Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.
The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of conventional quantum mechanics over $\mathbb{R}^n$ but it may fail in non-trivial phase spaces and also suffer from ordering problems. Affine quantization is an alternative method, similar to the canonical quantization, that may offer a positive result in situations for which canonical quantization fails. In this paper we revisit the affine quantization metho…
Gravity does not naturally fit well with canonical quantization. Affine quantization is an alternative procedure that is similar to canonical quantization but may offer a positive result when canonical quantization fails to offer a positive result. Two simple examples given initially illustrate the power of affine quantization. These examples clearly point toward an affine quantization procedure that vastly simplifies a successful quantization of general relativity.
In this paper the benefits of affine quantization method are highlighted through oscillation problems. We show how affine quantization is able to solve oscillation problems where canonical quantization fails.
Covariant affine integral quantization of the half-plane is studied and applied to the motion of a particle on the half-line. We examine the consequences of different quantizer operators built from weight functions on the half-plane. To illustrate the procedure, we examine two particular choices of the weight function, yielding thermal density operators and affine inversion respectively. The former gives rise to a temperature-dependent probability distribution on the half-plane whereas the later yields the usual canonical quantization and a quasi-probability distribution (affine Wigner functio…
Canonical quantization (CQ) is built around $[Q,P]=i\hbar1\!\!1$, while affine quantization (AQ) is built around $[Q,D]=i\hbar\,Q$, where $D\equiv(PQ+QP)/2$. The basic CQ operators must fit $-\infty< P, Q <\infty$, while the basic AQ operators can fit $-\infty<P<\infty$ and $ 0<Q<\infty$, $-\infty <Q<0$, or even $-\infty<Q\neq0<\infty$. AQ can also be the key to quantum gravity, as our simple outline demonstrates.
Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer operators built from weight functions on this phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner…
We present an original approach to quantization based on operator-valued measure that generalizes the so-called Berezin-Klauder-Toeplitz quantization, and more generally coherent state quantization approches.
Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein's gravity.
We prove through Monte Carlo analysis that the covariant euclidean scalar field theory, $φ^r_n$, where $r$ denotes the power of the interaction term and $n = s + 1$ where $s$ is the spatial dimension and $1$ adds imaginary time, such that $r = n = 4$ can be acceptably quantized using scaled affine quantization and the resulting theory is nontrivial and renormalizable even at low temperatures in the highly quantum regime.
We study canonical and affine versions of the quantized covariant Euclidean Higgs scalar field-theory for two real fields on four dimensional lattices through the Monte Carlo method. We calculate the two-point function near the continuum limit at finite volume.
We study canonical and affine versions of the quantized covariant Euclidean free real scalar field-theory on four dimensional lattices through the Monte Carlo method. We calculate the two-point function at small values of the bare coupling constant and near the continuum limit at finite volume. Our investigation shows that affine quantization is able to give meaningful results for the two-point function for which is not available an exact analytic result and therefore numerical methods are necessary.
A brief summary of the application of coherent states in the examination of quantum dynamics of cosmological models is given. We discuss quantization maps, phase space probability distributions and semiclassical phase spaces. The implementation of coherent states based on the affine group resolves the hardest singularities, renders self-adjoint Hamiltonians without boundary conditions and provides a completely consistent semi-classical description of the involved quantum dynamics. We consider three examples: the closed Friedmann model, the anisotropic Bianchi Type I model and the deep quantum …
Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows establishing a $c$-equivalence between the Moyal and the n…
In this paper we study a two-dimensional [2D] rotationally symmetric harmonic oscillator with time-dependent frictional force. At the classical level, we solve the equations of motion for a particular case of the time-dependent coefficient of friction. At the quantum level, we use the Lewis-Riesenfeld procedure of invariants to construct exact solutions for the corresponding time-dependent Schrödinger equations. The eigenfunctions obtained are in terms of the generalized Laguerre polynomials. By mean of the solutions we verify a generalization version of the Heisenberg's uncertainty relation a…
The transformation of a classical system into its quantum counterpart is usually done through the well known procedure of canonical quantization. However, on non-Cartesian domains, or on bounded Cartesian domains, this procedure can be plagued with theoretical inconsistencies. An alternative approach is {\it affine quantization} (AQ) Fantoni and Klauder (arXiv:2109.13447,Phys. Rev. D {\bf 103}, 076013 (2021)), resulting in different conjugate variables that lead to a more consistent quantization formalism. To highlight these issues, we examine a deceptively simple, but important, problem: that…
We present a new method for the consistent construction of time-continuous coherent-state path integrals using the theory of half-form quantization. Through the inversion of the quantization procedure we construct a de-quantization map taking first order operators to their corresponding path integrals. We generalize our results using functional techniques, allowing for the consistent path integral study of more general operators, including higher orders and interactions.
我们提出了一种在单位圆上构造相干态的可能性。我们的方法基于S^1上的Borel量子化，包括Aharonov-Bohm型量子描述。在量子相空间Z x S^1上，利用Perelomov方法构造了由Weyl算符产生的群相关相干态。由于正则坐标和动量的对偶性，即角变量和整数，这个公式也可以解释为无限周期链上的相干态。对于构造，我们使用与有限周期链上的量子化和相干态的类比，其中量子相空间为Z_M × Z_M。在这项工作中构建的相干态满足统一的分辨率。为了将它们与正则相干态进行比较，还研究了它们的一些进一步的性质，证明了它们的相似之处以及实质性的差异。
Canonical quantization has taught us great things. A common example is that of the harmonic oscillator, which is like swinging a ball on a string back and forth. However, the half-harmonic oscillator blocks the ball at the bottom and then it quickly bounces backwards. This second model cannot be correctly solved using canonical quantization. Now, there is an expansion of quantization, called affine quantization, that can correctly solve the half-harmonic oscillator, and offers correct solutions to a grand collection of other problems, which even reaches to field theory and gravity. This paper …
This work presents a selective review of results concerning the mathematical interface between the classical and quantum aspects encountered in problems such as the nuclear mean-field dynamics or quantum Brownian motion. It is shown that the main difference between classical and quantum behavior arises from the coherence properties of the phase-space distributions known as "action waves" and Wigner functions. The quantum wave functions appear as elementary degrees of freedom for the phase space granularity.
The ordinary Landau problem consists of describing a charged particle in time-independent magnetic field. In the present case the problem is generalized onto time-dependent uniform electric fields with time-dependent mass and harmonic frequency [1].The spectrum of a Hamiltonian describing this system is obtained. The configuration space wave functions of the model is expressed in terms of the generalised Laguerre polynomials. To diagonalize the time-dependent Hamiltonian we employ the Lewis-Riesenfeld method of invariants. To this end, we introduce an unitary transformation in the framework of…
