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 theory to be constructed perturbatively in terms of the soliton normal modes. We show how translation invariance may be perturbatively imposed. We apply this to construct the two-loop ground state of an arbitrary scalar kink.
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…
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''.
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.
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.
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…
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…
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.
The Lagrangian field-antifield formalism of Batalin and Vilkovisky (BV) is used to investigate the application of the collective coordinate method to soliton quantisation. It is shown how Noether identities and local symmetries of the Lagrangian arise when collective coordinates are introduced in order to avoid divergences related to zero modes. This transformation to collective and fluctuation degrees of freedom is interpreted as a canonical transformation in the symplectic field-antifield space which induces a time-local gauge symmetry. Separating the corresponding Lagrangian path integral o…
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 …
Recently it was shown that NCOS theories are part of a ten-dimensional theory known as Non-relativistic Wound string theory. We clarify the sense in which gravity is present in this theory. We show that Wound string theory contains exceptional unwound strings, including a graviton, which mediate the previously discovered instantaneous long-range interactions, but are negligible as asymptotic states. Unwound strings also provide the expected collective coordinates for the transverse D-branes in the theory. These and other results are shown to follow both from a direct analysis of the effect of …
We study the Das-Jevicki collective field description of arbitrary classical solutions in the c=1 matrix model, which are believed to describe nontrivial spacetime backgrounds in 2d string theory. Our analysis naturally includes the case of a Fermi droplet cosmology: a finite size droplet of Fermi fluid, made up of a finite number of eigenvalues. We analyze properties of the coordinates in which the metric in the collective field theory is trivial, and comment on the form of the interaction terms in these coordinates.
We pursue the construction of higher-spin theory in AdS_4 from CFT_3 of the O(N) vector model in terms of canonical collective fields. In null plane quantization an exact map is established between the two spaces. The coordinates of the AdS_4 space-time are generated from the collective coordinates of the bi-local field. This, in the light cone gauge, provides an exact one to one reconstruction of bulk AdS_4 space-time and higher-spin fields.
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…
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…
The matrix elements connected non physic angles space representation with the space of physic observables of alternative O(2,4) world found on explicit form. All such matrix elements are represented in terms of solutions of Gauss hypergeometric equations. From these results wave function of free moving particle in alternative world may be constructed. Definite relations between micro and macro label variables are presented.
A new instanton solution is found in the quantum-mechanical double-well potential with a four-fermion term. The solution has finite action and depends on four fermionic collective coordinates. We explain why in general the instanton action can depend on collective coordinates.
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…
Calorons of the SU(N) gauge group with non-trivial holonomy, i.e. periodic instantons with arbitrary eigenvalues of the Polyakov line at spatial infinity, can be viewed as composed of N Bogomolnyi--Prasad--Sommerfeld (BPS) monopoles or dyons. Using the metric of the caloron moduli space found previously we compute the integration measure over caloron collective coordinates in terms of the constituent monopole positions and their U(1) phases. In the limit of small separations between dyons and/or trivial holonomy, calorons reduce locally to the standard instantons whose traditional collective c…
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 discuss collective coordinate quantization of the half-BPS geometries of Lin, Lunin and Maldacena (hep-th/0409174). The LLM geometries are parameterized by a single function $u$ on a plane. We treat this function as a collective coordinate. We arrive at the collective coordinate action as well as path integral measure by considering D3 branes in an arbitrary LLM geometry. The resulting functional integral is shown, using known methods (hep-th/9309028), to be the classical limit of a functional integral for free fermions in a harmonic oscillator. The function $u$ gets identified with the cla…
The SU(2) collective coordinates quantization of the Born-Infeld Skyrmions Lagrangean is performed. The obtainment of the classical Hamiltonian from this special Lagrangean is made by using an approximate way: it is derived from the expansion of this non-polynomial Lagrangean up to second-order variable in the collective coordinates, using for this some causality arguments. Because this system presents constraints, we use the Dirac Hamiltonian method and the Faddeev-Jackiw Lagrangean approach to quantize this model.
It is shown that physical states of a non-abelian Yang-Mills-Higgs dyon are invariant under large gauge transformations that do not commute with its magnetic field. This result is established within an enlarged Hamiltonian formalism where surface terms are kept as dynamical variables. These additional variables are parameters of large gauge transformations, and are potential collective coordinates for the quantization of the monopole. Our result implies that there are no physical effects associated to some large gauge transformations and therefore their parameters should not be counted as coll…
In a wide class of supersymmetric theories degenerate families of the BPS-saturated domain walls exist. The internal structure of these walls can continuously vary, without changing the wall tension. This is described by hidden parameters (collective coordinates). Differentiating with respect to the collective coordinates one gets a set of the bosonic zero modes localized on the wall. Neither of them is related to the spontaneous breaking of any symmetry. Through the residual 1/2 of supersymmetry each bosonic zero mode generates a fermionic partner.
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.
In the Skyrme model, the Lagrangian can be quantized in several ways using the collective coordinate approach. Not all of which produce quantum states that can be interpreted as physical particles. For example the SU(2) collective coordinate approach produces both integral and half-integral spin and isospin states. Only half of these are the physical baryons. Less well known is the fact that it is equally possible to quantize the system using the SO(3) collective coordinates. This produces only unphysical integral spin and isospin states. To fulfill the goal of being able to express a classica…
The $\mathbb{C}P^N$ extended Skyrme-Faddeev model possesses planar soliton solutions. We consider quantum aspects of the solutions applying collective coordinate quantization in regime of rigid body approximation. In order to discuss statistical properties of the solutions we include an Abelian Chern-Simons term (the Hopf term) in the Lagrangian. Since $Π_3(\mathbb{C}P^1)=\mathbb{Z}$ then for $N=1$ the term becomes an integer. On the other hand for $N>1$ it became perturbative because $Π_3(\mathbb{C}P^N)$ is trivial. The prefactor of the Hopf term (anyon angle) $Θ$ is not quantized and its val…
For kink-antikink scattering within the ϕ^4 non--linear field theory in one space and one time dimension resonance type configurations emerge when the relative velocity between kink and antikink falls below a critical value. It has been conjectured that the vibrational excitation of the kink would be the source for these resonances because (simplified) collective coordinate calculations, that emphasized on this excitation, qualitatively reproduced those resonances. Surprisingly a numerical study in the ϕ^6 field theory also exhibited such resonances even though it does not contain the vibratio…
In this article, we shall focus on the collective dynamics of the fermions in a $ν= 1$ quantum Hall droplet. Specifically, we propose to look at the quantum Hall ferromagnet. In this system, the electron spins are ordered in the ground state due to the exchange part of the Coulomb interaction and the Pauli exclusion principle. The low energy excitations are ferromagnetic magnons. In order to obtain an effective Lagrangian for these magnons, we shall introduce bosonic collective coordinates in the Hilbert space of many-fermion systems. These collective coordinates describe a part of the fermion…
In this article, we shall focus on the collective dynamics of the fermions in a $ν= 1$ quantum Hall droplet. Specifically, we propose to look at the quantum Hall ferromagnet. In this system, the electron spins are ordered in the ground state due to the exchange part of the Coulomb interaction and the Pauli exclusion principle. The low energy excitations are ferromagnetic magnons. To provide a means for describing these magnons, we shall discuss a method of introducing collective coordinates in the Hilbert space of many-fermion systems. These collective coordinates are bosonic in nature. They m…
A recent method of constructing quantum mechanics in noncommutative coordinates, alternative to implying noncommutativity by means of star product is discussed. Within this approach we study Hall effect as well as quantum phases in noncommutative coordinates. The $θ$--deformed phases which we obtain are velocity independent.
