Functional Quantization Research Articles

Regularized quantum motion in a bounded set: Hilbertian aspects

It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.

Functional quantization of rough volatility and applications to volatility derivatives

We develop a product functional quantization of rough volatility. Since the optimal quantizers can be computed offline, this new technique, built on the insightful works by [Luschgy, H. and Pagès, G., Functional quantization of Gaussian processes. J. Funct. Anal., 2002, 196(2), 486–531; Luschgy, H. and Pagès, G., High-resolution product quantization for Gaussian processes under sup-norm distortion. Bernoulli, 2007, 13(3), 653–671; Pagès, G., Quadratic optimal functional quantization of stochastic processes and numerical applications. In Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 101–142, 2007 (Springer: Berlin Heidelberg)], becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis on the pricing of options on the VIX and realized variance in the rough Bergomi model [Bayer, C., Friz, P.K. and Gatheral, J., Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887–904] and compare our results to other benchmarks recently suggested.

Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes

In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described.

Riemannian exponential and quantization

This article continues and completes the previous one [18]. First of all, we present two methods of quantization associated with a linear connection given on a differentiable manifold, one of them being the one presented in [18]. The two methods allow quantization of functions that come from covariant tensor fields. The equivalence of both is demonstrated as a consequence of a remarkable property of the Riemannian exponential (Theorem 5.1) that, as far as we know, is new to the literature. In addition, we provide a characterization of the Schrödinger operators as the only ones that by quantization correspond to classical mechanical systems. Finally, it is shown that the extension of the above quantization to functions of a very broad type can be carried out by generalizing the method of [18] in terms of fields of distributions.

Modified gauge-unfixing formalism and gauge symmetries in the noncommutative chiral bosons theory

Abstract We use the gauge-unfixing (GU) formalism framework in a two-dimensional noncommutative chiral bosons (NCCB) model to disclose new hidden symmetries. That amounts to converting a second-class system to a first-class one without adding any extra degrees of freedom in phase space. The NCCB model has two second-class constraints —one of them turns out as a gauge symmetry generator while the other one, considered as a gauge-fixing condition, is disregarded in the converted gauge-invariant system. We show that it is possible to apply a conversion technique based on the GU formalism direct to the second-class variables present in the NCCB model, constructing deformed gauge-invariant GU variables, a procedure which we name here as modified GU formalism. For the canonical analysis in noncommutative phase space, we compute the deformed Dirac brackets between all original phase space variables. We obtain two different gauge-invariant versions for the NCCB system and, in each case, a GU Hamiltonian is derived satisfying a corresponding first-class algebra. Finally, the phase space partition function is presented for each case allowing for a consistent functional quantization for the obtained gauge-invariant NCCB.

A novel quantum model of mass function for uncertain information fusion

Understanding the uncertainty involved in a mass function is a central issue in Dempster–Shafer evidence theory for uncertain information fusion. Recent advances suggest to interpret the mass function from a view of quantum theory. However, existing studies do not truly implement the quantization of evidence. In order to solve the problem, a usable quantization scheme for mass function is studied in this paper. At first, a novel quantum model of mass function is proposed, which effectively embodies the principle of quantum superposition. Then, a quantum averaging operator is designed to obtain the quantum average of evidence, which not only retains many basic properties, for example idempotency, commutativity, and quasi-associativity, required by a rational approach for uncertain information fusion, but also yields some new characters, namely nonlinearity and globality, caused by the quantization of mass functions. At last, based on the quantum averaging operator, a new rule called quantum average combination rule is developed for the fusion of multiple pieces of evidence, which is compared with other representative average-based combination methods to show its performance. Numerical examples and applications for classification tasks are provided to demonstrate the effectiveness of the proposed quantum model, averaging operator, and combination rule.

Third quantization for scalar and spinor wave functions of the Universe in an extended minisuperspace

We consider the third quantization in quantum cosmology of a minisuperspace extended by the Eisenhart–Duval lift. We study the third quantization based on both Klein–Gordon type and Dirac-type equations in the extended minisuperspace. Spontaneous creation of ‘Universes’ is investigated upon the quantization of a simple model. We find that the quantization of the Dirac-type wave function reveals that the number density of universes is expressed by the Fermi–Dirac distribution. We also calculate the entanglement entropy of the multi-universe system.

BFV quantization and BRST symmetries of the gauge invariant fourth-order Pais-Uhlenbeck oscillator

We perform the BFV-BRST quantization of the fourth-order Pais-Uhlenbeck oscillator (PUO) for the first time. We show that although the PUO is not naturally constrained in the sense of Dirac-Bergmann, it is possible to profit from the introduction of suitable constraints in phase space in order to obtain a proper BRST invariant quantum system. Starting from its second-class constrained system description, we use the BFFT formalism to obtain first-class constraints as gauge symmetry generators. After the Abelianization of the constraints, we obtain the conserved BRST charge, the corresponding BRST transformations and proceed further to the BFV functional quantization of the model. We further construct appropriate finite field dependent BRST transformation to establish the interconnections between different BRST invariant effective theories of PUO in different gauges. Our approach sheds light on the open problem of the quantization of general higher derivative quantum field theories.

Classical and Quantum Spherical Pendulum

The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum system of the type we are looking for requires a new treatment of the classical action functions and their properties. For the sake of completeness we have provided an extensive presentation of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions between different quantum states. Moreover, we relate these shifting operators to quantization of functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study the properties of the quantum spherical pendulum, in particular, the boundary conditions for the shifting operators and quantum monodromy.

Diagnostic Data Integration Using Deep Neural Networks for Real-Time Plasma Analysis

Recent advances in acquisition equipment are providing experiments with growing amounts of precise, yet affordable sensors. At the same time, an improved computational power, coming from new hardware resources [GPU, field-programmable gate array (FPGA), adaptive compute acceleration platform (ACAP)] has been made available at relatively low costs. This led us to explore the possibility of completely renewing the chain of acquisition for a fusion experiment, where many high-rate sources of data, coming from different diagnostics, can be combined in a wide framework of algorithms. If, on the one hand, adding new data sources with different diagnostics enriches our knowledge about physical aspects, on the other hand, the dimensions of the overall model grow, making relations among variables more and more opaque. A new approach for integrating such heterogeneous diagnostics, based on the composition of deep variational autoencoders, could ease this problem, acting as a structural sparse regularizer. This has been applied to RFX-mod experimental data, integrating the soft X-ray linear images of plasma temperature with the magnetic state. However, to ensure a real-time signal analysis, these algorithmic techniques must be adapted to run in well-suited hardware. In particular, it is shown that, attempting a quantization of neuron transfer functions, such models can be adapted to run in an embedded programmable logic device. The resulting firmware, approximating the deep inference model to a set of simple operations, fits well with the simple logic units that are largely abundant in FPGAs. This is the key factor that permits the use of affordable hardware with complex deep neural topology and operates them in real-time.

Dynamic Quantization of Digital Filter Coefficients

The possibility of quantizing the coefficients of a digital filter in the concept of dynamic mathematical
 programming, as a dynamic process of step-by-step quantization of coefficients with their discrete optimization at
 each step according to the objective function, common to the entire quantization process, is considered. Dynamic
 quantization can significantly reduce the functional error when implementing the required characteristics of a lowbit digital filter in comparison with classical quantization. An algorithm is presented for step-by-step dynamic
 quantization using integer nonlinear programming methods, taking into account the specified signal scaling and the
 radius of the poles of the filter transfer function. The effectiveness of this approach is illustrated by dynamically
 quantizing the coefficients of a cascaded high-order IIR bandpass filter with a minimum bit depth to represent integer
 coefficients. A comparative analysis of functional quantization errors is carried out, as well as a test of the quantized
 filter performance on test and real signals.

BRST analysis and BFV quantization of the generalized quantum rigid rotor

We identify a strong similarity among several distinct originally second-class systems, including both mechanical and field theory models, which can be naturally described in a gauge-invariant way. The canonical structure of such related systems is encoded into a gauge-invariant generalization of the quantum rigid rotor. We perform the BRST symmetry analysis and the BFV functional quantization for the mentioned gauge-invariant version of the generalized quantum rigid rotor. We obtain different equivalent effective actions according to specific gauge-fixing choices, showing explicitly their BRST symmetries. We apply and exemplify the ideas discussed to two particular models, namely, motion along an elliptical path and the [Formula: see text] nonlinear sigma model, showing that our results reproduce and connect previously unrelated known gauge-invariant systems.

Duffin–Kemmer–Petiau equation and thermodynamic quantities

We investigate the generalized form of Duffin–Kemmer–Petiau (DKP) equation in the presence of both a position-dependent electrical field and curved spacetime for the 2-dimensional anti-de Sitter spacetime. Moreover, we derive both the asymptotic wave function and construct energy quantization with the help of the properties of gamma function. All thermodynamic quantities of the system have been calculated with the help of the Euler–MacLaurin formula in the final state.

Functional quantization of rough volatility and applications to the VIX

We develop a product functional quantization of rough volatility. Since the quantizers can be computed offline, this new technique, built on the insightful works by Luschgy and Pages, becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis to pricing VIX Futures in the rough Bergomi model and compare our results to other recently suggested benchmarks.

Fluctuation-dissipation theorem and fundamental photon commutation relations in lossy nanostructures using quasinormal modes

We provide theory and formal insight on the Green function quantization method for absorptive and dispersive spatial-inhomogeneous media in the context of dielectric media. We show that a fundamental Green function identity, which appears, e.g., in the fundamental commutation relation of the electromagnetic fields, is also valid in the limit of non-absorbing media. We also demonstrate how the zero-point field fluctuations yields a non-vanishing surface term in configurations without absorption, when using a more formal procedure of the Green function quantization method. We then apply the presented method to a recently developed theory of photon quantization using quasinormal modes [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)] for finite nanostructures embedded in a lossless background medium. We discuss the strict dielectric limit of the commutation relations of the quasinormal mode operators and present different methods to obtain them, connected to the radiative loss for non-absorptive but open resonators. We show exemplary calculations of a fully three-dimensional photonic crystal beam cavity, including the lossless limit, which supports a single quasinormal mode and discuss the limits of the commutation relation for vanishing damping (no material loss and no radiative loss).

Almost Sure Weyl Law for Quantized Tori

We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean $0$, variance $1$ and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by T. Christiansen and M. Zworski.

Landau-Hall states and Berezin-Toeplitz quantization of matrix algebras

We argue that the Landau-Hall states provide a suitable framework for formulating the Berezin-Toeplitz quantization of classical functions on a K\"ahler phase space. We derive the star-products for such functions in this framework and generalize the Berezin-Toeplitz quantization to matrix-valued classical functions. We also comment on how this is related to different calculations of the effective action for Hall systems.

Covariant integral quantization of the unit disk

We implement a SU(1, 1) covariant integral quantization of functions on the unit disk. The latter can be viewed as the phase space for the motion of a “massive” test particle on (1+1)-anti-de Sitter space-time, and the relevant unitary irreducible representations of SU(1, 1) corresponding to the quantum version of such motions are found in the discrete series and its lower limit. Our quantization method depends on the choice of a weight function on the phase space in such a way that different weight functions yield different quantizations. For instance, the Perelomov coherent states quantization is derived from a particular choice. Semi-classical portraits or lower symbols of main physically relevant operators are determined, and the statistical meaning of the weight function is discussed.

On Integrals Over a Convex Set of the Wigner Distribution

We provide an example of a normalized $$L^{2}({\mathbb {R}})$$ function u such that its Wigner distribution $${\mathcal {W}}(u,u)$$ has an integral $$>1$$ on the square $$[0,a]\times [0,a]$$ for a suitable choice of a. This provides a negative answer to a question raised by Flandrin (Proc IEEE Int Conf Acoustics 4(1):2176–2179, 1988). Our arguments are based upon the study of the Weyl quantization of the characteristic function of $${{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}}$$ along with a precise numerical analysis of its discretization.

Calculation of diffractive optical elements for the formation of thin light sheet

In the work, the calculation and study of diffractive optical elements (DOE) for the formation of a diffraction-free beam in the form of a thin light sheet, which can be used in planar microscopy, were performed. The calculation of phase DOEs is made on the basis of an iterative algorithm, taking into account the quantization of the phase function.