Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics

Camblong, Horacio E.; Epele, L.N.; Fanchiotti, H.; Canal, C.A. Garcia

doi:10.1006/aphy.2000.6092

Cited by 52 publications

(68 citation statements)

References 48 publications

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“…Even though this generalization is somewhat arbitrary, in this paper we follow the convenient prescription provided in Refs. [7,8,21,22]. Accordingly,…”

Section: Dimensional Regularizationmentioning

confidence: 98%

“…[4,7,8,11,22]. This can be achieved by properly renormalizing the theory in the strong-coupling regime, with the introduction of a scale-breaking parameter.…”

Section: B Anomalous Commutatorsmentioning

confidence: 99%

“…More precisely, even though the regularized wave functions satisfy the regular boundary conditions (A.9) and (A.10), the renormalized wave functions acquire a logarithmic singularity at the origin. For example, the Hilbert subspace of normalized bound-states of the two-dimensional δ-function interaction reduces to the one-dimensional space spanned by the renormalized ground state [4,7,8,22]…”

Section: The Two-dimensional δ-Function Interactionmentioning

confidence: 99%

“…where the physical dimensions of the original theory are preserved by changing the dimensions of the coupling according to g → g µ ǫ [7].…”

Section: Dimensional Regularizationmentioning

confidence: 99%

“…In addition, similar techniques and concepts have been used to analyze and renormalize the inverse square potential [7,8,[10][11][12], which can also be shown to be conformally invariant at the classical level [13,14]. These scale-invariant potentials may be regarded as the most outstanding examples of conformal quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

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Anomalous commutator algebra for conformal quantum mechanics

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The structure of the commutator algebra for conformal quantum mechanics is considered. Specifically, it is shown that the emergence of a dimensional scale by renormalization implies the existence of an anomaly or quantum-mechanical symmetry breaking, which is explicitly displayed at the level of the generators of the SO(2,1) conformal group. Correspondingly, the associated breakdown of the conservation of the dilation and special conformal charges is derived.

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“…Even though this generalization is somewhat arbitrary, in this paper we follow the convenient prescription provided in Refs. [7,8,21,22]. Accordingly,…”

Section: Dimensional Regularizationmentioning

confidence: 98%

“…[4,7,8,11,22]. This can be achieved by properly renormalizing the theory in the strong-coupling regime, with the introduction of a scale-breaking parameter.…”

Section: B Anomalous Commutatorsmentioning

confidence: 99%

Section: The Two-dimensional δ-Function Interactionmentioning

confidence: 99%

“…where the physical dimensions of the original theory are preserved by changing the dimensions of the coupling according to g → g µ ǫ [7].…”

Section: Dimensional Regularizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anomalous commutator algebra for conformal quantum mechanics

et al. 2003

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Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics

et al. 2001

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A thorough analysis is presented of the class of central fields of force that exhibit: (i) dimensional transmutation and (ii) rotational invariance. Using dimensional regularization, the twodimensional delta-function potential and the D-dimensional inverse square potential are studied. In particular, the following features are analyzed: the existence of a critical coupling, the boundary condition at the origin, the relationship between the bound-state and scattering sectors, and the similarities displayed by both potentials. It is found that, for rotationally symmetric scale-invariant potentials, there is a strong-coupling regime, for which quantummechanical breaking of symmetry takes place, with the appearance of a unique bound state as well as of a logarithmic energy dependence of the scattering with respect to the energy.

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Point-particle effective field theory I: classical renormalization and the inverse-square potential

Burgess

Hayman

Williams

et al. 2017

J. High Energ. Phys.

109

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Singular potentials (the inverse-square potential, for example) arise in many situations and their quantum treatment leads to well-known ambiguities in choosing boundary conditions for the wave-function at the position of the potential's singularity. These ambiguities are usually resolved by developing a self-adjoint extension of the original problem; a non-unique procedure that leaves undetermined which extension should apply in specific physical systems. We take the guesswork out of this picture by using techniques of effective field theory to derive the required boundary conditions at the origin in terms of the effective point-particle action describing the physics of the source. In this picture ambiguities in boundary conditions boil down to the allowed choices for the source action, but casting them in terms of an action provides a physical criterion for their determination. The resulting extension is self-adjoint if the source action is real (and involves no new degrees of freedom), and not otherwise (as can also happen for reasonable systems). We show how this effective-field picture provides a simple framework for understanding well-known renormalization effects that arise in these systems, including how renormalization-group techniques can resum non-perturbative interactions that often arise, particularly for non-relativistic applications. In particular we argue why the low-energy effective theory tends to produce a universal RG flow of this type and describe how this can lead to the phenomenon of reaction catalysis, in which physical quantities (like scattering cross sections) can sometimes be surprisingly large compared to the underlying scales of the source in question. We comment in passing on the possible relevance of these observations to the phenomenon of the catalysis of baryon-number violation by scattering from magnetic monopoles.

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Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics

Cited by 52 publications

References 48 publications

Anomalous commutator algebra for conformal quantum mechanics

Anomalous commutator algebra for conformal quantum mechanics

Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics

Point-particle effective field theory I: classical renormalization and the inverse-square potential

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