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 method on the half line. We formulate and solve some simple models, the free particle and the harmonic oscillator.
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…
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.
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.
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…
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.
We explain the powerful role that operator-valued measures can play in quantizing any set equipped with a measure, for instance a group (resp. group coset) with its invariant (resp. quasi-invariant) measure. Coherent state quantization is a particular case. Such integral quantizations are illustrated with two examples based on the Weyl-Heisenberg group and on the affine group respectively. An interesting application of the affine quantization in quantum cosmology is mentioned, and we sketch a construction of new coherent states for the hydrogen atom.
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.
The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the…
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.
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.
For a particle moving on a half-line or in an interval the operator $\hat p = - i \partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\hat p$ fails. Based upon a new concept for a self-adjoint momentum operator $\hat p_R$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\hat H$ and the momentum operator $\hat p_R$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\hat H)$ and $D(\hat p_R)$ in the H…
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…
The program of quantizing the gravitational field with the help of affine field variables is continued. For completeness, a review of the selection criteria that singles out the affine fields, the alternative treatment of constraints, and the choice of the initial (before imposition of the constraints) ultralocal representation of the field operators is initially presented. As analogous examples demonstrate, the introduction and enforcement of the gravitational constraints will cause sufficient changes in the operator representations so that all vestiges of the initial ultralocal field operato…
Quantum systems with adiabatic classical parameters are widely studied, e.g., in the modern holonomic quantum computation. We here provide complete geometric quantization of a Hamiltonian system with time-dependent parameters, without the adiabatic assumption. A Hamiltonian of such a system is affine in the temporal derivative of parameter functions. This leads to the geometric Berry factor phenomena.
A general dynamical system composed by two coupled sectors is considered. The initial time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum. The aim of this paper is to derive the dynamical evolution of a general half quantum system from its full quantum formulation. The standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data. The main problem is how can quantum mechanics be applied to a dy…
We give an efficient algorithm to evaluate a certain class of exponential sums, namely the periodic, quadratic, multivariate half Gauss sums. We show that these exponential sums become $\#\mathsf{P}$-hard to compute when we omit either the periodicity or quadraticity condition. We apply our results about these exponential sums to the classical simulation of quantum circuits, and give an alternative proof of the Gottesman-Knill theorem. We also explore a connection between these exponential sums and the Holant framework. In particular, we generalize the existing definition of affine signatures …
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 …
Today's quantum field theory (QFT) relies heavenly on canonical quantization (CQ), which fails for $φ^4_4$ leading only to a "free" result. Affine quantization (AQ), an alternative quantization procedure, leads to a "non-free" result for the same model. Perhaps adding AQ to CQ can improve the quantization of a wide class of problems in QFT.
Normally, the half-harmonic oscillator is active when $x>0$ and absent when $x<0$. From a canonical quantization perspective, this leads to odd eigenfunctions being present while even eigenfunctions are absent. In that case, only the usual odd eigenfunctions will appear if the wall slides to negative infinity. However, if an affine quantization is used, sliding the wall away shows that all the odd and even eigenfunctions are encountered, exactly like any full-harmonic oscillator. We provide numerical support for this.
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 …
