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Completing Canonical Quantization

Completing Canonical Quantization. John R. Klauder University of Florida Gainesville, FL. Something Strange. L. Landau, E.M. Lifshitz, Quantum mechanics: Non-relativistic theory , 3rd ed., Pergamon Press, 1977. "Thus quantum mechanics occupies a very unusual

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Completing Canonical Quantization

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  1. Completing Canonical Quantization John R. Klauder University of Florida Gainesville, FL

  2. Something Strange L. Landau, E.M. Lifshitz, Quantum mechanics: Non-relativistic theory, 3rd ed., Pergamon Press, 1977. "Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

  3. Classical & Quantum quantum classical

  4. DIRAC The Principles of Quantum Mechanics

  5. Classical Quantum quantum classical

  6. List of Topics 1 Classical/Quantum connection “Enhanced Quantization” Canonical & Affine quantization Enhanced classical theories 2 Two toy models 3 Rotationally symmetric models

  7. TOPIC 1 • Classical & Quantum formalism • Canonical coherent states • Classical Quantum formalism • Canonical transformations • Cartesian coordinates • Affine vs. canonical variables • Affine quantization as canonical quantization

  8. Action Principle Formulations VERY DIFFERENT

  9. Restricted Action Principle (half space) (Gaussians)

  10. Unification of Classical and Quantum (1) ? Macroscopic variations of Microscopic states: Basic state: Translated basic state: Translated Fourier state: Coherent states: s.a.

  11. Unification of Classical and Quantum (2) subset CLASSICAL MECHANICS IS QUANTUM MECHANICS RESTRICTED TO A CERTAIN TWO DIMENSIONAL SURFACE IN HILBERT SPACE

  12. Canonical Transformations

  13. Cartesian Coordinates

  14. Quantum/Classical Summary

  15. Is There More? • Are there other two-dimensional sheets of normalized Hilbert space vectors that may be used in restricting the quantum action and which lead to an enhanced classical canonical formalism?

  16. Is There More? • Are there other two-dimensional sheets of normalized Hilbert space vectors that may be used in restricting the quantum action and which lead to an enhanced classical canonical formalism? YES !

  17. Affine Variables [( s.a. also (q < 0 , Q < 0) U (q > 0 , Q > 0)

  18. Affine Quantization (1) subset

  19. Affine Quantization (2)

  20. The Q/C Connection : Summary • The classical action arises by a restriction of the quantum action to coherent states • Canonical quantization uses P and Q which must be self adjoint • Affine quantization uses D and Q which are selfadjoint when Q > 0 (and/or Q < 0) • Both canonical AND affine quantum versions are consistent with classical, canonical phase space variables p and q • Now for some applications!

  21. TOPIC 2 • Solutions of the first model have singularities • Canonical quantum corrections • Affine quantum corrections • Affine quantization resolves singularities! • A second classical model is similar

  22. Toy Model - 1

  23. Toy Model - 2

  24. Enhanced Toy Models : Summary • Classical toy models exhibit singular solutions for all positive energies • Enhanced classical theory with canonical quantum corrections still exhibits singularities • Enhanced classical theory with affine quantum corrections removes all singularities • Enhanced quantization can eliminate singularities

  25. TOPIC 3 • Rotationally symmetric models • Free quantum models for • Interacting quantum models for • Reducible operator representation is the key . . .

  26. Rotationally Sym. Models (1)

  27. Rotationally Sym. Models (2) A free theory!

  28. …Now, Do Some Hard Work…

  29. Rotationally Sym. Models (3) TEST REAL

  30. Rot. Sym. Models : Summary • Conventional quantization works if N is finite but leads totriviality if N is infinite • Enhanced quantization applies even for reducible operator representations • Using the Weak Correspondence Principle a nontrivial quantization results if N is finite or N is infinite---withNOdivergences! • Class. & Quant. formalism is similar for all N WHAT HAS BEEN ACCOMPLISHED ??

  31. Canonical vs. Enhanced • Canonical quantization requires Cartesian coordinates, but WHY is not clear • Canonical quantization works well for certain problems, but NOT for all problems • Enhanced quantization clarifies coordinate transformations and Cartesian coordinates • Enhanced quantization can yield canonical results -- OR provide proper results when canonical quantization fails

  32. Other Enh. Quant. Projects • Simple models of affine quantization eliminating classical singularities (on going) • Covariant scalar models (done) • Affine quantum gravity (started) • Incorporating constrained systems within enhanced quantization (started) • Additional sheets of vectors in Hilbert space relating quan. and class. models (started) • Extension to fermion fields (hints)

  33. Main Message of Today q q c c

  34. Thank You

  35. References • ``Enhanced Quantization: A Primer'', J.Phys. A: Math. Theor.45, 285304 (8pp) (2012); arXiv:1204.2870 • ``Enhanced Quantization on the Circle’’; Phys. Scr. 87 035006 (5pp) (2013); arXiv:1206.1180 • ``Enhanced Quantum Procedures that Resolve Difficult Problems’’; arXiv: 1206.4017 • ``Revisiting Canonical Quantization’’; arXiv: 1211.735

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