Deformation Quantization Research Articles

Quantum phase space description of a cosmological minimal massive bigravity model

Bimetric gravity theories describes gravitational interactions in the presence of an extra spin-2 field. The Hassan-Rosen (HR) nonlinear massive minimal bigravity theory is a ghost-free bimetric theory formulated with respect a flat, dynamical reference metric. In this work the deformation quantization formalism is applied to a HR cosmological model in the minisuperspace. The quantization procedure is performed explicitly for quantum cosmology in the minisuperspace. The Friedmann-Lema\^itre-Robertson-Walker (FLRW) model with flat metrics is worked out and the computation of the Wigner functions for theHartle-Hawking, Vilenkin and Linde wavefunctions are done numerically and, in the Hartle-Hawking case, also analytically. From the stability analysis in the quantum minisuper phase space it is found an interpretation of the mass of graviton as an emergent cosmological constant and as a measure of the deviation of classical behavior of the Wigner functions. Also, from the Hartle-Hawking case, an interesting relation between the curvature and the mass of graviton in a cusp catastrophe surface is discussed

Weakly associative algebras, Poisson algebras and deformation quantization

Weakly associative algebras are a class of Lie-admissible algebras that generalize associative algebras and useful to extend the class of formal deformations of associative commutative algebras. Thus we obtain a new method of determining quantization deformations of Poisson algebras which generalizes the classical process. The algebraic study of weakly associative algebras is also interesting, in particular the homology of free algebras with one generator that coincides with the free flexible algebra with one generator.

The Roger–Yang skein algebra and the decorated Teichmüller space

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper, we consider surfaces with punctures which are not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang’s Poisson algebra homomorphism is injective, and the skein algebra has no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.

Categorical Saito theory, I: A comparison result

In this paper, we present an explicit cyclic minimal A∞ model for the category of matrix factorizations MF(W) of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this idea further, we use the Tsygan formality map to obtain a comparison theorem that the categorical Variation of Semi-infinite Hodge Structure of MF(W) is isomorphic to Saito's original geometric construction in primitive form theory.

Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd [Formula: see text] describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number [Formula: see text]. In the limit [Formula: see text], Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

Deformation quantization on the cotangent bundle of a Lie group

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Fréchet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved, and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.

BMS algebras in 4 and 3 dimensions, their quantum deformations and duals

BMS symmetry is a symmetry of asymptotically flat spacetimes in vicinity of the null boundary of spacetime and it is expected to play a fundamental role in physics. It is interesting therefore to investigate the structures and properties of quantum deformations of these symmetries, which are expected to shed some light on symmetries of quantum spacetime. In this paper we discuss the structure of the algebra of extended BMS symmetries in 3 and 4 spacetime dimensions, realizing that these algebras contain an infinite number of distinct Poincaré subalgebras, a fact that has previously been noted in the 3 dimensional case only. Then we use these subalgebras to construct an infinite number of different Hopf algebras being quantum deformations of the BMS algebras. We also discuss different types of twist-deformations and the dual Hopf algebras, which could be interpreted as noncommutative, extended quantum spacetimes.

Confronting dual models of the strong interaction

Studies of the mathematical properties of Regge-pole and dual amplitudes are important both for their applications in high energy phenomenology and in their generalizations to strings, superstrings, branes, and other theoretical developments. In this paper, we investigate the similarities and differences between two classes of dual amplitudes: one with Mandelstam analyticity (DAMA) and another one with logarithmic trajectories (Dual-log). By using quantum (q-) deformations, new features of Dual-log amplitude are unveiled, in particular those concerning its asymptotic behavior and the spectrum of resonances. The two classes of dual amplitudes are compared in various kinematic regions: at fixed transferred momenta asymptotic, fixed angle asymptotic, and in the resonance region.

L∞ - derivations and the argument shift method for deformation quantization algebras

The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with respect to the Poisson bracket. In this notice we give an analogous construction, which gives one a way to create commutative subalgebras of a deformed algebra from its center (which is as it is well known describable in the terms of the center of the Poisson algebra) and an L∞-differentiation of the algebra of Hochschild cochains, verifying some additional conditions with respect to the Poisson structure.

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi–Civita connection of the Killing metric on G. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $$G = {\mathrm {SU}}(2)$$ the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the—in general singular—symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

On a Connection Used in Deformation Quantization

Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on the cotangent bundle of a Riemannian manifold is obtained by symplectification of the complete lift of the corresponding Levi-Civita connection, in the sense of Yano and Patterson. In terms of local coordinates, this has already been shown by Plebanski, Przanowski and Turrubiates.

Deformation quantization with minimal length

Deformation quantization with minimal length

Group actions in deformation quantization

This set of notes corresponds to a mini-course given in September 2018 in Będlewo; it does not contain any new result; it complements—with intersection—the introduction to formal deformation quantization and group actions published in 2017, correspond

Liouville quantum gravity \u2014 holography, JT and matrices

We study two-dimensional Liouville gravity and minimal string theory on spaces with fixed length boundaries. We find explicit formulas describing the gravitational dressing of bulk and boundary correlators in the disk. Their structure has a striking resemblance with observables in 2d BF (plus a boundary term), associated to a quantum deformation of SL(2, ℝ), a connection we develop in some detail. For the case of the (2, p) minimal string theory, we compare and match the results from the continuum approach with a matrix model calculation, and verify that in the large p limit the correlators match with Jackiw-Teitelboim gravity. We consider multi-boundary amplitudes that we write in terms of gluing bulk one-point functions using a quantum deformation of the Weil-Petersson volumes and gluing measures. Generating functions for genus zero Weil-Petersson volumes are derived, taking the large p limit. Finally, we present preliminary evidence that the bulk theory can be interpreted as a 2d dilaton gravity model with a sinh Φ dilaton potential.

The classical limit of mean-field quantum spin systems

The theory of strict deformation quantization of the two-sphere S2⊂R3 is used to prove the existence of the classical limit of mean-field quantum spin chains, whose ensuing Hamiltonians are denoted by HN, where N indicates the number of sites. Indeed, since the fibers A1/N=MN+1(C) and A0 = C(S2) form a continuous bundle of C*-algebras over the base space I={0}∪1/N*⊂[0,1], one can define a strict deformation quantization of A0 where quantization is specified by certain quantization maps Q1/N:Ã0→A1/N, with Ã0 being a dense Poisson subalgebra of A0. Given now a sequence of such HN, we show that under some assumptions, a sequence of eigenvectors ψN of HN has a classical limit in the sense that ω0(f) ≔ limN→∞⟨ψN, Q1/N(f)ψN⟩ exists as a state on A0 given by ω0(f)=1n∑i=1nf(Ωi), where n is some natural number. We give an application regarding spontaneous symmetry breaking, and moreover, we show that the spectrum of such a mean-field quantum spin system converges to the range of some polynomial in three real variables restricted to the sphere S2.

Deformation quantization and the tomographic representation of quantum fields

The tomographic representation of quantum fields within the deformation quantization formalism is constructed. By employing the Wigner functional we obtain the symplectic tomogram associated with quantum fields. In addition, the tomographic version of the Wigner map allows us to compute the symbols corresponding to field operators. Finally, the functional integral representation of the tomographic star product is determined. Some possible applications of the formalism to loop quantum cosmology and loop quantum gravity are briefly discussed.

Classical universe arising from quantum cosmology

In this paper, we study the classical limit and unitary evolution of quantum cosmology by applying the Weyl--Wigner--Groenewold--Moyal formalism of deformation quantization to quantum cosmology of a homogeneous and isotropic universe with positive spatial curvature and conformally coupled scalar field. The corresponding quantum cosmology (similar to the Schr\"{o}dinger interpretation in canonical quantization scheme of quantum cosmology) is described by the Moyal--Wheeler--DeWitt equation which has an exact solution in Moyal phase space, resulting in Wigner quasiprobability distribution function, peaking over the classical solutions. We show that for a large value of the quantum number $n$, the emerged classical universe is filled with radiation with quantum mechanical origin. Also, we introduce a canonical transformation on the scalar field sector of the model such that the conjugate momenta of the new canonical variable appear linearly in the transformed total Hamiltonian. Using this canonical transformation, we show that, it may lead to disentangle the time from the true dynamical variables. We obtain the time-dependent Wigner function for a coherent as well as for squeezed states. We show that the peak of these Wigner functions follows the classical trajectory in the phase space.

Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

Deformation-obstruction theory for diagrams of algebras and applications to geometry

Let X be an algebraic variety over an algebraically closed field of characteristic 0 and let \operatorname{Coh} (X) denote its Abelian category of coherent sheaves. By the work of W. Lowen and M. Van den Bergh, it is known that the deformation theory of \operatorname{Coh}(X) as an Abelian category can be seen to be controlled by the Gerstenhaber–Schack complex associated to the restriction of the structure sheaf \mathcal O_X \vert_{\mathfrak U} to a cover of affine open sets. We construct an explicit L _\infty algebra structure on the Gerstenhaber–Schack complex controlling the higher deformation theory of \mathcal O_X \vert_{\mathfrak U} or \operatorname{Coh}(X) in case X can be covered by two acyclic open sets, giving an explicit deformation-obstruction calculus for such deformations. For smooth X , such deformations recover the deformation of complex structures and deformation quantizations of X as degenerate cases, as we show by means of concrete examples.

Quantization of a Poisson structure on products of principal affine spaces

We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld–Jimbo quantum group, we obtain a deformation quantization \mathbb C_\hslash[(N \backslash G)^m] of a Poisson structure \pi^{(m)} on products (N \backslash G)^m of principal affine spaces of a connected and simply connected complex semisimple Lie group G . The Poisson structure \pi^{(m)} descends to a Poisson structure \pi_m on products (B \backslash G)^m of the flag variety of G which was introduced and studied by the Lu and the author. Any ample line bundle on (B \backslash G)^m inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of \mathbb C_\hslash[(N \backslash G)^m] . We define the notion of a strongly coisotropic subalgebra in a Hopf algebra, and explain how strong coisotropicity guarantees that any homogeneous coordinate ring of a homogeneous space of a Poisson Lie group can be quantized in the sense of Ciccoli, Fioresi, and Gavarini.