Moment-wavelet quantization: a first principles analysis of quantum mechanics through continuous wavelet transform theory

The space of polynomials is invariant under affine maps. This suggests that a moment based analysis can facilitate a first principles incorporation of continuous wavelet transform (CWT) theory into quantum mechanics. We show that this is indeed the case for a large class of Hamiltonians and mother wavelet functions. We establish the equivalence between moment quantization (MQ) and CWT. By so doing, we clearly demonstrate the inherent multiscale structure of MQ analysis with regards to determining the physical energies and corresponding wavefunctions.     

On the relativistic equation for particles with spinj in the 2[2j+1] — component formalism

The equation describing a relativistic particle with spinj and massm by a 2[2j+1] component wave function is derived using the method of boost transformations. The formalism developed in this paper allows us to find the wave functions satisfying the equation obtained and to construct the relativistically invariant quantities from these functions in an easy way. For the case of spin 3/2 the unitary equivalence with earlier results is demonstrated.     

Gaupta-Bleuler triplet for massless spin-3/2 field in de Sitter space

Physicists have been interested in quantization of spinor and vector free fields in 4-dimensional de Sitter space-time,in ambient space notation.The Gupta-Bleuler formalism has been extensively applied to the quantization of gauge invariant theories.The field equation of the massless spin-3/2 fields is gauge invariant in de Sitter space.In this paper,we study the quantization of massless spin-3/2 gauge fields in de Sitter space-time by the Gupta-Bleuler formalism.This triplet carries an indecomposable representation of the de Sitter group.     

From the BRST invariant Hamiltonian to the field-antifield formalism

We study the relation between the Lagrangian field-antifield formalism and the BRST invariant phase-space formulation of gauge theories. Starting from the Batalin-Fradkin-Vilkovisky unitarized action, we demonstrate in a deductive way the equivalence of the phase-space, and the Lagrangian field-antifield partition functions for the case of irreducible first rank theories.     

Conformal Orbifold Theories and Braided Crossed G-Categories

The Berezin-Toeplitz deformation quantization of an abelian variety is explicitly computed by the use of Theta-functions. An SL(2n,)-equivariant complex structure dependent equivalence E between the constant Moyal-Weyl product and this family of deformations is given. This equivalence is seen to be convergent on the dense subspace spanned by the pure phase functions. The Toeplitz operators associated to the equivalence E applied to a pure phase function produces a covariant constant section of the endomorphism bundle of the vector bundle of Theta-functions (for each level) over the moduli space of abelian varieties.Applying this to any holonomy function on the symplectic torus one obtains as the moduli space of U(1)-connections on a surface, we provide an explicit geometric construction of the abelian TQFT-operator associated to a simple closed curve on the surface. Using these TQFT-operators we prove an analog of asymptotic faithfulness (see [A1]) in this abelian case. Namely that the intersection of the kernels for the quantum representations is the Toreilli subgroup in this abelian case.Furthermore, we relate this construction to the deformation quantization of the moduli spaces of flat connections constructed in [AMR1] and [AMR2]. In particular we prove that this topologically defined *-product in this abelian case is the Moyal-Weyl product. Finally we combine all of this to give a geometric construction of the abelian TQFT operator associated to any link in the cylinder over the surface and we show the glueing axiom for these operators.This research was conducted in part for the Clay Mathematics Institute at University of California, Berkeley.This work was supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation     

An Explicit Formula for the Natural and Conformally Invariant Quantization

Lecomte (Prog Theor Phys Suppl 144:125–132, 2001) conjectured the existence of a natural and conformally invariant quantization. In Mathonet and Radoux (Existence of natural and conformally invariant quantizations of arbitrary symbols, math.DG 0811.3710), we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in Mathonet and Radoux and to tools already used in Radoux [Lett Math Phys 78(2):173–188, 2006] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in Radoux.     

BRST Cohomology and Phase Space Reduction in Deformation Quantization

In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a "quantum" Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is "sufficiently nice", e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.     

一种自对偶理论的规范不变形式及其量子化

对于一种新提出的自对偶场与规范场耦合的拉氏理论,本文给出相应的单上闭链,即Wess-Zumino项,构造了这种理论的规范不变的形式。利用正则量子化方法并通过选取适当的规范固定条件,证明了这规范不变的形式等价于原来的规范非不变的形式。此外,利用Batalin-Fradkin-Vilkovisky量子化方法,进一步指出这种等价性与规范固定条件的选择是无关的。 关键词：     

Classical geometry to quantum behavior correspondence in a virtual extra dimension

In the Lorentz invariant formalism of compact space–time dimensions the assumption of periodic boundary conditions represents a consistent semi-classical quantization condition for relativistic fields. In Dolce (2011) [18] we have shown, for instance, that the ordinary Feynman path integral is obtained from the interference between the classical paths with different winding numbers associated with the cyclic dynamics of the field solutions. By means of the boundary conditions, the kinematical information of interactions can be encoded on the relativistic geometrodynamics of the boundary, see Dolce (2012) [8]. Furthermore, such a purely four-dimensional theory is manifestly dual to an extra-dimensional field theory. The resulting correspondence between extra-dimensional geometrodynamics and ordinary quantum behavior can be interpreted in terms of AdS/CFT correspondence. By applying this approach to a simple Quark–Gluon–Plasma freeze-out model we obtain fundamental analogies with basic aspects of AdS/QCD phenomenology.     

Maxwell equations in conformal invariant electrodynamics

We consider a conformal invariant formulation of quantum electrodynamics. Conformal invariance is achieved with a specific mathematical construction based on the indecomposable representations of the conformal group associated with the electromagnetic potential and current. As a corollary of this construction modified expressions for the 3-point Green functions are obtained which both contain transverse parts. They make it possible to formulate a conformal invariant skeleton perturbation theory. It is also shown that the Euclidean Maxwell equations in conformal electrodynamics are manifestations of its kinematical structure: in the case of the 3-point Green functions these equations follow (up to constants) from the conformal invariance while in the case of higher Green functions they are equivalent to the equality of the kernels of the partial wave expansions. This is the manifestation of the mathematical fact of a (partial) equivalence of the representations associated with the potential, current and the field tensor.     

Gerbes and Duality

We describe a global approach to the study of duality transformations between antisymmetric fields with transitions and argue that the natural geometrical setting for the approach is that of gerbes; these objects are mathematical constructions generalizing U(1) bundles and are similarly classified by quantized charges. We address the duality maps in terms of the potentials rather than on their field strengths and show the quantum equivalence between dual theories which in turn allows a rigorous proof of a generalized Dirac quantization condition on the couplings. Our approach needs the introduction of an auxiliary form satisfying a global constraint which in the case of 1-form potentials coincides with the quantization of the magnetic flux. We apply our global approach to refine the proof of the duality equivalence between the d=11 supermembrane and d=10 IIA Dirichlet supermembrane.     

Quantization of Equivariant Vector Bundles

The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (A coadjoint orbit is a symplectic manifold with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail. This subject should not be confused with the quantization of the total space of a vector bundle such as the cotangent bundle. Received: 27 February 1998 / Accepted: 5 November 1998     

Note on duality on polarized symplectic manifolds

In this note we use some of the results of [3] to derive a general duality theorem for the cohomologies of foliated structures on a manifold. The result is applied to the special case of a symplectic manifold M on which the foliation is given by a complex polarization F in the sense of geometric quantization. We obtain, for example, a rigorous proof of the fact that for a smooth function ƒ on M whose Hamiltonian vector field leaves F invariant, the spectrum of the corresponding prequantization operator v(ƒ) coincides with the spectrum of its transpose, under the above duality. This latter result was obtained by Simms in [12] under certain hypotheses. Proofs of the validity of those hypotheses are now available in the literature; cf. [3] and [7].     

Non-equivalence of Faddeev-Jackiw method and Dirac-Bergmann algorithm and the modification of Faddeev-Jackiw method for keeping the equivalence

From the angle of the calculation of constraints, we compare the Faddeev-Jackiw method with Dirac-Bergmann algorithm, study the relations between the Faddeev-Jackiw constraints and Dirac constraints, and demonstrate that Faddeev-Jackiw method is not always equivalent to Dirac method. For some systems, under the assumption of no variables being eliminated in any step in Faddeev-Jackiw formalism, except for the Dirac primary constraints, we are possible to get some Dirac secondary constraints which do not appear in the corresponding Faddeev-Jackiw formalism, which will result in the contradiction between Faddeev-Jackiw quantization and Dirac quantization. At last, accordingly, we propose a modified Faddeev-Jackiw method which keeps the equivalence between Dirac-Bergmann algorithm and Faddeev-Jackiw method. However, one point must be stressed that the Faddeev-Jackiw method and quantization in this paper is these mentioned in [J. Barcelos-Neto, C. Wotzasek, Mod. Phys. Lett. A 7 (1992) 1737], not the initial Faddeev-Jackiw method mentioned in [L. Faddeev, R. Jackiw, Phys. Rev. Lett. 60 (1988) 1692], which is completely on basis of Darboux transformation, and must have the elimination of variables in every step of that, so it is reasonable that the constraints in this Faddeev-Jackiw method is fewer than the Dirac secondary constraints. Thus, we overcome the difficulty of the Non-equivalence of the Faddeev-Jackiw method and Dirac-Bergmann algorithm, and make the equivalence of the Faddeev-Jackiw method and Dirac-Bergmann algorithm restored.     

Manifestly-covariant canonical formalism of Poincaré gauge theories

We present a general framework for manifestly-covariant canonical formulation of Poincaré gauge theories. We construct a general class of action that is invariant under two kinds of BRS transformations—translation and internal Lorentz—and suitable for manifestly-covariant canonical quantization. This theory contains a great number of conserved quantities, which we investigate systematically. It is also pointed out that a canonical formulation of higher-derivative theories may be obtained as a limiting case in this framework.     

On the Higgs feature of gravity

The gauge gravitation theory, based on the equivalence principle besides the gauge principle, is formulated in the fibre bundle terms. The correlation between gauge geometry on spinor bundles describing Dirac fermion fields and space-time geometry on a tangent bundle is investigated. We show that field functions of fermion fields in presence of different gravitational fields are always written with respect to different reference frames. Therefore, the conventional quantization procedure is applicable to fermion fields only if gravitational field is fixed. Quantum gravitational fields violate the above mentioned correlation between two geometries.     

Nonassociative Strict Deformation Quantization of C*-Algebras and Nonassociative Torus Bundles

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with H-flux. In particular, the Octonions fit nicely into our theory.     

Schrödinger operators with symmetries

We consider the stationary Schrödinger operator H of a many-body system M with two-body rotation invariant interactions. The operator H is reduced with respect to the symmetries of permutation of identical particles, rotations and reflections, into a direct sum of operators H^τ̃, where τ̃ is an index of the irreducible representations of the symmetry group of the system.The spectra of the operators H^τ̃ were investigated in a series of papers of G.M. Zislin and A.G. Sigalov ([20], [21], [31]-[35]). In a recent paper [3] we have developed the spectral theory of these operators on the basis of the Weinberg equations.In the present work we complete and simplify this theory. In particular we treat in detail the case where the given system can be decomposed into two identical subsystems. For such systems there is a certain coupling between permutation and rotation-reflection symmetries, because a permutation, which interchanges the two subsystems, imposes a reflection on the relative position vector of the two centers of mass. This requires a modification of the theorem on essential spectrum as formulated in [3] in the case where such a division is not possible. The importance of this special case, as exemplified by diatomic molecules, fully justifies such a detailed treatment.This special case was treated by Zislin [34] under the assumption that the interactions are essentially multiplicative, relatively compact two-body interactions. Our method allows for general relatively compact two-body interactions, and can without difficulty be generalized to many-body interactions.Moreover, the method based on the Weinberg equation is suitable for a further analysis of the spectra of these operators.     

A local approach to dimensional reduction: I. General formalism

We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (“fields”) and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact.

In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a “non-canonical” reduction.

In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.