The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energydependent s-wave scattering matrix element.PACS numbers: 03.65. Ge, 03.65.Nk, 11.10.Gh, The quantum-mechanical inverse square potential is a singular problem that has generated controversy for decades. For instance, the solution proposed in Ref.[1] failed to give a Hamiltonian bounded from below, and this led to a number of alternative regularization techniques [2-4] based on appropriate parametrizations of the potentialincluding the replacement [5] of self-adjointness by an interpretation of the "fall of the particle to the center" [6]. However, it is generally recognized that the singular nature of this problem lies in that its Hamiltonian, being symmetric but not self-adjoint, admits self-adjoint extensions [7]. Recently, a renormalized solution was presented using field-theoretic techniques [8], but it was just limited to the one-dimensional case and cutoff renormalization.In this Letter (i) we generalize the results of Ref.[8] to D dimensions (including the all-important D = 3 case) using cutoff regularization in configuration space; (ii) present a complete picture of the renormalized theory; and (iii) confirm the same conclusions using dimensional regularization [9]. This problem is crucial for the analysis and interpretation of the point dipole interaction of molecular physics [10,11], and may be relevant in polymer physics [12]. In addition (i) it displays remarkable similarities with the two-dimensional δ-function potential [13][14][15]; (ii) it provides another example of dimensional transmutation [16] in a system with a finite number of degrees of freedom; and (iii) it illustrates the relevance of field-theoretic concepts in quantum mechanics [13][14][15]17].This problem is ideally suited for implementation in configuration space [18], where the radial Schrödinger equation for a particle subject to the r −2 potential in D dimensions [19] reads (withh = 1 and 2m = 1)which is explicitly scale-invariant because λ is dimensionless [20]. In Eq. (1), l is the angular momentum quantum number and λ > 0 corresponds to an attractive potential; with the transformation R l (r) = r −(D−1)/2 u l (r), Eq. (1) If λ were allowed to vary, one would see that the nature of the solutions changes around the critical value λ ( * ) l , for each angular momentum state. For λ < λ ( * ) l (including repulsive potentials), the order s l of the Bessel functions is real, so that the solution regular at the origin is proportional to the Bessel function of the first kind J s l √ E r . However, the same solution fails to satisfy the required behavior at ...

The interaction of an electron with a polar molecule is shown to be the simplest realization of a quantum anomaly in a physical system. The existence of a critical dipole moment for electron capture and formation of anions, which has been confirmed experimentally and numerically, is derived. This phenomenon is a manifestation of the anomaly associated with quantum symmetry breaking of the classical scale invariance exhibited by the point-dipole interaction. Finally, analysis of symmetry breaking for this system is implemented within two different models: point dipole subject to an anomaly and finite dipole subject to explicit symmetry breaking.

A number of physical systems exhibit a particular form of asymptotic conformal invariance: within a particular domain of distances, they are characterized by a long-range conformal interaction (inverse square potential), the apparent absence of dimensional scales, and an SO(2,1) symmetry algebra. Examples from molecular physics to black holes are provided and discussed within a unified treatment. When such systems are physically realized in the appropriate strong-coupling regime, the occurrence of quantum symmetry breaking is possible. This anomaly is revealed by the failure of the symmetry generators to close the algebra in a manner shown to be independent of the renormalization procedure.

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.

This is the first in a series of papers addressing the phenomenon of dimensional transmutation in nonrelativistic quantum mechanics within the framework of dimensional regularization. Scale-invariant potentials are identified and their general properties are derived. A strategy for dimensional renormalization of these systems in the strong-coupling regime is presented, and the emergence of an energy scale is shown, both for the bound-state and scattering sectors. Finally, dimensional transmutation is explicitly illustrated for the two-dimensional delta-function potential.