Lagrangian Field Theory Research Articles

Field theory with the Maxima computer algebra system

The Maxima computer algebra system, the open-source successor to MACSYMA, the first general-purpose computer algebra system that was initially developed at the Massachusetts Institute of Technology in the late 1960s and later distributed by the United States Department of Energy, has some remarkable capabilities, some of which are implemented in the form of add-on, “share” packages that are distributed along with the core Maxima system. One such share package is itensor, for indicial tensor manipulation. One of the more remarkable features of itensor is functional differentiation. Through this, it is possible to use itensor to develop a Lagrangian field theory and derive the corresponding field equations. In this paper, we demonstrate this capability by deriving Maxwell’s equations from the Maxwell Lagrangian, and exploring the properties of the system, including current conservation.

EFTofLSS meets simulation-based inference: σ 8 from biased tracers

Cosmological inferences typically rely on explicit expressions for the likelihood and covariance of the data vector, which normally consists of a set of summary statistics. However, in the case of nonlinear large-scale structure, exact expressions for either likelihood or covariance are unknown, and even approximate expressions can become very cumbersome, depending on the scales and summary statistics considered. Simulation-based inference (SBI), in contrast, does not require an explicit form for the likelihood but only a prior and a simulator, thereby naturally circumventing these issues. In this paper, we explore how this technique can be used to infer σ 8 from a Lagrangian effective field theory (EFT) based forward model for biased tracers. The power spectrum and bispectrum are used as summary statistics to obtain the posterior of the cosmological, bias and noise parameters via neural density estimation. We compare full simulation-based inference with cases where the data vector is drawn from a Gaussian likelihood with sample and analytical covariances. We conclude that, for k max = 0.1hMpc-1 and 0.2hMpc-1, the form of the covariance is more important than the non-Gaussianity of the likelihood, although this conclusion is expected to depend on the cosmological parameter inferred, the summary statistics considered and range of scales probed.

The field theory of collective Cherenkov radiation associated with electron beams

Classical Cherenkov radiation is a celebrated physics phenomenon of electromagnetic (EM) radiation stimulated by an electric charge moving with constant velocity in a three-dimensional dielectric medium. Cherenkov radiation has a wide spectrum and a particular distribution in space similar to the Mach cone created by a supersonic source. It is also characterized by the energy transfer from the charge's kinetic energy to the EM radiation. In the case of an electron beam passing through the middle of an EM waveguide, the radiation is manifested as collective Cherenkov radiation. In this case, the electron beam can be viewed as a one-dimensional non-neutral plasma, whereas the waveguide can be viewed as a slow wave structure. This collective radiation occurs, in particular, in traveling wave tubes (TWTs), and it features the energy transfer from the electron beam to the EM radiation in the waveguide. Based on a first principles Lagrangian field theory, we develop a convincing argument that the collective Cherenkov effect in TWTs is, in fact, a convective instability, that is, amplification. We also recover Pierce's theory as a high-frequency limit of our generalized Lagrangian theory. Finally, we derive for the first time expressions identifying low- and high-frequency cutoffs for amplification in TWTs.

Constraint characterization and degree of freedom counting in Lagrangian field theory

We present a Lagrangian approach to counting degrees of freedom in first-order field theories. The emphasis is on the systematic attainment of a complete set of constraints. In particular, we provide the first comprehensive procedure to ensure the functional independence of all constraints and discuss in detail the possible closures of the constraint algorithm. We argue degrees of freedom can but need not correspond to physical modes. The appendix comprises fully worked out, physically relevant examples of varying complexity. Published by the American Physical Society 2024

Field theory description of the non-perturbative optical nonlinearity of epsilon-near-zero media

In this paper, we introduce a fully non-perturbative approach for the description of the optical nonlinearity of epsilon-near-zero (ENZ) media. In particular, based on the rigorous Feynman path integral method, we develop a dressed Lagrangian field theory for light–matter interactions and discuss its application to dispersive Kerr-like media with order-of-unity light-induced refractive index variations. Specifically, considering the relevant case of Indium Tin Oxide (ITO) nonlinearities, we address the novel regime of non-perturbative refractive index variations in ENZ media and establish that it follows naturally from a scalar field theory with a Born–Infeld Lagrangian. Moreover, we developed a predictive model that includes the intrinsic saturation effects originating from the light-induced modification of the Drude terms in the linear dispersion of ITO materials. Our results extend the Huttner–Barnett–Bechler electrodynamics model to the case of non-perturbative optical Kerr-like media providing an intrinsically nonlinear, field-theoretic framework for understanding the exceptional nonlinearity of ITO materials beyond traditional perturbation theory.

Higher-Order Corrections to the Effective Field Theory of Low-Energy Axions

Dark matter (DM) can be composed of a collection of axions, or axion-like particles (ALPs), whose existence is due to the spontaneous breaking of the Peccei–Quinn U(1) symmetry, which is the most compelling solution of the strong CP-problem of quantum chromodynamics (QCD). Axions must be spin-0 particles with very small masses and extremely weak interactions with themselves as well as with the particles that constitute the Standard Model. In general, the physics of axions is detailed by a quantum field theory of a real scalar field, ϕ. Nevertheless, it is more convenient to implement a nonrelativistic effective field theory with a complex scalar field, ψ, to characterize the mentioned axions in the low-energy regime. A possible application of this equivalent description is for studying the collapse of cold dark matter into more complex structures. There have been a few derivations of effective Lagrangians for the complex field ψ, which were all equivalent after a nonlocal space transformation between ϕ and ψ was found, and some other corrections were introduced. Our contribution herein is to further provide higher-order corrections; in particular, we compute the effective field theory Lagrangian up to order (ψ*ψ)5, also incorporating the fast-oscillating field fluctuations into the dominant slowly varying nonrelativistic field.

Generalization of Noether Theorem and action principle for non-Lagrangian theories

A non-Lagrangian field theory is a theory whose field equations cannot be derived from the action principle. The action principle is described by holonomic variational equations. These equations can be used only for standard field theories, equations of which can be represented as Euler–Lagrange equations for some Lagrangian density. In the general case, one should use non-holonomic variational equations to obtain a wider class of field equations. In this paper, it is proposed to use the Sedov non-holonomic variational equation as a generalization of action principle. Examples of application of non-holonomic variational equations to derive various field equations are suggested. The standard Noether theorem is formulated for the standard (Lagrangian) field theory with an action functional and expresses the invariance of the Lagrangian with respect to some continuous group of transformations. In this paper, a generalization of Noether’s theorem for non-Lagrangian field theories is proposed and proved by using the non-holonomic variational equation. The expression of Noether current is described in the general case of non-Lagrangian field theory. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory. An important result of this paper is the proof of possibility of existence of some analogues of dissipative structures in non-Lagrangian field theories and some properties of such structures are suggested.

Particle-dimer approach for the Roper resonance in a finite volume

We propose a new finite-volume approach which implements two- and three-body dynamics in a transparent way based on an Effective Field Theory Lagrangian. The formalism utilizes a particle-dimer picture and formulates the quantization conditions based on the self-energy of the decaying particle. The formalism is studied for the case of the Roper resonance, using input from lattice QCD and phenomenology. Finally, finite-volume energy eigenvalues are predicted and compared to existing results of lattice QCD calculations. This crucially provides initial guidance on the necessary level of precision for the finite-volume spectrum.

Analytic theory of coupled-cavity traveling wave tubes

Coupled-cavity traveling wave tube (CCTWT) is a high power microwave vacuum electronic device used to amplify radio frequency signals. CCTWTs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. Microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT, particularly the amplification mechanism, are similar to those of a multicavity klystron. We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange field equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs, and our theory applies to them also.

Axion-like ALPs

The Effective Field Theory (EFT) Lagrangian for a generic axion-like particle (ALP) has many free parameters, and leaves quite some freedom for the expected phenomenology. In this work, we set up more constrained EFTs by enforcing true axion-like properties for the ALP. Indeed, though the Peccei-Quinn symmetry of the QCD axion is anomalous, it is so in specific ways, and this shows up as consistency conditions between the gauge boson and fermion couplings. We propose to enforce such conditions, as inspired from the DFSZ and KSVZ scenarios, on the generic ALP EFTs. These truly axion-like ALP EFTs are then particularly well-suited as benchmark scenarios, to be used in the search for ALPs both at colliders and at low-energy experiments.

BV equivalence with boundary

An extension of the notion of classical equivalence of equivalence in the Batalin–Vilkovisky (BV) and Batalin–Fradkin–Vilkovisky (BFV) frameworks for local Lagrangian field theory on manifolds possibly with boundary is discussed. Equivalence is phrased in both a strict and a lax sense, distinguished by the compatibility between the BV data for a field theory and its boundary BFV data, necessary for quantisation. In this context, the first- and second-order formulations of nonabelian Yang–Mills and of classical mechanics on curved backgrounds, all of which admit a strict BV–BFV description, are shown to be pairwise equivalent as strict BV–BFV theories. This in particular implies that their BV complexes are quasi-isomorphic. Furthermore, Jacobi theory and one-dimensional gravity coupled with scalar matter are compared as classically equivalent reparametrisation-invariant versions of classical mechanics, but such that only the latter admits a strict BV–BFV formulation. They are shown to be equivalent as lax BV–BFV theories and to have isomorphic BV cohomologies. This shows that strict BV–BFV equivalence is a strictly finer notion of equivalence of theories.

A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian

<abstract><p>We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms, we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.</p></abstract>

Cotangent bundle reduction and Routh reduction for polysymplectic manifolds

We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by means of the Routhian function and its associated Legendre transformation. Throughout the paper we provide examples that illustrate various aspects of the results.

Momentum, angular momentum, and spin of waves in an isotropic collisionless plasma

We examine the momentum and angular momentum (including spin) properties of linear waves, both longitudinal (Langmuir) and transverse (electromagnetic), in an isotropic nonrelativistic collisionless electron plasma. We focus on conserved quantities associated with the translational and rotational invariance of the wave fields with respect to the homogeneous medium; these are sometimes called pseudomomenta. There are two types of the momentum and angular momentum densities: (i) the kinetic ones associated with the energy flux density and the symmetrized (Belinfante) energy-momentum tensor and (ii) the canonical ones associated with the conserved Noether currents and canonical energy-momentum tensor. We find that the canonical momentum and spin densities of Langmuir waves are similar to those of sound waves in fluids or gases; they are naturally expressed via the electron velocity field. In turn, the momentum and spin densities of electromagnetic waves can be written either in the forms known for free-space electromagnetic fields, involving only the electric field, or in the dual-symmetric forms involving both electric and magnetic fields, as well as the effective permittivity of plasma. We derive these properties both within the phenomenological macroscopic approach and microscopic Lagrangian field theory for the coupled electromagnetic fields and electrons. Finally, we explore implications of the canonical momentum and spin densities in transport and electrodynamic phenomena: the Stokes drift, the wave-induced magnetization (inverse Faraday effect), etc.

Non-Lorentzian SU(1, n) Spacetime Symmetry In Various Dimensions

We discuss non-Lorentzian Lagrangian field theories in 2n − 1 dimensions that admit an SU(1, n) spacetime symmetry which includes a scaling transformation. These can be obtained by a conformal compactification of a 2n-dimensional Minkowskian conformal field theory. We discuss the symmetry algebra, its representations including primary fields and unitarity bounds. We also give various examples of free theories in a variety of dimensions and a discussion of how to reconstruct the parent 2n-dimensional theory.

Analytic theory of multicavity klystrons

Multicavity Klystron (MCK) is a high power microwave vacuum electronic device used to amplify radio frequency (RF) signals. MCKs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in klystrons take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Importantly, there is no electromagnetic coupling between cavities. The cavities are coupled only by the flow of bunched electrons drifting from one cavity to the next. We advance here a Lagrangian field theory of MCKs with the space being represented by a one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop the instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, the dispersion relations and the frequency dependent gain foundational to the RF signal amplification. We assume that MCKs operate in the voltage amplification mode associated with the maximal gain.

Scotogenic A5 → A4 Dirac neutrinos with freeze-in dark matter

Radiative Dirac neutrino masses and their mixing are linked to dark matter through the non-Abelian discrete symmetry A5 of the 4-dimensional pentatope, softly broken to A4 of the 3-dimensional tetrahedron. This unifying understanding of neutrino family structure from dark matter is made possible through the interplay of gauge symmetry, renormalizable Lagrangian field theory, and softly broken discrete symmetries. Dark neutral fermions are produced through Higgs decay.

The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility

A second-order generalization of the fundamental Lepage form of geometric calculus of variations over fibered manifolds with 2-dimensional base is described by means of insisting on (i) an equivalence relation “Lepage differential 2-form is closed if and only if the associated Lagrangian is trivial” and (ii) the principal component of Lepage form, extending the well-known Poincaré–Cartan form, preserving order prescribed by a given Lagrangian. This approach completes several attempts of finding a Lepage equivalent of a second-order Lagrangian possessing condition (i), which is well-known for first-order Lagrangians in field theory due to Krupka and Betounes.

Skinner–Rusk formalism for k-contact systems

In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of k-contact Hamiltonian systems, which is based on the k-symplectic formulation of field theories as well as on contact geometry. In this work we present the Skinner–Rusk unified setting for these kinds of theories, which encompasses both the Lagrangian and Hamiltonian formalisms into a single picture. This unified framework is specially useful when dealing with singular systems, since: (i) it incorporates in a natural way the second-order condition for the solutions of field equations, (ii) it allows to implement the Lagrangian and Hamiltonian constraint algorithms in a unique simple way, and (iii) it gives the Legendre transformation, so that the Lagrangian and the Hamiltonian formalisms are obtained straightforwardly. We apply this description to several interesting physical examples: the damped vibrating string, the telegrapher's equations, and Maxwell's equations with dissipation terms.

Half-lives of α-decay from nuclei with Z = 92 − 118 using the double folding model with relativistic NN interactions

The α-decay half-lives of nuclei with Z=92−118 have been calculated using the WKB semi-classical approximation. The α-daughter nuclear interaction potentials are obtained from the double folding model, where the effective nucleon-nucleon (NN) interactions are obtained from relativistic mean field theory Lagrangian (R3Y). In addition to using R3Y NN interactions containing only linear σ, ω, and ρ mesons (R3Y-HS), R3Y parametrizations containing non-linear terms (NL2 and NL-SH) are also employed. In fact, they are found to give better descriptions of the α-decay half-lives than R3Y-HS. Moreover, two empirical formulas are employed in the study. They have been included to further test the accuracy of the R3Y models.