Constrained Quantization Research Articles

Constrained quantization for the Cantor distribution

The theory of constrained quantization has been recently introduced by Pandey and Roychowdhury. In this paper, they have further generalized their previous definition of constrained quantization and studied the constrained quantization for the classical Cantor distribution. Toward this, they have calculated the optimal sets of n -points, n th constrained quantization errors, the constrained quantization dimensions, and the constrained quantization coefficients, taking different families of constraints for all n\in \mathbb{N} . The results in this paper show that both the constrained quantization dimension and the constrained quantization coefficient for the Cantor distribution depend on the underlying constraints. It also shows that the constrained quantization coefficient for the Cantor distribution can exist and be equal to the constrained quantization dimension. These facts are not true in the unconstrained quantization for the Cantor distribution.

Gravitational path integral from the T2 deformation

We study a T2 deformation of large N conformal field theories, a higher dimensional generalization of the Toverline{T} deformation. The deformed partition function satisfies a flow equation of the diffusion type. We solve this equation by finding its diffusion kernel, which is given by the Euclidean gravitational path integral in d + 1 dimensions between two boundaries with Dirichlet boundary conditions for the metric. This is natural given the connection between the flow equation and the Wheeler-DeWitt equation, on which we offer a new perspective by giving a gauge-invariant relation between the deformed partition function and the radial WDW wave function. An interesting output of the flow equation is the gravitational path integral measure which is consistent with a constrained phase space quantization. Finally, we comment on the relation between the radial wave function and the Hartle-Hawking wave functions dual to states in the CFT, and propose a way of obtaining the volume of the maximal slice from the T2 deformation.

A quantum approach to electromagnetic wave propagation inside a dielectric

As the quantum treatment of field propagation inside a dielectric becomes more important due to developing quantum technologies, we present a detailed study of quantum field propagation through a dielectric when up to third-order dispersion is included. Initially, a proper canonical Lagrangian is defined, leading to Maxwell’s equations and the classical energy. Then, by employing a constrained quantization approach and making use of Ostrogradski’s theorem for higher-order field derivatives in the Lagrangian, the final Hamiltonian is expanded in terms of properly defined annihilation and creation operators. These operators are applied to obtain the quantum fields describing electromagnetic waves propagation inside a dispersive and nonlinear dielectric. The number of photon–polariton pairs in the medium is defined by the number operator. The creation and the annihilation operators were used to describe the nonlinearity of the medium by adding the proper perturbation terms. The present quantum theory is applied to a single-dimensional slab when a light signal is propagating along it.

Path integral quantization of the relativistic Hopfield model

The path-integral quantization method is applied to a relativistically covariant version of the Hopfield model, which represents a very interesting mesoscopic framework for the description of the interaction between quantum light and dielectric quantum matter, with particular reference to the context of analogue gravity. In order to take into account the constraints occurring in the model, we adopt the Faddeev-Jackiw approach to constrained quantization in the path-integral formalism. In particular, we demonstrate that the propagator obtained with the Faddeev-Jackiw approach is equivalent to the one which, in the framework of Dirac canonical quantization for constrained systems, can be directly computed as the vacuum expectation value of the time-ordered product of the fields. Our analysis also provides an explicit example of quantization of the electromagnetic field in a covariant gauge and coupled with the polarization field, which is a novel contribution to the literature on the Faddeev-Jackiw procedure.

A novel enhancement for hierarchical image coding

Hierarchical image coding usually codes a down-sampled version of an original image and then the difference between the original image and a reconstructed version that is interpolated from the down-sampled layer. In this paper, we demonstrate, for the first time, that when the bit-rate used to code the residual layer falls into a critical region (which covers almost all typical bit-rates used in practice), it often happens that all pixels in the down-sampled layer would be deteriorated if the corresponding coded residuals are added into them. To avoid this problem, we first propose a “naive” solution: no coded residuals will be added back into the down-sampled layer; whereas coded residuals will be added only into the interpolated pixels. Then, we propose to apply a constrained quantization technique during the coding of the residual layer so that all residual pixels at the interpolated positions will end up with an improved quality. To verify its effectiveness, we conduct extensive tests to show that the gap between the hierarchical coding scheme and its single-level counterpart (which is typically around 2–3dB in the 2-level hierarchy) will be filled up by a rather big percentage.

On Constrained Randomized Quantization

Randomized (dithered) quantization is a method capable of achieving white reconstruction error independent of the source. Dithered quantizers have traditionally been considered within their natural setting of uniform quantization. In this paper we extend conventional dithered quantization to nonuniform quantization, via a subterfage: dithering is performed in the companded domain. Closed form necessary conditions for optimality of the compressor and expander mappings are derived for both fixed and variable rate randomized quantization. Numerically, mappings are optimized by iteratively imposing these necessary conditions. The framework is extended to include an explicit constraint that deterministic or randomized quantizers yield reconstruction error that is uncorrelated with the source. Surprising theoretical results show direct and simple connection between the optimal constrained quantizers and their unconstrained counterparts. Numerical results for the Gaussian source provide strong evidence that the proposed constrained randomized quantizer outperforms the conventional dithered quantizer, as well as the constrained deterministic quantizer. Moreover, the proposed constrained quantizer renders the reconstruction error nearly white. In the second part of the paper, we investigate whether uncorrelated reconstruction error requires random coding to achieve asymptotic optimality. We show that for a Gaussian source, the optimal vector quantizer of asymptotically high dimension whose quantization error is uncorrelated with the source, is indeed random. Thus, random encoding in this setting of rate-distortion theory, is not merely a tool to characterize performance bounds, but a required property of quantizers that approach such bounds.

Optimal vector quantization in terms of Wasserstein distance

The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Rényi- α -entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained ( α = 1 ) and memory-size constrained ( α = 0 ) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by György and Linder (2002, 2003) [11,12].

Constrained Quantization in the Transform Domain With Applications in Arbitrarily-Shaped Object Coding

In any block-based transform coding of image/video signals, it is well-known that the mean square error (MSE) distortion measured in the pixel domain is exactly equal to the MSE distortion resulted from quantization in the transform domain if the involved transform matrix is unitary. However, such a property no longer exists if the pixel-domain distortion is measured only on a selected part of pixels within one image block. This provides us an opportunity of dynamically shaping the quantization errors so as to make the selected pixels (much) better than the unselected ones. In this paper, we first develop a reversed iterative algorithm to guide us to perform a highly constrained quantization so that the coding quality of the selected pixels in each image block is significantly higher than what can be achieved by using the normal quantization. Then, we apply this intelligent quantization in one practical scenario-coding of arbitrarily-shaped image blocks in MPEG-4, showing remarkable improvements in comparison with the original MPEG-4.

Constrained quantization

Discrete functions over a continuous domain are approximated by discrete functions with fewer levels. These quantizations are endowed with different types of constraints such as monotonicity and variational constraints. Quantization functions exactly or approximately minimize the squared error to a given discrete function. Exact algorithms are derived from dynamic programming with finite horizon. All algorithms have polynomial run time.

Constrained Quantization of Charged Strings in a Background B Field and g-Factors

The Dirac quantization is performed for the constrained system of the open string with different charges located at both ends in the constant background B field. Noncommutativity reveals to commutators [X, X], [P, P] and also [X,P] at both ends of the string. We consider a dependence on the change of the "cyclotron frequency" of the charged string. The g-factor of the charged string is also calculated in the framework of our formulation.

Constrained quantization of open string in background B field and non-commutative D-brane

In a previous paper we provided a consistent quantization of open strings ending on D-branes with a background B field. In this letter, we show that the same result can also be obtained using the more traditional method of Dirac's constrained quantization. We also extend the discussion to the fermionic sector.

Representations of the Infinite Unitary Group from Constrained Quantization

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U 0(ℋ) of all unitary operators U on a separable Hilbert space ℋ for which U − I is compact, originally found by Kirillov and Ol’shanskii, through constrained quantization of its coadjoint orbits. For this purpose the coadjoint orbits are realized as Marsden-Weinstein quotients. The unconstrained system, given as a Weinstein dual pair, is quantized by a corresponding Howe dual pair. Constrained quantization is then performed in replacing the classical procedure of symplectic reduction by the C ∗-algebraic method of Rieffel induction. Reduction and induction have to be performed with respect to either U(M), which is straightforward, or U(M, N). In the latter case one induces from holomorphic discrete series representations, and the desired result is obtained if one ignores half-forms, and induces from a representation, ‘half’ of whose highest weight is shifted relative to the naive orbit correspondence. This is only possible when ℋ is finite-dimensional.

Constrained quantization and θ-angles

We apply a new and mathematically rigorous method for the quantization of constrained systems to two-dimensional gauge theories. In this method, which quantizes Marsden-Weinstein symplectic reduction, the inner product on the physical state space is expressed through a certain integral over the gauge group. The present paper, the first of a series, specializes to the Minkowski theory defined on a cylinder. The integral in question is then constructed in terms of the Wiener measure on a loop group. It is shown how θ-angles emerge in the new method, and the abstract theory is illustrated in detail in an example.

Constrained quantization on symplectic manifolds and quantum distribution functions

A quantization scheme based on the extension of phase space with application of constrained quantization technique is considered. The obtained method is similar to the geometric quantization. For constrained systems the problem of scalar product on the reduced Hilbert space is investigated and a possible solution to this problem is given. Generalization of the Gupta–Bleuler-like conditions is done by minimization of quadratic fluctuations of quantum constraints. The scheme for the construction of generalized coherent states is considered and relation with Berezin quantization is found. The quantum distribution functions are introduced and their physical interpretation is discussed.

The Stückelberg–Kibble model as an example of quantized symplectic reduction

Recently, it has been observed that a certain class of classical theories with constraints can be quantized by a mathematical procedure known as Rieffel induction. After a short exposition of this idea, we apply the new quantization theory to the Stückelberg–Kibble model. We explicitly construct the physical state space ℋphys, which carries a massive representation of the Poincaré group. The longitudinal one-particle component arises from a particular Bogoliubov transformation of the five (unphysical) degrees of freedom one has started with. Our discussion exhibits the particular features of the proposed constrained quantization theory in great clarity.

Faddeev-Jackiw quantization in superspace

We consider the constrained Faddeev-Jackiw geometric quantization approach in superspace. We deal with a supersymmetric quantum mechanical model both in components and in superfield language.

Constrained quantization of the electromagnetic Maxwell theory in the second-order derivative formalism

We canonically quantize the electromagnetic Maxwell theory in the second-order derivative formalism. We work in the radiation gauge, extended to an enlarged phase space.

Constrained quantization of a topologically massive gauge theory in (2+1)-dimensions

We present a canonical quantization for a gauge field theory in a (2+1) dimensional space-time in both Dirac brackets and Schwinger action principle formalisms.