A complete perturbative expansion for quantum mechanics with constraints

Maraner, Paolo

doi:10.1088/0305-4470/28/10/021

Cited by 47 publications

(94 citation statements)

References 13 publications

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“…The analysis in this section is along the line of what is usually known as constrained quantum system in the literature. A partial list of references is [21][22][23][24][25][26][27][28]. Here one considers a nonrelativistic classical system in an ambient space with a potential that tries to confine the motion into a submanifold.…”

Section: Isrn High Energy Physicsmentioning

confidence: 99%

“…Here we consider the transverse degrees of freedom to be frozen in the harmonic oscillator ground state and derive the effective Hamiltonian, as will be defined in (22) below, for the longitudinal degree of freedom at the leading order. This will give us the linearized tachyon effective equation (see (24)) at this order. We will explain this analogy later in Section 5.…”

Section: Analogue Of Linearized Tachyon Effective Equation Atmentioning

confidence: 99%

“…(b) Hamiltonian is a constraint. The effective form of this constraint on the submanifold obtained by integrating out the transverse (internal) degrees of freedom is supposed to give the linearized equation of motion for the string field components on M. This explains why (24) has been interpreted to be analogue of the linearized tachyon effective equation.…”

Section: Analogy With Finite Dimensional Modelmentioning

confidence: 99%

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On a Semiclassical Limit of Loop Space Quantum Mechanics

Mukhopadhyay

2013

ISRN High Energy Physics

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Following earlier work, we view two-dimensional nonlinear sigma model as single particle quantum mechanics in the free loop space of the target space. In a natural semiclassical limit of this model, the wavefunction localizes on the submanifold of vanishing loops. One would expect that the semiclassical expansion should be related to the tubular expansion of the theory around the submanifold and effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. We develop a framework to carry out such an analysis at the leading order. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar. The steps are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semiclassical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in loop space using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of target space which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model, we arrive at the final result for LSQM.

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Section: Isrn High Energy Physicsmentioning

confidence: 99%

Section: Analogue Of Linearized Tachyon Effective Equation Atmentioning

confidence: 99%

Section: Analogy With Finite Dimensional Modelmentioning

confidence: 99%

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On a Semiclassical Limit of Loop Space Quantum Mechanics

Mukhopadhyay

2013

ISRN High Energy Physics

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“…We give only a short sketch of the proof and refer to [10,8] for the details. Introducing Fermi coordinates [11] in a small tubular neighbourhood of each level set, the Hamiltonian (2.1) takes the form…”

Section: Thermodynamic Integrationmentioning

confidence: 99%

Comment on two distinct notions of free energy

Hartmann¹,

Schütte²

2007

Physica D: Nonlinear Phenomena

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Abstract. In classical molecular dynamics free energy is arguably one of the most important quantities in analyzing a molecular system. In addition to the standard free energy there is a related free energy concept that is prevalent in transition state theory, but which relies on a different ensemble concept. We show that problems that rely on either definition can be treated in a uniform way using constrained molecular dynamics with no need for unbiasing the respective probability ensembles. Not only proves this useful in designing algorithms that sample the free energy landscape, but it also clarifies the relation between various results that are available in the literature. In particular we explain that the famous Blue Moon formula is an instance of Federer's co-area formula, and can easily be generalized to phase space observables. We moreover argue that Blue Moon reweighting also becomes an issue for first-order dynamical systems (e.g., Brownian motion).

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“…These geometrical terms where first written down by Maraner [10]. The dynamical terms denoted by U depend upon derivatives of the potential.…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanics on moduli spaces

Moss¹,

Shiiki²

2000

Nuclear Physics B

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It has been assumed that it is possible to approximate the interactions of quantized BPS solitons by quantising a dynamical system induced on a moduli space of soliton parameters. General properties of the reduction of quantum systems by a Born-Oppenheimer approximation are described here and applied to sigma models and their moduli spaces in order to learn more about this approximation. New terms arise from the reduction proceedure, some of them geometrical and some of them dynamical in nature. The results are generalised to supersymmetric sigma models, where most of the extra terms vanish.Pacs numbers: 03.70.+k, 98.80.Cq

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A complete perturbative expansion for quantum mechanics with constraints

Abstract: A complete perturbative expansion for the Hamiltonian describing the motion of a quantomechanical system constrained to move on an arbitrary submanifold of its configuration space R n is obtained.

Cited by 47 publications

References 13 publications

On a Semiclassical Limit of Loop Space Quantum Mechanics

On a Semiclassical Limit of Loop Space Quantum Mechanics

Comment on two distinct notions of free energy

Quantum mechanics on moduli spaces

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