Models are developed for the motion of charge-2 Abelian Higgs vortices through the 2-vortex moduli space $M$, with the vortices excited by their shape mode oscillations. The models simplify to the well-known geodesic flow on $M$, modified by a potential, when the mode oscillations are fast relative to the moduli space motion and their amplitudes are small. When the lowest-frequency mode is excited with a large amplitude, the geodesic flow is not a correct description. Instead, a chaotic, or even fractal, multi-bounce structure in vortex-vortex collisions is predicted.
We connect the translation modes of the instanton in the two-dimensional Abelian Higgs model with local translations of the vortex of the related model in (3+1) dimensions, the Nielsen-Olesen vortex. In this context these modes describe collective oscillations of the string. We construct the wave function of this mode and we derive, via a virial theorem, an effective action for these oscillations, which is consistent with the action constructed by Nielsen and Olesen using general arguments. We discuss some aspects of renormalization, based on a recent computation of one loop corrections to str…
Laughlin's Ansatz to explain the fractional Quantum Hall effect is derived by coupling a particle associated with ``exotic'' the two-fold central extension of the planar Galilei group. The reduced system is identical to the one used to describe the dynamics of vortices in an incompressible planar fluid.
Vortex dynamics in a thin superfluid ${}^4$He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the ``exotic'' extension of the planar Galilei group.
We show that the system where $CP^1$ model coupled to Hopf term can reveal fractional spin in a collective coordinate quantization scheme, provided one makes a transition to physically inequivalent sector within a same solitonic sector characterized by a nonvanishing topological number
We develope a formalism for the scattering off D-branes that includes collective coordinates. This allows a systematic expansion in the string coupling constant for such processes, including a worldsheet calculation for the D-brane's mass.
The $CP^2$ model, with and without a generalized Hopf term, is studied using the collective coordinate approximation. In the spirit of this approximation, an ansatz is given which in previous numerical studies was seen to give a good parameterization of the numerical solution. The equations of motion for the collective coordinates are then solved analytically, for solitons close together and for solitons far apart. The solutions show how the generalized Hopf term changes the scattering angle which in its absence is $90^{\circ}$.
A field theoretical perturbation theory in inverse powers of coupling constant is developed which is manifestly covariant in every order of the expansion. A dilatation operator serves as an evolution dynamical one in a scale non-invariant theory.
Collective coordinate quantization of Dirichlet branes is discussed. Utilizing Polchinski's combinatoric rule, semiclassical D-brane wave functional is given in proper-time formalism. D-brane equation of motion is then identified with renormalization group equation of defining Dirichlet open string theory. Quantum mechanical size of macroscopically charged D-brane is illustrated and striking difference from ordinary field theory BPS particle is emphasized.
The Becchi-Rouet-Stora-Tyutin (BRST) treatment for the quantization of collective coordinates is considered in the Lagrangian formalism. The motion of a particle in a Riemannian manifold is studied in the case when the classical solutions break a non-abelian global invariance of the action. Collective coordinates are introduced, and the resulting gauge theory is quantized in the BRST antifield formalism. The partition function is computed perturbatively to two-loops, and it is shown that the results are independent of gauge-fixing parameters.
Various soliton-obstruction systems have been studied from analytical perspective. We have used collective coordinate to approach the dynamics of solitons as they meet a potential obstruction in a form of square barriers and holes for three models in (1+1) dimensions, namely: $λϕ^{4}$ model, deformed Sine-Gordon model, and a model that give rise to Q-ball solution. We have shown that our approximated field solution is valid enough to describe the behaviour of solitons scattering off a potential obstruction.
A Collective coordinate variable for adding a space dependent potential to the sine-Gordon model is presented. Interaction of solitons with a delta function potential barrier and also delta function potential well is investigated. Most of the characters of interaction are derived analytically. We will find that the behaviour of a solitonic solution is like a point particle which is moved under the influence of a complicated effective potential. The effective potential is a function of the field initial conditions and also parameters of added potential.
In this paper I describe a collective coordinate approach to the study of giant graviton states and their excitations in various field theories. The method simplifies considerably the understanding of emergent gauge symmetry of these configurations, as well as the calculation of the spectrum of strings stretched between the giant gravitons. There is a limit where these results reproduce the one loop dispersion relation for giant magnons. I also show that this method gives rise to a simple geometric interpretation of a Higgs mechanism for the emergent gauge symmetry which parallels the holograp…
Collective coordinates provide a powerful tool for separating collective and elementary excitations, allowing both to be treated in the full quantum theory. The price is a canonical transformation which leads to a complicated starting point for subsequent calculations. Sometimes the collective behavior of a soliton is simple but nontrivial, and one is interested in the elementary excitations. We show that in this case an alternative prescription suffices, in which the canonical transformation is not necessary. The use of a nonperturbative operator which creates a soliton state allows the theor…
We apply The Batalin-Tyutin constraint formalism of converting a second class system into a first class system for the rotational quantisation of the SU(2) Skyrme model. We obtain the first class constraint and the Hamiltonian in the extended phase space. The vacuum functional is constructed and evaluated in the unitary gauge and a multiplier dependent gauge. Finally, we discuss the spectrum of the extended theory. The use of the BT formalism on the collective coordinates quantisation of the SU(2) Skyrme model leads an additional term in the usual quantum Hamiltonian that can improve the pheno…
The fractal velocity pattern in symmetric kink-antikink collisions in $ϕ^4$ theory is shown to emerge from a dynamical model with two effective moduli, the kink-antikink separation and the internal shape mode amplitude. The shape mode usefully approximates Lorentz contractions of the kink and antikink, and the previously problematic null-vector in the shape mode amplitude at zero separation is regularized.
The problem of construction of fiber bundle over the moduli space of the Skyrme model is considered. We analyse an extension of the original Skyrme model which includes the minimal interaction with fermions. An analogy with modili space of the fermion-monopole system is used to construct a fiber bundle structure over the skyrmion moduli space. The possibility of the non-trivial holonomy appearance is considered. It is shown that the effect of the fermion interaction turns the $n$-skyrmion moduli space into a real vector bundle with natural $SO(2n+1)$ connection.
The emergence of violations of conformal invariance in the form of non-local operators in the two-dimensional action describing solitons inevitably leads to the introduction of collective coordinates as two dimensional ``wormhole parameters''.
In this Letter the method of Lund is applied to formulate a variational principle for the motion of charged vortices in an effective non-linear Schrödinger field theory describing finite size two-dimensional quantum Hall samples under the influence of an arbitrary perpendicular magnetic field. Freezing out variations in the modulus of the effective field yields a $U(1)$ sigma-model. A duality transformation on the sigma-model reduces the problem to finding the Green function for a related electrostatics problem. This duality illuminates the plasma analogy to the Laughlin wave function.
The collective coordinates approximation for the kink/anti-kink scattering in the $1+1$ dimensional $ϕ^4$ model is considered and we discuss how the results found in the current literature on the topic can be improved by giving the analytical expression for the lagrangian in the space of parameters. A comprehensive discussion of the role of the collective coordinates approximation in this particular situation is given for completeness.
We compare numerical solutions to the full field equations to simplified approaches based on implementing three collective coordinates for kink-antikink interactions within the $φ^4$ and $ϕ^6$ models in one time and one space dimensions. We particularly pursue the question whether the collective coordinate approximation substantiates the conjecture that vibrational modes are important for resonance structures to occur in kink-antikink scattering.
The collective coordinates expansion of the Skyrme soliton particle model gives rise to the second class constraints. We use the non-abelian BFFT formalism to convert this system into the one with only first class constraints. Choosing two different structure functions of the non-abelian algebra, we obtain simplified algebraic expressions for the first class non-abelian Hamiltonians. This result shows that the non-abelian BFFT method is, in many aspects, richer than the abelian BFFT formalism. For both of the first class Hamiltonians, we derive the Lagrangians which lead to the new theory. Whe…
Collective coordinate methods are frequently applied to study dynamical properties of solitons. These methods simplify the field equations - typically partial differential equations - to ordinary differential equations for selected excitations. More importantly though, collective coordinates provide a practical means to focus on particular modes of otherwise complicated dynamical processes. We review the application of collective coordinate methods in the analysis of the kink-antikink interaction within the $φ^4$ soliton model and illuminate discrepancies between these methods and the exact re…
We develop a consistent relativistic generalization of collective coordinate quantization of field theory solitons. Our principle of introducing collective coordinates is that the equations of motion of the collective coordinates ensure those of the original field theory. We illustrate this principle with the quantization of spinning degrees of freedom of Skyrmion representing baryons. We calculate the leading relativistic corrections to the static properties of nucleons, and find that the corrections are non-negligible ones of 10% to 20%.
We present a collective coordinate approximation to model the dynamics of two interacting nonlinear Schrödinger (NLS) solitons. We discuss the accuracy of this approximation by comparing our results to those of the full numerical simulations and find that it is remarkably accurate not only when the solitons are some distance apart but also during their interaction.
A string theory in $3$ euclidean spacetime dimensions is found to describe the semiclassical behavior of a certain exact physical state of quantum general relativity in $4$ dimensions. Both the worldsheet and the three dimensional metric emerge as collective coordinates that describe a sector of the solution space of quantum general relativity. Additional collective coordinates exist which are interpreted as worldsheet degrees of freedom. The construction may be extended to the case in which the Kalb-Ramond field is included in the non-perturbative dynamics. It is possible that this mechanism …
The introduction and quantization of a center-of-mass coordinate is demonstrated for the one-soliton sector of nonlinear field theories in (1+1) dimensions. The present approach strongly emphazises the gauge and BRST-symmetry aspects of collective coordinate quantization. A gauge is presented which is independent of any approximation scheme and which allows to interpret the new degree of freedom as the {\em quantized} center of mass coordinate of a soliton. Lorentz invariance is used from the beginning to introduce fluctuations of the collective coordinate in the {\em rest frame} of the {\em m…
We consider constructing the relativistic system of collective coordinates of a field theory soliton on the basis of a simple principle: The collective coordinates must be introduced into the static solution in such a way that the equation of motion of the collective coordinates ensures that of the original field theory. As an illustration, we apply this principle to the quantization of spinning motion of the Skyrmion by incorporating the leading relativistic correction to the rigid body approximation. We calculate the decay constant and various static properties of nucleons, and find that the…
We write a Ginzburg-Landau Hamiltonian for a charged order parameter interacting with a background electromagnetic field in 2+1 dimensions. Using the method of Lund we derive a collective coordinate action for vortex defects in the order parameter and demonstrate that the vortices are charged. We examine the classical dynamics of the vortices and then quantize their motion, demonstrating that their peculiar classical motion is a result of the fact that the quantum motion takes place in the lowest Landau level. The classical and quantum motion in two dimensional regions with boundaries is also …
We develop the formulation of turbulence in terms of the functional integral over the phase space configurations of the vortex cells. The phase space consists of Clebsch coordinates at the surface of the vortex cells plus the Lagrange coordinates of this surface plus the conformal metric. Using the Hamiltonian dynamics we find an invariant probability distribution which satisfies the Liouville equation. The violations of the time reversal invariance come from certain topological terms in effective energy of our Gibbs-like distribution. We study the topological aspects of the statistics and use…
We investigate the validity of collective coordinate approaximations to the scattering of solitons in several classes of models in (1+1) dimensional field theory models. We look at models which are deformations of the sine-Gordon (SG) or the nonlinear Schrödinger (NLS) model as they posses solitons which are topological (SG) or non-topological (NLS). Our deformations preserve their topology (SG), but changes their integrability properties, either completely or partially (models become `quasi-integrable'). As our collective coordinate approximation does not allow for the radiation of energy out…
Moduli spaces - finite-dimensional, collective coordinate manifolds - for kinks and antikinks in $ϕ^4$ theory and sine-Gordon theory are reconsidered. The field theory Lagrangian restricted to moduli space defines a reduced Lagrangian, combining a potential with a kinetic term that can be interpreted as a Riemannian metric on moduli space. Moduli spaces should be metrically complete, or have an infinite potential on their boundary. Examples are constructed for both kink-antikink and kink-antikink-kink configurations. The naive position coordinates of the kinks and antikinks sometimes need to b…
We thoroughly compare the bound state and rigid rotator approaches to three--flavored chiral solitons. We establish that these two approaches yield identical results for the baryon spectrum and kaon--nucleon $S$--matrix in the limit that the number of colors ($N_C$) tends to infinity. After proper subtraction of the background phase shift the bound state approach indeed exhibits a clear resonance behavior in the strangeness $S=+1$ channel. We present a first dynamical calculation of the widths of the $Θ^+$ and $Θ^*$ pentaquarks for finite $N_C$ in a chiral soliton model.
