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Mathematics of the USSR-Sbornik

QUANTUM SCATTERING IN GAUGE FIELDS OF ADIABATIC REPRESENTATIONS

Yu. A. Kuperin, Yu. B. Melnikov

Published Feb 28, 1992• 1 citation
Publisher

Abstract

A geometric approach to the method of adiabatic representations is developed for a class of relativistic Hamiltonians. The theory is used to analyze the associated dynamical equations with effective nonabelian interactions that can be regarded as gauge fields, induced by dimensional reduction of the initial problem in a special representation. It is shown that the approach can be used to study 2→(2, 3) quantum scattering processes in a three-body system, and a one-to-one relation between the complete and the effective S -matrices is derived. Asymptotic expressions are found for the solutions of the effective dynamical equation and for the gauge fields in the adiabatic representations. The method is illustrated for systems admitting adiabatic representations with a one-dimensional base; in several cases the field operator is proved to be Hilbert-Schmidt.

DOI
10.1070/sm1992v072n01abeh001412

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A geometric approach to the method of adiabatic representations is developed for a class of relativistic Hamiltonians. The theory is used to analyze the associated dynamical equations with effective nonabelian interactions that can be regarded as gauge fields, induced by dimensional reduction of the initial problem in a special representation. It is shown that the approach can be used to study 2→(2, 3) quantum scattering processes in a three-body system, and a one-to-one relation between the complete and the effective S -matrices is derived. Asymptotic expressions are found for the solutions of the effective dynamical equation and for the gauge fields in the adiabatic representations. The method is illustrated for systems admitting adiabatic representations with a one-dimensional base; in several cases the field operator is proved to be Hilbert-Schmidt.
Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately S d (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.
We introduce and analyze several low-dimensional scattering systems that exhibit geometric phase phenomena. The systems are fully solvable and we compare accurate solutions of them with those obtained in a Born-Oppenheimer projection approximation. We illustrate how geometric magnetism manifests in them, and explore the relationship between solutions obtained in the diabatic and adiabatic pictures. We provide an example, involving a neutral atom dressed by an external field, in which the system mimics the behavior of a charged particle that interacts with, and is scattered by, a ferromagnetic material. We also introduce a similar system that exhibits Aharonov-Bohm scattering. We propose possible practical applications. We provide a theoretical approach that underscores universality in the appearance of geometric gauge forces. We do not insist on degeneracies in the adiabatic Hamiltonian, and we posit that the emergence of geometric gauge forces is a consequence of symmetry breaking in the latter.
Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian:
The conventional formulation of the nonadiabatic (Aharonov–Anandan) phase is based on the equivalence class {eiα(t)ψ(t,x)} which is not a symmetry of the Schrödinger equation. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the nonadiabatic phase. This reformulation reproduces all the successful aspects of the nonadiabatic phase in a manner manifestly consistent with the conventional notion of rays and the superposition principle. The hidden local symmetry is thus identified as the natural origin of the gauge symmetry in both of the adiabatic and nonadiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. The nonadiabatic phase may well be regarded as a special case of the adiabatic phase in this reformulation, contrary to the customary understanding of the adiabatic phase as a special case of the nonadiabatic phase. Some explicit examples of geometric phases are discussed to illustrate this reformulation.
In this paper high-order adiabatic approximate method is proposed to study the evolution of degenerate quantum systums with slowly-changing parameters and varying symmetry. We not only prove the quantum adiabatic theorem for degenerate cases and discuss the corresponding non-abelian induced gauge structure, but also point out the non-adiabatic corrections resulting from higher-order approximations. With nuclear quadrupole resonance as an explicit example, we use this method to analyse observable effects of non-abelian Berry's phases in adiabatic and non-adiabatic experimental process.
A realistic analysis shows that constraining a quantomechanical system produces the effective dynamics to be coupled with abelian/nonabelian gauge fields and quantum potentials induced by the intrinsic and extrinsic geometrical properties of the constraint’s surface. This phenomenon is observable in the effective rotational motion of some simple polyatomic molecules. By considering specific examples it is shown that the effective Hamiltonians for the nuclear rotation of linear and symmetric top molecules are equivalent to that of a charged system moving in a background magnetic-monopole field. For spherical top molecules an explicit analytical expression of a non-abelian monopole-like field is found. Quantum potentials are also relevant for One of the first questions addressed by researchers who first studied the properties of surfaces embedded in the three-dimensional euclidean space, at