x bundle L x over T ∗M associated with geometric quantization theory is identified with a fiber bundle associated to the principal bundle of affine frames AM of the manifold M. Viewing AM as a principal R n bundle over LM we show that the connection on L x used in geometric quantization theory is induced from a canonical connection on AM that is constructed from the R n -valued n-symplectic potential. We then show that the connection preserving vector fields on L x that are related to linear polynomial observables on T ∗M are also induced from connection preserving vector fields on AM.

Second quantization, projective modules, and local gauge invariance

Bundles and bundle structures have gained wide currency in modern approaches to certain topics in quantum physics, significant applications appearing in connection with gauge theories (e.g., Atiyah and Jones, 1978), theories of geometric quantization (e.g., Kostant, 1970; Sniatycki, 1974), and in numerous other contexts. In this paper we argue that such structures can already be discerned in the most elementary notions of second quantization, albeit in disguised form. An examination of the methods traditionally used by physicists in dealing quantum mechanically with systems exhibiting an infinite number of degrees of freedom reveals, almost from the outset, the implicit use of module structures over algebras of functions (Section 2). By making these structures explicit and adapting some results of perturbation theory we arrive at an association between bare particles and finitely generated projective modules (Sections 3 and 4). In particular, rank one modules emerge naturally, for algebraic reasons, as the appropriate descriptors of bosons in this association. (This provides a possible setting for the development of standard geometric quantization theory.) As a first application of the formalism we show the existence of phononlike excitations in general many-fermion systems. When these ideas are further specialized (local) gauge theoretical notions arise in a natural way out of a consideration of the bundles obtained via Swan's theorem. These theories emerge moreover equipped with an interpretation linked directly to the geometrical entities associated with the underlying bundles. Thus for example in the line bundle (or rank one) case,

Geometric quantization and the Bogoliubov transformation

This paper treats some simple applications of geometric quantization theory to problems involving the interaction of quantum fields with ex­ternal classical potentials. The main new result is a representation of the transition amplitudes for a boson field in terms of functional integrals over the classical phase space (the space of solutions of the classical field equations). An analogous representation for fermion fields is also con­structed.

Group actions on quantum bundles

Suppose we are given a group G acting through canonical transformations on a symplectic manifold ( M, ω). If there is a quantum bundle over ( M, ω), a carrier for wave functions in the geometric quantization theory, then G acts infinitesimally on the bundle in a natural way. We give a necessary and sufficient condition for the infinitesimal G-action to integrate up to a global G-action. This is used for an investigation how the choice of the quantum bundle over ( M, ω) influences the integrability of the corresponding infinitesimal G-action. The relationship to group representations is briefly mentioned.

On geometric quantization and its connections with the Maslov theory

A modification of the Konstant-Souriau geometric quantization theory is proposed. It includes the case of a multi-odd-dimensional harmonic oscillator which is unquantizable in the Konstant-Souriau theory. The connections between geometric quantization and the Maslov theory of first term in the WKB asymptotic series are considered. Examples of a multidimensional harmonic oscillator, the three-dimensional Kepler problem and a relativistic mass-spin particle are computed.

Geometric quantization and quantum field theory in curved space-time

The Kostant-Souriau geometric quantization theory is applied to the problem of constructing a generally covariant quantum field theory. The occupation number formalism for a scalar field is introduced as a semiclassical approximation which is valid in low curvature regions of space-time and which depends on making a particular choice of polarization in the classical phase space of a single massive particle. The application of the formalism to particle creation problems is outlined.

Geometric quantization and Hamilton-Jacobi theory

Geometric quantization and Hamilton-Jacobi theory