Polymer quantum mechanics some examples using path integrals

Parra, Lorena; Vergara, J. David

doi:10.1063/1.4861963

Cited by 7 publications

(22 citation statements)

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“…This changed recently: a path integral approach was considered [18][19][20] in order to provide a more detailed link between loop quantum cosmology and the covariant spin foam models [5], and a polymer path integral in quantum field theory and its relation to Lorentz invariance has also been studied in [21]. Also the Feynman formula for other mechanical examples has been worked out [22], and explicit polymer propagators have been obtained [23]. An interesting aspect of this so-called Feynman approach is that semiclassical approximations can be at hand to investigate important gravitating systems through the saddle point approximation of its path integral description.…”

Section: Introductionmentioning

confidence: 99%

Polymer quantization and the saddle point approximation of partition functions

Morales‐Técotl

Orozco-Borunda

Rastgoo³

2015

Phys. Rev. D

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The saddle point approximation of the path integral partition functions is an important way of deriving the thermodynamical properties of black holes. However, there are certain black hole models and some mathematically analog mechanical models for which this method cannot be applied directly. This is due to the fact that their action evaluated on a classical solution is not finite and its first variation does not vanish for all consistent boundary conditions. These problems can be dealt with by adding a counterterm to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation.In this work we study the effects of polymer quantization on a mechanical model presenting the aforementioned difficulties and contrast it with the above counterterm method. This type of quantization for mechanical models is motivated by the loop quantization of gravity which is known to play a role in the thermodynamics of black hole systems.The model we consider is a nonrelativistic particle in an inverse square potential, and analyze two polarizations of the polymer quantization in which either the position or the momentum is discrete. In the former case, Thiemann's regularization is applied to represent the inverse power potential but we still need to incorporate the Hamilton-Jacobi counterterm which is now modified by polymer corrections. In the latter, momentum discrete case however, such regularization could not be implemented. Yet, remarkably, owing to the fact that the position is bounded, we do not need a Hamilton-Jacobi counterterm in order to have a well-defined saddle point approximation. Further developments and extensions are commented upon in the discussion.

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Section: Introductionmentioning

confidence: 99%

Polymer quantization and the saddle point approximation of partition functions

Morales‐Técotl

Orozco-Borunda

Rastgoo³

2015

Phys. Rev. D

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“…Nevertheless, it also renders periodic the semiclassical description, via path integral formulation [28][29][30] which is, in this sense, a desired result. This feature results from the fact that (1) is a particular case of an almost periodic Schrödinger operator [31,32] i.e., operators formed by a finite sum of holonomies.…”

Section: Each H (λ)mentioning

confidence: 99%

“…Summarizing this section, we have obtained an instanton solution P + given in (33) for the Euclidean action (29). The Euclidean action evaluated on this solution gives (35) and the instanton fulfills the conditions P + (+∞) = 2πp c and P + (−∞) = 0.…”

Section: Fig 2: Graph Of the Potential V (P) To Notice The Presence mentioning

confidence: 99%

Instanton solutions on the polymer harmonic oscillator

Olivares¹,

García-Chung²,

Vergara³

2017

Class. Quantum Grav.

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It is computed, using instanton methods, the first allowed energy band for the polymer harmonic oscillator. The result is consistent with the band structure of the standard quantum pendulum but with pure point spectrum. An effective infinite degeneracy emerges in the formal limit µ/l 0 → 0 where l 0 is the characteristic length of the vacuum eigenfunction of a quantum harmonic oscillator. As an additional result, it is shown along the article the role played by the lattice reference point λ in the full quantization of the polymer harmonic oscillator.

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“…A diferencia de lo que es propuesto en [33,30,32,31], nosotros no obtendremos la fórmula de Feynman, escribir la amplitud de transición en términos de la acción, en este caso la del OAP, sino que trataremos de evaluar la expresión (4.8) de una manera directa. Por este motivo realizaremos la teoría de perturbaciones, la cual nos permite calcular la amplitud de transición del OAP en el régimen de rotor, como veremos en las siguientes secciones.…”

Section: 8)unclassified

“…Segundo, En el trabajo [31] sólo se formula la amplitud de transición en términos de la acción, en nuestro trabajo calculamos de forma explícita el propagador. Por otra parte, también se han estudiado sistemas mecánicos en la integral de trayectoria polimérica, como lo son el oscilador armónico polimérico (OAP) [32,33], una partícula libre relativista [34], así como el caso del potencial 1 r 2 [35]. En el contexto de la TCC, algunos de los modelos que se han explorado ha sido a través de un esquema hamiltoniano y parten de un espacio-tiempo clásico preestablecido, así el análisis se ha centrado en las consecuencias que resultan de emplear la teoría polimérica exclusivamente en los grados de libertad de materia.…”

Section: Introductionunclassified

Teoría cuántica de campos polimérica en el esquema perturbativo y el propagador del campo escalar en el régimen de altas energías

González

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Polymer quantum mechanics some examples using path integrals

Cited by 7 publications

References 0 publications

Polymer quantization and the saddle point approximation of partition functions

Polymer quantization and the saddle point approximation of partition functions

Instanton solutions on the polymer harmonic oscillator

Teoría cuántica de campos polimérica en el esquema perturbativo y el propagador del campo escalar en el régimen de altas energías

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