Renormalization in non-relativistic quantum mechanics

Adhikari, Sadhan K.; Ghosh, Angsula

doi:10.1088/0305-4470/30/18/029

Cited by 37 publications

(37 citation statements)

References 21 publications

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“…Strictly, even though a more careful treatment with dimensional regularization changes Eq. (17), the difference appears only at the level of the finite parts (linear in ǫ) and is immaterial to the arguments presented here. These corrections, as well as a detailed analysis of the scattering sector of the theory, will be presented elsewhere.…”

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confidence: 89%

“…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.…”

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Renormalization of the Inverse Square Potential

et al. 2000

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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 ...

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Renormalization of the Inverse Square Potential

et al. 2000

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“…A number of papers 22,23,24,25,26,27 discussed the peculiar features which arise in this EFT when one naively implements different regularization schemes. Ultimately though, the issue here was not associated with the use of a specific regularization scheme.…”

Section: You Can't Choose Your Neighbors: Living Near a Fixed Pointmentioning

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From Hadrons to Nuclei: Crossing the Border

Beane

Bedaque

Haxton

et al. 2001

At the Frontier of Particle Physics

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190

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The study of nuclei predates by many years the theory of quantum chromodynamics. More recently, effective field theories have been used in nuclear physics to "cross the border" from QCD to a nuclear theory. We are now entering the second decade of efforts to develop a perturbative theory of nuclear interactions using effective field theory. This chapter describes the current status of these efforts.

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“…As a case in point, we treat the density matrix of the Lieb-Liniger gas [9], an object of considerable interest to both cold gases [10,11] and integrable systems [12] communities.The GML-RG suggests itself very naturally in problems where a physical quantity at one value of a param-eter is given as an expansion in terms of the value of the same quantity at another value of the parameter, a situation usually caused by regularization and renormalization [5]. A simple single-parameter example is Twhich is the Born series for the scattering problem for the 2D delta-function potential [13]; T (k) is the complex-valued function one is trying to determine (Tmatrix), µ > 0 is a constant (mass), and the expansion holds for all k a , k b > 0 (momenta). The expansion would not seem to be useful unless k b /k a ≈ 1; however, we may try a "divide and conquer" approach: instead of going from k a to k b in a single large step, we use many smaller steps, k a = k 0 → k 1 → k 2 → · · · → k N = k b , for each of which k j+1 /k j ≈ 1.…”

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A Hermite–Padé perspective on the renormalization group, with an application to the correlation function of Lieb–Liniger gas

Dunjko¹,

Olshanii²

2011

J. Phys. A: Math. Theor.

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While Padé approximation is a general method for improving convergence of series expansions, Gell-Mann-Low renormalization group normally relies on the presence of special symmetries. We show that in the single-variable case, the latter becomes an integral Hermite-Padé approximation, needing no special symmetries. It is especially useful for interpolating between expansions for small values of a variable and a scaling law of known exponent for large values. As an example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb-Liniger gas. Using a new result for the 4th-order term in the short-distance expansion, we find a remarkable agreement with known ab initio numerical results.Quantitative information very often comes in the form of a series expansion in a dimensionless variable [1]. Often one knows only the first few terms; but even if one knows many, the raw series is usually useful only for a limited range of values of the expansion parameter. One may therefore turn to various methods for improving the convergence properties. This is justified if one expects that the solution, in the region of interest of the expansion variable, is an analytic continuation of the solution in the region in which one is performing the expansion. The best-known method is the Padé approximation [2] and its various Hermite-Padé generalizations [3].On the other hand, when extrapolations from one parameter region to another were called for within particle physics, the solution was the Gell-Mann-Low renormalization group (GML-RG) [4,5]. This method is normally not comparable to the Padé approximation because it takes as input not only the known terms of the expansion, but also certain known nontrivial symmetries in the problem, an approach that was later formalized and extended to the study of differential equations using Lie group-theoretic methods [6]. We should also mention a related body of work in Ref. [7].We analyze the mechanism of operations of GML-RG in the single-variable (SV) case. We show that the SV GML-RG requires no special symmetry and is in fact an integral Hermite-Padé approximation [8], a connection that remains largely unexplored. The method is applicable to any series expansion, but is especially good for interpolation between a power-series expansion (for small values of a variable) and a scaling law (for large values of the variable), where the scaling law exponent is a known continuous function of a parameter, and one seeks, e.g., an approximation for the prefactor. As a case in point, we treat the density matrix of the Lieb-Liniger gas [9], an object of considerable interest to both cold gases [10,11] and integrable systems [12] communities.The GML-RG suggests itself very naturally in problems where a physical quantity at one value of a param-eter is given as an expansion in terms of the value of the same quantity at another value of the parameter, a situation usually caused by regularization and renormalization [5]. A simple single-parameter example is Twhich is the Born series f...

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Renormalization in non-relativistic quantum mechanics

Cited by 37 publications

References 21 publications

Renormalization of the Inverse Square Potential

Renormalization of the Inverse Square Potential

From Hadrons to Nuclei: Crossing the Border

A Hermite–Padé perspective on the renormalization group, with an application to the correlation function of Lieb–Liniger gas

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