Abstract
Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger’s representation and Schroedinger’s equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characterize such favored variables. This focus leans heavily toward affine variables with a constant negative curvature, which leads to a surprisingly accommodating analysis of non-renormalizable scalar models as well as Einstein’s general relativity.
Highlights
Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity
A favored pair of phase-space variables, i.e. p and q, for which −∞ < p, q < ∞, and which are Cartesian coordinates, arising from a flat surface [1], i.e. a constant zero curvature surface, to become P and Q, the basic pair of quantum variables, with [Q, P] = i 1l. Another familiar approach deals with the SU (2) or SO (3) groups, and its favored classical variable pair arises from a spherical surface, i.e. a constant positive curvature surface of fixed radius determined by the Hilbert space dimension
If the reader can accept that an “harmonic oscillator” for which 0 < q < ∞ cannot be quantized by canonical quantization but can be quantized by affine quantization, it is a natural step to examine the affine quantization of non-renormalizable scalar fields and Einstein’s gravity, with both not having been generally accepted as being successfully quantized by canonical quantization
Summary
Canonical quantization is traditionally used to quantize most classical theories. For a simple system, a favored pair of phase-space variables, i.e. p and q, for which −∞ < p, q < ∞ , and which are Cartesian coordinates, arising from a flat surface [1], i.e. a constant zero curvature surface, to become P and Q, the basic pair of quantum variables, with [Q, P] = i 1l. A third example, which is less well known, involves affine quantization, that, in one example, involves a favored pair of phase-space variables, p and q, for −∞ < p < ∞ while 0 < q < ∞ , and the geometric surface is that of a constant negative curvature [3], along with the basic pair of operators 0 < Q < ∞ and. Favored Classical Variables Favored phase-space coordinates promoted to quantum operators apply to all three quantization procedures. ( ) cillator Hamiltonian, say H ( p,= q) p2 + q2 2 , in one set of coordinates, can be described by alternative phase-space coordinates, say p and q , as one example, where p = p q 2 and q = q 3 3 It follows that ( ) H= ( p, q) H = ( p , q ) p 2 q 4 + q 6 9 2. It is essential to identify the favored classical variables, and only promote them to quantum operators; otherwise you risk a false quantization! We focus on affine quantization
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