Exact solutions ofn-level systems and gauge theories

Abstract

We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-1 2 particles in a magnetic field B i (t) can be formulated in a natural way as an SU (2) gauge theory, with the magnetic field B i (t) playing the role of the gauge potential A i . The present approach can also be applied to systems of n levels with time-dependent potentials, U (n) being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schrödinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed. 03.65.Ca, The discovery of the quantum nature of matter fields is one of the most important events in physics in this century. It has been realized that the quantum nature of matter rests on a more fundamental level than the corresponding classical limit. The last one emerges as a suitable limit case of quantum behavior [1]. Also, gauge as well as geometric aspects of quantum systems are one of the most striking results in quantum theory [2][3][4][5]. Gauge theories appear in many branches of physics [6]. They are the natural language to incorporate internal symmetries of systems.On the other hand, the physics of systems of two levels has been one of the most interesting research lines followed up to now. In particular, the dynamics of spin-1 2 particles in time-dependent magnetic fields is a common physical situation in neutron interferometry [7,8]. This type of physical application belongs to n-level systems with time-dependent potentials in quantum mechanics. Nevertheless, to find exact solutions of the Schrödinger equation with time-dependent potentials for n-level systems is a hard problem. The standard theoretical techniques employed to solve this type of physical situation are limited.Here, we combine the topics of exact solutions and gauge theories. The final result is twofold. First, we exhibit the geometric structure underlying unitary transformations associated with n-level quantum systems characterized by time-dependent Hamiltonians. Second, this formalism provides a powerful method to find exact solutions to a very general class of time-dependent Hamiltonians. Moreover, because the present formalism is exact, it is a serious alternative for the standard techniques allowed in the literature to study this type of physical situation, namely, unitary evolution and time-dependent perturbation theory. Now, let us go to the nonrelativistic dynamics of spin-1 2 particles (we mean the spinorial part of the Hamiltonian of spin-1 2 particles). The nonrelativistic dynamics of spin-1 2 particles in a magnetic field B(t) = (B 1 (t), B 2 (t), B 3 (t)) is given by the Schrödinger-Pauli equationwhere µ B = eh 2mec is the Bohr magneton, S i =h 2 σ i with σ i the Pauli matrices, and g s depends on the spe...

Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model

“…The exact solutions and the geometric phase factor of the time-dependent spin model have been extensively investigated by many authors [12][13][14][15][16][17][18][19]. Bouchiat and Gibbons discussed the geometric phase for the spin-1 system [12].…”

“…The dynamical algebraic structure of a type of general case of spin model such as the time-dependent L − S coupled system has been investigated and a set of SU (N ) generators was constructed to linearize the Hamiltonian by Cen et al [15]. The relationship between the unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and the gauge theories has been found by Montesinos et al [17]. They discussed the time evolution of the Heisenberg spin system and obtained the formally exact solutions of the Schrödinger equation in a time-dependent magnetic field.…”

Non-Commutative Worlds and Classical Constraints

“…Much of the present work is the fruit of a long series of discussions with Pierre Noyes. paper [27] also works with minimal coupling for the FeynmanDyson derivation. The first remark about the minimal coupling occurs in the original paper by Dyson [1], in the context of Poisson brackets.…”

From Feynman proof of Maxwell equations to noncommutative quantum mechanics

“…[8]. Other works [10,11,12] have provided new insights ino the Feynman's derivation of the Maxwell's equations. Recently [13], some of the authors embedded Feynman's derivation of the Maxwell's equation in the framework of noncommutative geometry.…”

“…Although with this approach, a Lagrangian or Hamiltonian structure is unnecessary, Hojman and Shepley [3] showed that by using a Helmholtz inverse variational problem under certain conditions, an action can be associated to these Feynman commutation relations. The interpretation of the Feynman's derivation of Maxwell's equations has generated [3,4,5,6,7,8,9,10,11,12] a great interest among physicists. In particular, Tanimura [4] has generalized the Feynman's derivation in a Lorentz covariant form with a scalar time evolution parameter.…”