Obstructions to Quantization

@article{Gotay1998ObstructionsTQ,
  title={Obstructions to Quantization},
  author={Mark J. Gotay},
  journal={Journal of Nonlinear Science},
  year={1998},
  volume={6},
  pages={171-216}
}
  • M. J. Gotay
  • Published 10 September 1998
  • Mathematics
  • Journal of Nonlinear Science
Quantization is not a straightforward proposition, as demonstrated by Groenewold’s and Van Hove’s discovery, more than fifty years ago, of an “obstruction” to quantization. Their “no-go theorems” assert that it is in principle impossible to consistently quantize every classical polynomial observable on the phase space R 2n in a physically meaningful way. Similar obstructions have been recently found for S 2 and T*S 1, buttressing the common belief that no-go theorems should hold in some… 
View PDF on arXiv
57 Citations
Highly Influential Citations
3
Background Citations
30
Methods Citations
14
Results Citations
3

57 Citations

No-Go theorems and quantization

In this review, we would like to highlight the three known no-go theorems in quantum physics in relation to the process of quantization that maps classical observables to quantum ones. The…

A No-Go Theorem for Nonlinear Canonical Quantization

We want to point out the following strengthening of the classical theorem of Groenewold and van Hove: There exists no mapping Op from polynomial observables on the phase space into linear operators…

Localized Quantum States

Let X be a symplectic manifold and Aut(L) the automorphism group of a Kostant-Souriau line bundle on X. *Quantum states for X*, as defined by J.-M. Souriau in the 1990s, are certain positive-definite…

Higher Category Theory and Hilbert’s Sixth Problem

In 1900 David Hilbert published his famous list of 23 problems. The sixth of them - the axiomatization of Physics - remains partially unsolved. In this work we will give a gentle introduction and a…

Towards An Approach to Hilbert's Sixth Problem: A Brief Review

In 1900 David Hilbert published his famous list of 23 problems. The sixth of them-the axiomatization of Physics-remains partially unsolved. In this work we will give a gentle introduction and a brief…

Unfinished history and paradoxes of quantum potential. II. Relativistic point of view

This is the second of the two related papers analyzing the origins and possible explanations of a paradoxical phenomenon of the quantum potential (QP). It arises in quantum mechanics (QM) of a…

Unfinished history and paradoxes of quantum potential. II. Relativistic point of view

This is the second of the two related papers analyzing the origins and possible explanations of a paradoxical phenomenon of the quantum potential (QP). It arises in quantum mechanics (QM) of a…

On the Groenewold-Van Hove problem for R 2n

We discuss the Groenewold–Van Hove problem for R2n, and completely solve it when n=1. We rigorously show that there exists an obstruction to quantizing the Poisson algebra of polynomials on R2n,…

Unfinished history and paradoxes of quantum potential. I. Non-relativistic origin, history and paradoxes

This is the first of two related papers analyzing and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic points of…

Beables/Observables in Classical and Quantum Gravity

Observables 'are observed' whereas beables just 'are'. This gives beables more scope in the cosmological and quantum domains. Both observables and beables are enti- ties that form 'brackets' with…
...

References

SHOWING 1-10 OF 75 REFERENCES

On quantizing T ∗ S 1

Fronsdal *-quantization and Fell inducing

  • M. Hennings
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1986
Abstract The process of Fronsdal inducing is discussed as a method of quantizing physical systems, and is seen to reduce to a technique for obtaining representations of *-algebras. It is asked…

Some ideas about quantization

A GROENEWOLD -VAN HOVE THEOREM FOR S

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S2 which is irreducible on the su(2) subalgebra generated by the components {S1, S2, S3}…

Deformation theory and quantization. II. Physical applications

Mathematical obstructions to quantization

Quantization is commonly viewed as a mapping of functions on classical phase space to operators on Hilbert space, preserving the Lie algebra structure and satisfying some additional physically…

A Groenewold-van Hove theorem for S**2

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S2 which is irreducible on the su(2) subalgebra generated by the components {S1, S2, S3}…

On the Principles of Elementary Quantum Mechanics

Questions on quantization

The present manuscript is based on informal notes which I distributed at the Dartmouth Workshop on E-theory, Quantization, and Deformations, September 1997, and a number of people subsequently gave me suggestions for improvements, which are incorporated here.

On the Groenewold-Van Hove problem for R 2n

We discuss the Groenewold–Van Hove problem for R2n, and completely solve it when n=1. We rigorously show that there exists an obstruction to quantizing the Poisson algebra of polynomials on R2n,…
...