Nonlinear Scalar Field Theory Research Articles

A scalar field theory of 1+1-dimensional laser wakefield accelerators

Abstract A relativistic non-linear scalar field theory is developed from a 2+2-dimensional decomposition of the cold plasma field equations, and the theory is used to investigate a 1+1-dimensional description of a laser wakefield accelerator. The relationship between the properties of a compact laser pulse and its wake is explored. Non-linear solutions are sought describing a regular (i.e. unbroken) wake driven by a prescribed rectangular circularly-polarised laser pulse. An upper bound on the dimensionless amplitude a 0 of the laser pulse is determined as a function of the phase speed v of the wake. The asymptotic behaviour of the upper bound on a 0 as v → c is shown to agree with well-established, but approximate, results obtained using the conventional encoding of the plasma degrees of freedom. Our approach leads to a closed-form expression for the upper bound on a 0 which is exact for all values of the phase speed of the wake, unlike conventional results that are applicable only when v is sufficiently close to c.

Quantum Solitodynamics: Non-linear Wave Mechanics and Pilot-Wave Theory

In 1927 Louis de Broglie proposed an alternative approach to standard quantum mechanics known as the double solution program (DSP) where particles are represented as bunched fields or solitons guided by a base (weaker) wave. DSP evolved as the famous de Broglie-Bohm pilot wave interpretation (PWI) also known as Bohmian mechanics but the general idea to use solitons guided by a base wave to reproduce the dynamics of the PWI was abandoned. Here we propose a nonlinear scalar field theory able to reproduce the PWI for the Schrödinger and Klein–Gordon guiding waves. Our model relies on a relativistic ‘phase harmony’ condition locking the phases of the solitonic particle and the guiding wave. We also discuss an extension of the theory for the N particles cases in presence of entanglement and external (classical) electromagnectic fields.

Fifth-force screening around extremely compact sources

Many non-linear scalar field theories possess a screening mechanism that can suppress any associated fifth force in dense environments. As a result, these theories can evade local experimental tests of new forces. Chameleon-like screening, which occurs because of non-linearities in the scalar potential or the coupling to matter, is well understood around extended objects. However, many experimental tests of these theories involve objects with spatial extent much smaller than the scalar field's Compton wavelength, and which could therefore be considered point-like. In this work, we determine how the fifth forces are screened in the limit that the source objects become extremely compact.

Modeling dark matter halos with nonlinear field theories

In the present work, we adopt a nonlinear scalar field theory coupled to the gravity sector to model galactic dark matter. We found analytical solutions for the scalar field coupled to gravity in the Newtonian limit, assuming an isotropic spacetime and a field potential, with a position dependent form of the superpotential, which entails the nonlinear dynamics of the model with self-interactions. The model introduces a position dependent enhancement of the self-interaction of the scalar fields towards the galaxy center, and while going towards the galaxy border the interaction tends to vanish building a non self-interacting DM scenario. The developed approach is able to provide a reasonable analytical description of the rotation curves in both dwarf and low surface brightness late-type galaxies, with parameters associated with the dynamics of the scalar field.

A Sufficient Condition for Asymptotic Stability of Kinks in General (1+1)-Scalar Field Models

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models $$\begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$ The orbital stability of kinks under general assumptions on the potential W is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential W for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $$P(\phi )_2$$ theories and the double sine-Gordon theory.

Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

The functional Schrödinger representation of a nonlinear scalar quantum field theory in curved space-time is shown to emerge as a singular limit from the formulation based on precanonical quantization. The previously established relationship between the functional Schrödinger representation and precanonical quantization is extended to arbitrary curved space-times. In the limiting case when the inverse of the ultraviolet parameter ϰ introduced by precanonical quantization is mapped to the infinitesimal invariant spatial volume element, the canonical functional derivative Schrödinger equation is derived from the manifestly covariant partial derivative precanonical Schrödinger equation. The Schrödinger wave functional is expressed as the trace of the multidimensional spatial product integral of Clifford-algebra-valued precanonical wave function or the product integral of a scalar function obtained from the precanonical wave function by a sequence of transformations. In non-static space-times, the transformations include a nonlocal transformation given by the time-ordered exponential of the zero-th component of spin-connection.

Particle level screening of scalar forces in 1+1 dimensions

We investigate how non-linear scalar field theories respond to point sources. Taking the symmetron as a specific example of such a theory, we solve the non-linear equation of motion in one spatial dimension for (i) an isolated point source and (ii) two identical point sources with arbitrary separation. We find that the mass of a single point source can be screened by the symmetron field, provided that its mass is above a critical value. We find that two point sources behave as independent, isolated sources when the separation between them is large, but, when their separation is smaller than the symmetron's Compton wavelength, they behave much like a single point source with the same total mass. Finally, we explore closely related behavior in a toy Higgs-Yukawa model, and find indications that the maximum fermion mass that can be generated consistently via a Yukawa coupling to the Higgs in 1+1 dimensions is roughly the mass of the Higgs itself, with potentially intriguing implications for the hierarchy problem.

Klein–Gordon field from the XXZ Heisenberg model

We examine the recently introduced idea of Spin-Field Correspondence (SFC) focusing on the example of the spin system described by the XXZ Heisenberg model with external magnetic field. The Hamiltonian of the resulting nonlinear scalar field theory is derived for arbitrary value of the anisotropy parameter [Formula: see text]. We show that the linear scalar field theory is reconstructed in the large spin limit. For [Formula: see text], a nonrelativistic scalar field theory satisfying the Born reciprocity principle is recovered. As expected, for the vanishing anisotropy parameter [Formula: see text], the standard relativistic Klein–Gordon field is obtained. Various aspects of the obtained class of the scalar fields are studied, including the fate of the relativistic symmetries and the properties of the emerging interaction terms. We show that, in a certain limit, the so-called polymer quantization of the field variables is recovered. This and other discussed properties suggest a possible relevance of the considered framework in the context of quantum gravity.

Self-gravitating oscillons and new critical behavior

The dynamical evolution of self-interacting scalars is of paramount importance in cosmological settings, and can teach us about the content of Einstein's equations. In flat space, nonlinear scalar field theories can give rise to localized, non-singular, time-dependent, long-lived solutions called {\it oscillons}. Here, we discuss the effects of gravity on the properties and formation of these structures, described by a scalar field with a double well potential. We show that oscillons continue to exist even when gravity is turned on, and we conjecture that there exists a sequence of critical solutions with infinite lifetime. Our results suggest that a new type of critical behavior appears in this theory, characterized by modulations of the lifetime of the oscillon around the scaling law and the modulations of the amplitude of the critical solutions.

Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos

We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.

Screening fifth forces in k-essence and DBI models

New fifth forces have not yet been detected in the laboratory or in the solar system, hence it is typically difficult to introduce new light scalar fields that would mediate such forces. In recent years it has been shown that a number of non-linear scalar field theories allow for a dynamical mechanism, such as the Vainshtein and chameleon ones, that suppresses the strength of the scalar fifth force in experimental environments. This is known as screening, however it is unclear how common screening is within non-linear scalar field theories. k-essence models are commonly studied examples of non-linear models, with DBI as the best motivated example, and so we ask whether these non-linearities are able to screen a scalar fifth force. We find that a Vainshtein-like screening mechanism exists for such models although with limited applicability. For instance, we cannot find a screening mechanism for DBI models. On the other hand, we construct a large class of k-essence models which lead to the acceleration of the Universe in the recent past for which the fifth force mediated by the scalar can be screened.

A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: I. Self-force through third order

The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the self-force on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g. black hole) spacetime. We use well-known renormalization methods and demonstrate the consistency of the formalism in rendering the self-force finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higher-order self-force effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higher-order self-force numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object’s motion for precise determinations of the parameters of detected EMRI sources.

A nonlinear scalar model of extreme mass ratio inspirals in effective field theory: II. Scalar perturbations and a master source

The motion of a small compact object (SCO) in a background spacetime is investigated further in the context of a class of model nonlinear scalar field theories that have a perturbative structure analogous to the general relativistic description of extreme mass ratio inspirals. We derive regular expressions for the scalar perturbations generated by the motion of the compact object that are valid through third order in ε the size of the SCO to the background curvature length scale. Our results for the field perturbations are compared to those calculated through second order in ε by Rosenthal (2005 Class. Quantum Grav. 22 S859) and found to agree. However, our procedure for regularizing the scalar perturbations is considerably simpler. Following the Detweiler–Whiting scheme, we use our results for the regular expressions for the field and derive the regular self-force corrections through third order. We find agreement with our previous derivation based on a variational principle of an effective action for the worldline associated with the SCO thereby demonstrating the internal consistency of our formalism. This also explicitly demonstrates that the Detweiler–Whiting decomposition of Green’s functions is a valid and practical method of self-force computation at higher orders in perturbation theory and more generally, at all orders in perturbation theory, as we show in an appendix. Finally, we identify a central quantity, which we call a master source, from which all other physically relevant quantities are derivable. Specifically, knowing the master source through some order in ε allows one to construct the waveform measured by an observer, the regular part of the field and its derivative on the worldline, the regular part of the self-force and various orbital quantities such as shifts of the innermost stable circular orbit, etc when restricting to conservative dynamics. The existence of a master source together with the regularization methods implemented in this series should be indispensable for derivations of higher order gravitational self-force corrections in the future.

Effective metric in nonlinear scalar field theories

We discuss several features of the propagation of perturbations in nonlinear scalar field theories using the effective metric. It is shown that the effective metric can be classified according to whether the gradient of the scalar field is timelike, null, or spacelike, and this classification is illustrated with two examples. We shall also show that different signatures for the effective metric are allowed.

Towards effective Lagrangians for adelic strings

Abstractp‐Adic strings are important objects of string theory, as well as of p‐adic mathematical physics and nonlocal cosmology. By a concept of adelic string one can unify and simultaneously study various aspects of ordinary and p‐adic strings. By this way, one can consider adelic strings as a very useful instrument in the further investigation of modern string theory. It is remarkable that for some scalar p‐adic strings exist effective Lagrangians, which are based on real instead of p‐adic numbers and describe not only four‐point scattering amplitudes but also all higher ones at the tree level. In this work, starting from p‐adic Lagrangians, we consider some approaches to construction of effective field Lagrangians for p‐adic sector of adelic strings. It yields Lagrangians for nonlinear and nonlocal scalar field theory, where spacetime nonlocality is determined by an infinite number of derivatives contained in the operator‐valued Riemann zeta function. Owing to the Riemann zeta function in the dynamics of these scalar field theories, obtained Lagrangians are also interesting in themselves.

Extended Born-Infeld dynamics and cosmology

We introduce an extension of the Born-Infeld action for a scalar field and show that it can act as unifying dark matter, providing an explanation for both structure formation and the accelerated expansion of the universe. We investigate the cosmological dynamics of this theory in a particular case, referred to as the ``Milne-Born-Infeld'' (MBI) Lagrangian. We show that this model, whose equation of state has effectively a single free parameter, is consistent with recent type Ia supernovae data, providing a fit as good as for the $\ensuremath{\Lambda}\mathrm{C}\mathrm{D}\mathrm{M}$ model with the same number of degrees of freedom. Furthermore, this parameter is tightly constrained by current data, making the model easily testable with other observables. Contrary to previous candidates for unifying dark matter, the sound velocity of the MBI model is vanishing both close to the dark-matter state as well as near the cosmological constant state. This could avoid the problems on the matter power spectrum that were present in previous adiabatic dark-matter/dark-energy unification models. We also present a short discussion on the causal propagation in nonlinear scalar field theories such as the one proposed here.

Oscillons: Resonant configurations during bubble collapse.

Oscillons are localized, non-singular, time-dependent, spherically-symmetric solutions of nonlinear scalar field theories which, although unstable, are extremely long-lived. We show that they naturally appear during the collapse of subcritical bubbles in models with symmetric and asymmetric double-well potentials. By a combination of analytical and numerical work we explain several of their properties, including the conditions for their existence, their longevity, and their final demise. We discuss several contexts in which we expect oscillons to be relevant. In particular, their nucleation during cosmological phase transitions may have wide-ranging consequences.

Particlelike solutions to nonlinear classical real scalar field theories

A classical Lorentz-covariant nonlinear model scalar field theory, which admits an exact, closed-form, spatially localized, nonsingular solution, is considered. This solution corresponds to a dynamically unstable extended ‘‘particle’’ of finite positive ‘‘rest mass.’’ An approximate two-particle solution is constructed, and the associated effective two-particle Lagrangian is derived by means of a Rayleigh–Ritz procedure.

Compactness criterion for scalar solitons.

The compactness property for topological and nontopological solitons in nonlinear scalar field theories is studied. A simple criterion for soliton compactness is derived in 1+1 dimensions. For nontopological solitons, it is shown that the criterion also applies for any number of space-time dimensions. On the other hand, topological solitons can also have the special property of semicompactness.

Nonlinear classical scalar field theory in curved space-time

A generic Lagrangian based classical field theory is formulated for any space-time manifold in which certain postulated conditions remain valid. The choice of a specific field Lagrangian leads to a nonlinear model theory that admits a rigorous closed-form particlelike solution in isotropic homogeneous space-time of positive spatial curvature. This metastable solution, of finite positive energy, is discussed in relation to its counterpart in isotropic homogeneous space-time of negative spatial curvature.