Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics

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

doi:10.1006/aphy.2000.6093

Cited by 47 publications

(74 citation statements)

References 28 publications

(25 reference statements)

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“…8 As many have observed [28][29][30][31][32][33][34][35][36][37][38][39][40] because of this the Schrödinger Hamiltonian can fail to be self-adjoint, depending the boundary conditions that hold at r = . Selecting a choice of boundary condition to secure its self-adjointness -not a unique construction -is known as constructing its self-adjoint extension [41][42][43][44][45][46][47][48].…”

Section: Jhep07(2017)072mentioning

confidence: 99%

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Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Burgess

Hayman

Rummel

et al. 2017

J. High Energ. Phys.

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We apply point-particle effective field theory (PPEFT) to compute the leading shifts due to finite-sized source effects in the Coulomb bound energy levels of a relativistic spinless charged particle. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees with standard calculations in several ways. Most notably we find there are two effective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to δE arises linearly (rather than quadratically) in the small length scale, , characterizing the finite-size effects, and is suppressed by (Zα) 5 . We argue that standard calculations miss the contributions of this second operator because they err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source's charge radius, as well as on the orbiting particle's mass. Its contribution turns out to be crucial if the charge radius satisfies (Zα) 2 a B , where a B is the Bohr radius, because then relativistic effects become important for the boundary condition. We show how the problem is equivalent to solving the Schrödinger equation with competing Coulomb, inverse-square and delta-function potentials, which we solve explicitly. A similar enhancement is not predicted for the hyperfine structure, due to its spin-dependence. We show how the charge-radius effectively runs due to classical renormalization effects, and why the resulting RG flow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We discuss how this flow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these effects would require more precise measurement of energy levels for mesonic atoms than are now possible.

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Section: Jhep07(2017)072mentioning

confidence: 99%

“…To be trusted for any UV scale on the RG flow we must askλ( ) to be such that . Takingλ 0 → ∞ in the RG flow (2.22) impliesλ 32) and so demanding impliesλ( ) −ζ s (1 + δ) with 0 < δ 1, and it is only for such couplings in the UV that a macroscopic bound state of the form (2.28) can be trusted.…”

Section: Jhep07(2017)072mentioning

confidence: 99%

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Burgess

Hayman

Rummel

et al. 2017

J. High Energ. Phys.

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“…In addition, the ubiquity of the Calogero model [18], from black holes [19] to applications in condensed-matter physics [20,21], has led to alternative applications of a formally identical algebra of conformal generators. Finally, the use of fieldtheory renormalization techniques has promoted novel methods for the treatment of singular interactions, including those within the conformal quantum mechanics class, by means of Hamiltonian [5,22,23,24,25,26] as well as path-integral techniques [27,28,29].…”

Section: Introductionmentioning

confidence: 99%

SO(2,1) conformal anomaly: Beyond contact interactions

2003

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The existence of anomalous symmetry-breaking solutions of the SO(2,1) commutator algebra is explicitly extended beyond the case of scale-invariant contact interactions. In particular, the failure of the conservation laws of the dilation and special conformal charges is displayed for the two-dimensional inverse square potential. As a consequence, this anomaly appears to be a generic feature of conformal quantum mechanics and not merely an artifact of contact interactions. Moreover, a renormalization procedure traces the emergence of this conformal anomaly to the ultraviolet sector of the theory, within which lies the apparent singularity.

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“…The quantum-mechanical inverse square potential has been extensively dealt with before via the Schrödinger equation [8,9,10,11], and to a lesser extent with path integrals [12,13,14,15,16]. A recent proposal for a comprehensive treatment of this problem was advanced within the Schrödinger-equation approach, properly combined with regularization and renormalization concepts borrowed from quantum field theory.…”

Section: Introductionmentioning

confidence: 99%

“…A recent proposal for a comprehensive treatment of this problem was advanced within the Schrödinger-equation approach, properly combined with regularization and renormalization concepts borrowed from quantum field theory. This was done first for the one-dimensional case with a real-space regulator [9]; later, it was generalized to a complete analysis of the D-dimensional case using different regularization schemes [10,11].…”

Section: Introductionmentioning

confidence: 99%

Path Integral Treatment of Singular Problems and Bound States

Camblong

Ordóñez

2004

Int. J. Mod. Phys. A

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A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential-possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analogue of ultraviolet regularization is required for the interpretation of these singular problems.

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Dimensional Transmutation and Dimensional Regularization in Quantum Mechanics

Cited by 47 publications

References 28 publications

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

SO(2,1) conformal anomaly: Beyond contact interactions

Path Integral Treatment of Singular Problems and Bound States

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