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

Completing Canonical Quantization

John R. Klauder

University of Florida

Gainesville, FL


Something strange

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

Classical & Quantum

quantum

classical


Completing canonical quantization

DIRAC

The Principles of Quantum Mechanics


Classical quantum1

Classical Quantum

quantum

classical


List of topics

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

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

Action Principle Formulations

VERY DIFFERENT


Restricted action principle

Restricted Action Principle

(half space)

(Gaussians)


Unification of classical and quantum 1

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

Unification of Classical and Quantum (2)

subset

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


Canonical transformations

Canonical Transformations


Cartesian coordinates

Cartesian Coordinates


Quantum classical summary

Quantum/Classical Summary


Is there more

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 more1

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

Affine Variables

[(

s.a.

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


Affine quantization 1

Affine Quantization (1)

subset


Affine quantization 2

Affine Quantization (2)


The q c connection summary

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

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 - 1


Toy model 2

Toy Model - 2


Enhanced toy models summary

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

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 (1)


Rotationally sym models 2

Rotationally Sym. Models (2)

A free theory!


Now do some hard work

…Now, Do Some Hard Work…


Rotationally sym models 3

Rotationally Sym. Models (3)

TEST

REAL


Rot sym models summary

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 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

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

Main Message of Today

q

q

c

c


Completing canonical quantization

Thank You


References

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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