Time-dependent scale-gauge transformations and the Adiabatic Principle

We discuss how symmetries and conservation laws are affected when Berry’s phase occurs in a quantum system: symmetry transformations of coordinates have to be supplemented by gauge transformations of Berry’s connection, and consequently constants of motion acquire terms beyond the familiar kinematical ones. We show how symmetries of a problem determine Berry’s connection, curvature and, once a specific path is chosen, the phase as well. Moreover, higher order corrections are also fixed. We demonstrate that in some instances Berry’s curvature and phase can be removed by a globally well-defined, time-dependent canonical transformation. Finally, we describe how field theoretic anomalies may be viewed as manifestations of Berry’s phase.

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GAUGE TRANSFORMATIONS AND GAUGE-FIXING CONDITIONS IN CONSTRAINT SYSTEMS

International Journal of Modern Physics A ◽

10.1142/s0217751x92000041 ◽

1992 ◽

Vol 07 (01) ◽

pp. 61-89 ◽

Cited By ~ 11

Author(s):

REIJI SUGANO ◽

YUSHO KAGRAOKA ◽

TOSHIEI KIMURA

Keyword(s):

Degrees Of Freedom ◽

Time Dependent ◽

Lagrangian Formalism ◽

Gauge Transformations ◽

Constraint Systems ◽

Gauge Fixing ◽

Stationarity Conditions ◽

Gauge Conditions

Gauge transformations and gauge-fixing conditions in the total Hamiltonian (H T ) and extended Hamiltonian (H E ) formalisms are investigated. For gauge-fixing conditions χα, only the condition det ({ϕα, χβ}) ≠ 0 is usually imposed, where ϕα are first class constraints. This condition is not sufficient and one should (i) employ H T and (ii) choose the gauge-fixing conditions χα to be stationary under H T . Gauge degrees of freedom in the Lagrangian formalism are equal in number to the primary first class constraints [Formula: see text]. Hence the number of arbitrarily chosen primary gauge conditions [Formula: see text] is the same as that of [Formula: see text]. Secondary gauge-fixing conditions associated with secondary first class constraints should be determined by the stationarity conditions of [Formula: see text]. If a canonical Hamiltonian (weakly) vanishes, χα must be explicitly time-dependent, otherwise we have the trivial result. Further, it is pointed out that the H E formalism has discrepancies with the H T formalism in many aspects. As illustrations of these properties, a few typical models are examined.

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ADVANCES IN TERNARY AND OCTONIONIC GAUGE FIELD THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x11053778 ◽

2011 ◽

Vol 26 (18) ◽

pp. 2997-3012 ◽

Cited By ~ 3

Author(s):

CARLOS CASTRO

Keyword(s):

Field Theory ◽

Gauge Field ◽

Lie Algebra ◽

Space Time ◽

Time Dependent ◽

Gauge Field Theory ◽

Gauge Fields ◽

Gauge Transformations ◽

Quadratic Action ◽

Rigid Transformations

A ternary gauge field theory is explicitly constructed based on a totally antisymmetric ternary-bracket structure associated with a 3-Lie algebra. It is shown that the ternary infinitesimal gauge transformations do obey the key closure relations [δ1, δ2] = δ3. Invariant actions for the 3-Lie algebra-valued gauge fields and scalar fields are displayed. We analyze and point out the difficulties in formulating a nonassociative octonionic ternary gauge field theory based on a ternary-bracket associated with the octonion algebra and defined earlier by Yamazaki. It is shown that a Yang–Mills-like quadratic action is invariant under global (rigid) transformations involving the Yamazaki ternary octonionic bracket, and that there is closure of these global (rigid) transformations based on constant antisymmetric parameters Λab = - Λba. Promoting the latter parameters to space–time dependent ones Λab(xμ) allows one to build an octonionic ternary gauge field theory when one imposes gauge covariant constraints on the latter gauge parameters leading to field-dependent gauge parameters and nonlinear gauge transformations. In this fashion one does not spoil the gauge invariance of the quadratic action under this restricted set of gauge transformations and which are tantamount to space–time dependent scalings (homothecy) of the gauge fields.

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Asymptotic channels and gauge transformations of the time-dependent Dirac equation for extremely relativistic heavy-ion collisions

Physical Review A ◽

10.1103/physreva.59.346 ◽

1999 ◽

Vol 59 (1) ◽

pp. 346-357 ◽

Cited By ~ 12

Author(s):

J. C. Wells ◽

B. Segev ◽

J. Eichler

Keyword(s):

Dirac Equation ◽

Heavy Ion Collisions ◽

Heavy Ion ◽

Relativistic Heavy Ion Collisions ◽

Time Dependent ◽

Gauge Transformations ◽

Ion Collisions

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THE QUANTUM TUNNELLING BETWEEN TWO-COMPONENT BOSE–EINSTEIN CONDENSATES IN A DOUBLE-WELL CONFIGURATION

International Journal of Modern Physics B ◽

10.1142/s0217979205031985 ◽

2005 ◽

Vol 19 (20) ◽

pp. 3285-3292

Author(s):

GUO-FENG ZHANG

Keyword(s):

Berry Phase ◽

Time Dependent ◽

Hamilton Operator ◽

Gauge Transformations ◽

Quantum Tunnelling ◽

Interacting Species ◽

Particle Population ◽

Two Component ◽

Population Imbalance ◽

Bose Einstein

We examine in terms of exact solutions of the time-dependent Schrödinger equation, the quantum tunnelling process in Bose–Einstein condensates of two interacting species trapped in a double well configuration. Based on the two series of time-dependent SU(2) gauge transformations, we diagonalize the Hamilton operator and obtain analytic time-evolution formulas of the population imbalance and the berry phase. The particle population imbalance [Formula: see text] of species A between the two wells is studied analytically.

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INVARIANCE OF THE GENERALIZED HANNAY ANGLE UNDER GAUGE TRANSFORMATIONS: APPLICATION TO THE TIME-DEPENDENT GENERALIZED HARMONIC OSCILLATOR

International Journal of Modern Physics B ◽

10.1142/s021797929300281x ◽

1993 ◽

Vol 07 (11) ◽

pp. 2147-2162

Author(s):

DONALD H. KOBE

Keyword(s):

Harmonic Oscillator ◽

Time Derivative ◽

Classical Mechanics ◽

Rate Of Change ◽

Time Dependent ◽

Gauge Transformations ◽

Correct Energy ◽

The One ◽

Time Dependent Problems ◽

Generalized Harmonic Oscillator

The Hannay angle of classical mechanics is generalized so that it is invariant under gauge transformations, which are a restricted class of canonical transformations. A distinction between the Hamiltonian and the energy is essential to make in time-dependent problems. A time-dependent generalized harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. The energy, the time derivative of which is the power, is obtained from the equation of motion and related to the action variable. Hamilton’s equations give the time rate of change of the angle and action variables. The generalized Hannay angle is shown to be zero, and remains invariant under gauge transformations. On the other hand, if the original Hamiltonian is chosen as the energy, a nonzero generalized Hannay angle is obtained, but the power is given incorrectly. Nevertheless in the adiabatic limit, the total angle, which is the sum of the dynamical and Hannay angles, is equal to the one calculated from the correct energy.

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