• DOI:10.4236/jhepgc.2022.81013
• Corpus ID: 238634803

Solving Major Problems Using Vector Affine Quantization

@article{Klauder2022SolvingMP,
  title={Solving Major Problems Using Vector Affine Quantization},
  author={John Klauder},
  journal={Journal of High Energy Physics, Gravitation and Cosmology},
  year={2022}
}

• J. Klauder
• Published 12 October 2021
• Mathematics
• Journal of High Energy Physics, Gravitation and Cosmology

Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with special problems. Vector affine quantization introduces multiple degrees of freedom which find that working together create novel tools suitable to eliminate typical difficulties encountered in more conventional approaches. 1 An Introduction to Affine Quantization 1.1 Basics of canonical and affine quantization The usual variables of classical physics are p and q, and a common Hamiltonian is H(p, q) = 1 2 p+V (q). If, for some reason q > 0, which is an example of our interest, then we prefer to use the dilation d = pq and q. For them, the classical Hamiltonian becomes §H (d, q) = 1 2 d/q+V (q). However, canonical quantization (CQ) requires that −∞ < p& q < ∞, and promotes its proper variables to operators, p → P (= P ) and q → Q (= Q) with [Q,P ] = ih̄1 . Affine quantization (AQ) chooses to promote proper variables to operators, d = pq → D = (P Q + QP )/2 (= D), q → Q 6= 0, and [Q,D] = ih̄Q. The reason that Q 6= 0 is because if Q = 0 then D = 0 and P can not help. The fact that Q 6= 0 leads to either Q > 0, Q < 0, or simply both, Q 6= 0. Because Q is ‘incomplete’, the operator P † 6= P , and so P † was used for D