On the Inequivalence of Renormalization and Self-Adjoint Extensions for Quantum Singular Interactions

A unified S-matrix framework of quantum singular interactions is presented for the comparison of self-adjoint extensions and physical renormalization. For the long-range conformal interaction the two methods are not equivalent, with renormalization acting as selector of a preferred extension and regulator of the unbounded Hamiltonian.Comment: 19 pages, including 2 figures. The title and abstract were changed to more accurately reflect the content. The text was rearranged into sections, with several equations and multiple paragraphs added for clarity; and a few typos were corrected. The central equations and concepts remain unchanged

Anomalous Commutator Algebra for Conformal Quantum Mechanics

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.Comment: 23 pages. A few typos corrected in the final version (which agrees with the published Phys. Rev. D article

Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions

It is demonstrated that a nonrelativistic quantum scale anomaly manifests itself in the appearance of composite operators with complex scaling dimensions. In particular, we study nonrelativistic quantum mechanics with an inverse square potential and consider a composite s-wave operator O=\psi\psi. We analytically compute the scaling dimension of this operator and determine the propagator . The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.Comment: 18 pages, 3 figures; published versio

Magnetic Multilayers: Quasiclassical Transport Via the Kubo Formula

A real‐space quantum approach, based on the Kubo formula, is used to describe the quasiclassical transport behavior of metallic multilayers. We emphasize the differences between the cases of current in the plane of the layers, for which size effects play a dominant role and current perpendicular to the planes, for which we provide several proofs that the so‐called series resistor model holds for all length scales

Anomalous Commutator Algebra for Conformal Quantum Mechanics

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

Electron Transport in Magnetic Inhomogeneous Media

Giant magnetoresistance has been observed in both magnetic multilayers and magnetic granular solids. We develop a framework for unifying these particular realizations of inhomogeneous magnetic media, based on the real-space Kubo formula. It constitutes a spin-dependent form of linear response theory, associated with internal spin-dependent fields arising from spin accumulation; moreover, we discuss the physical meaning of these spin dependences. For magnetic multilayers we discuss the particular cases of collinear and noncollinear configurations, and we consider limiting cases of the elastic mean-free path to inhomogeneity-scale ratio for granular solids. Furthermore, we introduce the concept of magnetically self-averaging systems, which include the current perpendicular to the plane geometry of multilayers and granular solids. In the limit of no spin-flip scattering, we show that there are no length scales associated with the magnetoresistance of self-averaging structures

Renormalization of Singular Potentials and Power Counting

We use a toy model to illustrate how to build effective theories for singular potentials. We consider a central attractive 1/r^2 potential perturbed by a 1/r^4 correction. The power-counting rule, an important ingredient of effective theory, is established by seeking the minimum set of short-range counterterms that renormalize the scattering amplitude. We show that leading-order counterterms are needed in all partial waves where the potential overcomes the centrifugal barrier, and that the additional counterterms at next-to-leading order are the ones expected on the basis of dimensional analysis.Comment: 23 pages, 6 figure

Black Hole Thermodynamics From Near-horizon Conformal Quantum Mechanics

The thermodynamics of black holes is shown to be directly induced by their near-horizon conformal invariance. This behavior is exhibited using a scalar field as a probe of the black hole gravitational background, for a general class of metrics in D spacetime dimensions (with D≥4). The ensuing analysis is based on conformal quantum mechanics, within a hierarchical near-horizon expansion. In particular, the leading conformal behavior provides the correct quantum statistical properties for the Bekenstein-Hawking entropy, with the near-horizon physics governing the thermodynamics from the outset. Most importantly: (i) this treatment reveals the emergence of holographic properties; (ii) the conformal coupling parameter is shown to be related to the Hawking temperature; and (iii) Schwarzschild-like coordinates, despite their “coordinate singularity,” can be used self-consistently to describe the thermodynamics of black holes