The logical quantization of algebraic groups

In a previous paper we introduced a highly abstract framework within which the theory of manuals initiated by Foulis and Randall is to be developed. The framework enabled us in a subsequent paper to quantize the notion of a set. Following these lines, this paper is devoted to quantizing algebraic groups viewed from Grothendieck's functorial standpoint.     

The algebraic structure of BRST quantization

It is shown that a system of first-class bosonic constraints obeying a Lie algebra has associated with it a natural superalgebra. BRST quantization arises as a non-linear representation of this superalgebra. Two distinct superalgebras are explicitly constructed and their associated BRST quantizations presented. The first BRST quantization is the canonical one with the BRST charge a grassmannian scalar. The second is new — the BRST charge is a grassmannian spinor transforming in the fundamental representation of the appropriate superalgebra. Generalizations are briefly discussed.     

Universal metaplectic structures and geometric quantization

The recently developed concepts of generalized and universal spin structures are carried over from the orthogonal to the symplectic and unitary cases. It turns out that the analogues ofSpin ^c-structures, namely theMp ^c-structures andMU ^c-structures, are sufficient to avoid topological obstructions to their existence. It is indicated how this fact can be used in the geometric quantization of certain suitably polarized symplectic manifolds with arbitrary second Stiefel-Whitney class, where the usual Kostant-Souriau quantization scheme breaks down.Supported by the Studienstiftung des deutschen Volkes     

Weak quantization of Poisson structures

In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira.     

On the lie algebraic structure of a new field quantization

It is shown that a new field quatization algebra with 2n generators is isomorphic to the O(2n, 1) algebra. The SL(2, C) algebra is realized by the new quantization algebra with two generators only (n = 1).     

Remarks on unified field structures,spin structures and canonical quantization

Unified field structures are defined and reviewed. Under certain conditions these are shown to be dynamical systems. And quantizable dynamical systems are shown to be unified field structures with invariant Riemannian metric. Spin structure is reviewed and manifoldsM ^8k+4 with spin structure are shown to be symplectic.This research was supported in part by NSF GP-13375.     

Linear momentum quantization in periodic structures II

Fraunhofer interference of a single particle by a periodic array of scatterers, usually treated with a wave picture, can be fully explained on the basis of linear momentum quantization, as pointed out in a previous study by Van Vliet (1967) [4]. This analysis is now extended to scattering (or passing through slits) involving a finite number N of equidistantly spaced entities comprising the interferometer. The usual intensity probability distribution for W(sinθ) is obtained, noting that total momentum is conserved (as in the Compton effect), while the interferometer is treated as a quantum object—rather than a classical measuring apparatus, as perceived in the Copenhagen interpretation. Various aspects of the ‘orthodox view’ are examined and renounced.     

Local and global algebraic structures in general relativity

The extent to which the well-known pointwise algebraic canonical forms used for the energy-momentum tensor, the Weyl tensor, etc., can be regarded as smooth relations over some open subset of (possibly the whole of) space-time is investigated.     

Translation-invariant quantizations and algebraic structures on phase space

A class of quantizations, including that of Weyl, called translation-invariant is defined and the phase space formulations of quantum mechanics arising from such quantizations are described. It is then shown that these formulations are co-extensive with non-commutative translation-invariant involutive associative algebraic structures in the linear space of complex polynomials on phase space, in which polynomials with real coefficients are self-adjoint.     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

Relational symplectic groupoid quantization for constant poisson structures

As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results are also addressed. In particular, the paper includes an extension to space–times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a “differential” version of the classical and quantum master equations. The quantization constructed in this paper induces Kontsevich’s deformation quantization on the underlying Poisson manifold, i.e., the Moyal product, which is known in full details. This allows focussing on the BV-BFV technology and testing it. For the inexperienced reader, this is also a practical and reasonably simple way to learn it.     

On algebraic structures in supersymmetric principal chiral model

Using the Poisson current algebra of the supersymmetric principal chiral model, we develop the algebraic canonical structure of the model by evaluating the fundamental Poisson bracket of the Lax matrices that fits into the r–s matrix formalism of non-ultralocal integrable models. The fundamental Poisson bracket has been used to compute the Poisson bracket algebra of the monodromy matrix that gives the conserved quantities in involution. PACS 11.30.Pb; 02.30.Ik     

Deformation theory and quantization. I. Deformations of symplectic structures

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C^∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C^∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.     

From Wigner-Inönü group contraction to contractions of algebraic structures

The development of the notion of group contraction first introduced by Inönü and Wigner in 1953 is briefly reviewed. The fundamental role of the idea of degenerate transformations is stressed. The significance of contractions of algebraic structures for exactly solvable problems of mathematical physics is noticed.     

On the gate capacitance of MOS structures of kane-type semiconductors under magnetic quantization

An attempt is made to formulate the gate capacitance of MOS structures of Kane-type semiconductors under magnetic quantization, without any approximations of weak or strong electric field limits, on the basis of the fourth-order effective mass theory and taking into account the interactions of the conduction, light-hole, heavy-hole, and split-off bands. It is found, taking n-channel Hg_1–x Cd _x Te as an example, that the gate capacitance exhibits spiky oscillations with changing magnetic field, which is in qualitative agreement with experimental observations, reported elsewhere, in MOS structures of the same semiconductor. The corresponding results for n-channel inversion layers on parabolic semiconductors are also obtained from the expressions derived.     

Equivalence of Dirac quantization and Schwinger's action principle quantization

We show that the method of Dirac quantization is equivalent to Schwinger's action principle quantization. The relation between the Lagrange undetermined multipliers in Schwinger's method and Dirac's constraint bracket matrix is established and it is explicitly shown that the two methods yield identical (anti)commutators. This is demonstrated in the non-trivial example of supersymmetric quantum mechanics in superspace.     

Geometry and quantization

Gravity is treated geometrically in terms of nonlinear realizations ofGL(4, ) with particular reference to almost complex structures. This approach is used to carry out a Bargmann-Segal type quantization of space-time via the vector and spinor structures of the tangent space that formulates the theory of measurement as a quantum theory quantized in terms of a basic unit of length that appears in a new uncertainty relation. The theory is also used to discuss the gauge conditions for quantum gravity and the Kostant theory of quantization applied using a line bundle with structure groupGL(2, )/SL(2, ).