Completing canonical quantization
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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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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

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."


Classical & Quantum

quantum

classical


DIRAC

The Principles of Quantum Mechanics


Classical Quantum

quantum

classical


List of Topics

1 Classical/Quantum connection

“Enhanced Quantization”

Canonical & Affine quantization Enhanced classical theories

2 Two toy models

3 Rotationally symmetric models


TOPIC 1

  • Classical & Quantum formalism

  • Canonical coherent states

  • Classical Quantum formalism

  • Canonical transformations

  • Cartesian coordinates

  • Affine vs. canonical variables

  • Affine quantization as canonical quantization


Action Principle Formulations

VERY DIFFERENT


Restricted Action Principle

(half space)

(Gaussians)


Unification of Classical and Quantum (1)

?

Macroscopic variations of Microscopic states:

Basic state: Translated basic state: Translated Fourier state: Coherent states:

s.a.


Unification of Classical and Quantum (2)

subset

CLASSICAL MECHANICS IS QUANTUM MECHANICS RESTRICTED TO A CERTAIN TWO DIMENSIONAL SURFACE IN HILBERT SPACE


Canonical Transformations


Cartesian Coordinates


Quantum/Classical Summary


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?


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 !


Affine Variables

[(

s.a.

also (q < 0 , Q < 0) U (q > 0 , Q > 0)


Affine Quantization (1)

subset


Affine Quantization (2)


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!


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


Toy Model - 1


Toy Model - 2


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


TOPIC 3

  • Rotationally symmetric models

  • Free quantum models for

  • Interacting quantum models for

  • Reducible operator representation is the key

. . .


Rotationally Sym. Models (1)


Rotationally Sym. Models (2)

A free theory!


…Now, Do Some Hard Work…


Rotationally Sym. Models (3)

TEST

REAL


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 ??


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


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)


Main Message of Today

q

q

c

c


Thank You


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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