Canonical Field Theory Research Articles

A vielbein formalism for SHP general relativity

The 4+1 formalism in general relativity expresses the Einstein equations as a manifestly covariant initial value problem, resulting in a pair of first order evolution equations for the metric γμv and intrinsic curvature Kμv of spacetime geometry (μ, v = 0, 1, 2, 3). This approach extends the Stueckelberg-Horwitz-Piron (SHP) framework, a covariant approach to canonical particle mechanics and field theory employing a Lorentz scalar Hamiltonian K and an external chronological parameter τ. The SHP Hamiltonian generates τ-evolution of spacetime events xμ (τ) or ψ (x, τ) in an a priori unconstrained phase space; standard relativistic dynamics can be recovered a posteriori by imposing symmetries that express the usual mass shell constraint for individual particles and fields as conservation laws. As a guide to posing field equations for the evolving metric, we generalize the structure of SHP electrodynamics, with particular attention to O(3,1) covariance. Thus, the 4+1 method first defines a 5D pseudo-spacetime as a direct product of spacetime geometry and chronological evolution, poses 5D field equations whose symmetry must be broken to 4D, and then implements the implied 4+1 foliation to obtain evolution equations. In this paper, we sharpen and clarify the interpretation of this decomposition by introducing a fixed orthonormal quintrad frame and a 5D vielbein field that by construction respects the preferred 4+1 foliation. We show that for any diagonal metric, this procedure enables the evolution equation for the metric to be replaced by an evolution equation for the vielbein field itself, simplifying calculation of the spin connection and curvature.

Formula omitted] quantum corrected scalar field inflation

String theory enjoys an elevated role among quantum gravity theories, since it seems to be the most consistent UV completion of general relativity and the Standard Model. However, it is hard to verify the existence of this underlying theory on terrestrial accelerators. One way to probe string theory is to study its imprints on the low-energy effective inflationary Lagrangian, which are quantified in terms of high energy correction terms. It is highly likely, thus, to find higher order curvature terms combined with string moduli, that is scalar fields, since both these types of interactions and matter fields appear in string theory. In this work we aim to stress the probability that the inflationary dynamics are controlled by the synergy of scalar fields and higher order curvature terms. Specifically, we shall consider a well motivated quantum corrected canonical scalar field theory, with the quantum corrections being of the R2 type. The reason for choosing minimally coupled scalar theory is basically because if scalar fields are evaluated in their vacuum configuration, they will either be minimally coupled or conformally coupled. Here we choose the former case, and the whole study shall be performed in the string frame (Jordan frame), in contrast to similar studies in the literature where the Einstein frame two scalar theory is considered. We derive the field equations of the quantum-corrected theory at leading order and we present the form the slow-roll indices obtain for the quantum corrected theory. We exemplify our theoretical framework by using the quadratic inflation model, and as we show, the R2 quantum corrected quadratic inflation model produces a viable inflationary phenomenology, in contrast with the simple quadratic inflation model.

Rescaled Einstein–Hilbert gravity: Inflation and the Swampland criteria

In this work, we shall study a class of [Formula: see text] gravity models which during the inflationary era, which is the large curvature regime, result to an effective inflationary Lagrangian that contains a rescaled Einstein–Hilbert term [Formula: see text] in the presence of a canonical minimally coupled scalar field. The dimensionless parameter [Formula: see text] is chosen to take values in the range [Formula: see text] and the main motivation for studying these rescaled Einstein–Hilbert [Formula: see text] gravities, is the fact that the rescaled action may render an otherwise incompatible canonical scalar field theory with the Swampland criteria, to be compatible with the Swampland criteria and specifically the distance conjecture and the de Sitter conjecture, without considering the refined de Sitter conjecture however. As we will show, by studying a large number of inflationary potentials appearing in the 2018 Planck collaboration paper for the constraints on inflation, the simultaneous compatibility with both the Planck constraints and the aforementioned two Swampland criteria is achieved for some models, and the main characteristic of the models for which this is possible is the small value that the parameter [Formula: see text] must take.

Soliton, breather and shockwave solutions of the Heisenberg and the Toverline{T} deformations of scalar field theories in 1+1 dimensions

In this note we study soliton, breather and shockwave solutions in certain two dimensional field theories. These include: (i) Heisenberg’s model suggested originally to describe the scattering of high energy nucleons (ii) Toverline{T} deformations of certain canonical scalar field theories with a potential. We find explicit soliton solutions of these models with sine-Gordon and Higgs-type potentials. We prove that the Toverline{T} deformation of a theory of a given potential does not correct the mass of the soliton of the undeformed one. We further conjecture the form of breather solutions of these models. We show that certain Toverline{T} deformed actions admit shockwave solutions that generalize those of Heisenberg’s Lagrangian.

Gauge invariant canonical symplectic algorithms for real-time lattice strong-field quantum electrodynamics

A class of high-order canonical symplectic structure-preserving geometric algorithms are developed for high-quality simulations of the quantized Dirac-Maxwell theory based strong-field quantum electrodynamics (SFQED) and relativistic quantum plasmas (RQP) phenomena. With minimal coupling, the Lagrangian density of an interacting bispinor-gauge fields theory is constructed in a conjugate real fields form. The canonical symplectic form and canonical equations of this field theory are obtained by the general Hamilton’s principle on cotangent bundle. Based on discrete exterior calculus, the gauge field components are discreted to form a cochain complex, and the bispinor components are naturally discreted on a staggered dual lattice as combinations of differential forms. With pull-back and push-forward gauge covariant derivatives, the discrete action is gauge invariant. A well-defined discrete canonical Poisson bracket generates a semi-discrete lattice canonical field theory (LCFT), which admits the canonical symplectic form, unitary property, gauge symmetry and discrete Poincaré subgroup, which are good approximations of the original continuous geometric structures. The Hamiltonian splitting method, Cayley transformation and symmetric composition technique are introduced to construct a class of high-order numerical schemes for the semi-discrete LCFT. These schemes involve two degenerate fermion flavors and are locally unconditional stable, which also preserve the geometric structures. Admitting Nielsen-Ninomiya theorem, the continuous chiral symmetry is partially broken on the lattice. As an extension, a pair of discrete chiral operators are introduced to reconstruct the lattice chirality. Equipped with statistically quantization-equivalent ensemble models of the Dirac vacuum and non-trivial plasma backgrounds, the schemes are expected to have excellent performance in secular simulations of relativistic quantum effects, where the numerical errors of conserved quantities are well bounded by very small values without coherent accumulation. The algorithms are verified in detail by numerical energy spectra. Real-time LCFT simulations are successfully implemented for the nonlinear Schwinger mechanism induced e-e+ pairs creation and vacuum Kerr effect, where the nonlinear and non-perturbative features captured by the solutions provide a complete strong-field physical picture in a very wide range, which open a new door toward high-quality simulations in SFQED and RQP fields.

Non-equilibrium phase separation with reactions: a canonical model and its behaviour

Materials undergoing both phase separation and chemical reactions (defined here as all processes that change particle type or number) form an important class of non-equilibrium systems. Examples range from suspensions of self-propelled bacteria with birth–death dynamics, to bio-molecular condensates, or ‘membraneless organelles’, within cells. In contrast to their passive counterparts, such systems have conserved and non-conserved dynamics that do not, in general, derive from a shared free energy. This mismatch breaks time-reversal symmetry and leads to new types of dynamical competition that are absent in or near equilibrium. We construct a canonical scalar field theory to describe such systems, with conserved and non-conserved dynamics obeying model B and model A, respectively (in the Hohenberg–Halperin classification), chosen such that the two free energies involved are incompatible. The resulting minimal model is shown to capture the various phenomenologies reported previously for more complicated models with the same physical ingredients, including microphase separation, limit cycles and droplet splitting. We find a low-dimensional subspace of parameters for which time-reversal symmetry is accidentally recovered, and show that here the dynamics of the order parameter field (but not its conserved current) is exactly the same as an equilibrium system in which microphase separation is caused by long-range attractive interactions.

Yang–Mills Theory of Gravity

The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Σ embedded in a four-dimensional space–time manifold. This implies continuous symmetries and conserved currents by Noether’s theorem on that surface. We construct a three-form E i ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b i is its energy density. We show that there is conserved spin current that couples to A i , and show that we have to include the term F μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.

Canonical scalar field inflation with a Woods–Saxon potential

This paper focuses on the realization of an inflationary model from a canonical scalar field theory with a Woods–Saxon potential, in the slow-roll approximation. Our analysis indicates that the observable quantities derived theoretically from our model, namely the spectral index of the primordial scalar curvature perturbations and the tensor-to-scalar ratio, are compatible with the latest Planck collaboration data. We also discuss the qualitative features of the potential, and we show that the value of the scalar field for which the graceful exit occurs, coincides with the inflection point of the scalar potential. We also attempt to study the post-inflation reheating phase of the model, in order to further examine the viability of the Woods–Saxon scalar field model, and as we demonstrate the results indicate viability of the model for this era too, however the instantaneous reheating is not allowed for the model at hand.

Viable inflation in scalar-Gauss-Bonnet gravity and reconstruction from observational indices

In this paper the focus is on inflationary dynamics in the context of Einstein Gauss-Bonnet gravitational theories. We investigate the implications of the slow-roll condition on the slow-roll indices and we investigate how the inflationary dynamical evolution is affected by the presence of the Gauss-Bonnet coupling to the scalar field. For exemplification of our analysis we investigate how the dynamics of inflationary cubic, quartic order and also exponential scalar potentials are affected by the non-trivial Gauss-Bonnet coupling to the scalar field. As we demonstrate it is possible to obtain a viable phenomenology compatible with the observational data, although the canonical scalar field theory with cubic and quartic order potentials does not yield phenomenologically acceptable results. In addition, with regard to the exponential potential example, the Einstein Gauss-Bonnet extension of the single canonical scalar field model has an inherent mechanism that can trigger the graceful exit from inflation. Furthermore we introduce a bottom-up reconstruction technique, in the context of which by fixing the tensor-to-scalar ratio and the Hubble rate as a function of the $e$-foldings number, one is capable of reproducing the Einstein Gauss-Bonnet theory which generates the aforementioned quantities. We illustrate how the method works by using some relatively simple examples.

Interpolating quantum electrodynamics between instant and front forms

The instant form and the front form of relativistic dynamics proposed by Dirac in 1949 can be linked by an interpolation angle parameter $\delta$ spanning between the instant form dynamics (IFD) at $\delta =0$ and the front form dynamics which is now known as the light-front dynamics (LFD) at $\delta =\pi/4$. We present the formal derivation of the interpolating quantum electrodynamics (QED) in the canonical field theory approach and discuss the constraint fermion degree of freedom which appears uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator, which is genuinely distinguished from the ordinary equal-time forward and backward propagation of relativistic fermion degrees of freedom. As discussed in our previous work, the helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not. To exemplify the characteristic difference of the fermion propagator between IFD and LFD, we compute the helicity amplitudes of typical QED processes such as $e^+ e^- \to \gamma \gamma$ and $e \gamma \to e \gamma$ and present the whole landscape of the scattering amplitudes in terms of the frame dependence or the scattering angle dependence with respect to the interpolating angle dependence. Our analysis clarifies any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame approach and the LFD.

Kinks in higher derivative scalar field theory

We study static kink configurations in a type of two-dimensional higher derivative scalar field theory whose Lagrangian contains second-order derivative terms of the field. The linear fluctuation around arbitrary static kink solutions is analyzed. We find that, the linear spectrum can be described by a supersymmetric quantum mechanics problem, and the criteria for stable static solutions can be given analytically. We also construct a superpotential formalism for finding analytical static kink solutions. Using this formalism we first reproduce some existed solutions and then offer a new solution. The properties of our solution is studied and compared with those preexisted. We also show the possibility in constructing twinlike model in the higher derivative theory, and give the consistency conditions for twinlike models corresponding to the canonical scalar field theory.

The Unruh effect for eccentric uniformly rotating observers

It is common to use Galilean rotational transformation (GRT) to investigate the Unruh effect for uniformly rotating observers. However, the rotating observer in this subject is an eccentric observer while GRT is only valid for centrally rotating observers. Thus, the reliability of the results of applying GRT to the study of the Unruh effect might be considered as questionable. In this work, the rotational analog of the Unruh effect is investigated by employing two relativistic rotational transformations corresponding to the eccentric rotating observer, and it is shown that in both cases, the detector response function is nonzero. It is also shown that although consecutive Lorentz transformations cannot give a frame within which the canonical construction can be carried out, the expectation value of particle number operator in canonical approach will be zero if we use modified Franklin transformation. These conclusions reinforce the claim that correspondence between vacuum states defined via canonical field theory and a detector is broken for rotating observers. Some previous conclusions are commented on and some controversies are also discussed.

A smooth constant-roll to a slow-roll modular inflation transition

In this work, we investigate how a smooth transition from a constant-roll to a slow-roll inflationary era may be realized in the context of a canonical scalar field theory. We study in some detail the dynamical evolution of the cosmological system, and we investigate whether a stable attractor exists, both numerically and analytically. We also investigate the slow-roll era and as we demonstrate, the partially compatibility of the resulting scalar theory may be achieved with the potential of the latter belonging to a class of modular inflationary potentials. The novel features of the constant-roll to slow-roll transition which we achieved are firstly that it is not compelling for the slow-roll era to last for [Formula: see text]–60 [Formula: see text]-foldings, but it may last for a smaller number of [Formula: see text]-foldings, since some [Formula: see text]-foldings may occur during the constant-roll era. Secondly, when the slow-roll era occurs after the constant-roll era, the graceful exit from inflation may occur, a feature absent in the constant-roll scenario, due to the stability properties of the final attractor in the constant-roll case.

Extended canonical field theory of matter and space‐time

AbstractAny physical theory that follows from an action principle should be invariant in its form under mappings of the reference frame in order to comply with the general principle of relativity. The required form‐invariance of the action principle implies that the mapping must constitute a particular extended canonical transformation. In the realm of the covariant Hamiltonian formulation of field theory, the term “extended” implies that not only the fields but also the space‐time geometry is subject to transformation. A canonical transformation maintains the general form of the action principle by simultaneously defining the appropriate transformation rules for the fields, the conjugate momentum fields, and the transformation rule for the Hamiltonian. Provided that the given system of fields exhibits a particular global symmetry, the associated extended canonical transformation determines an amended Hamiltonian that is form‐invariant under the corresponding local symmetry. This will be worked out for a Hamiltonian system of scalar and vector fields that is presupposed to be form‐invariant under space‐time transformations x µ ∼→ X µ with δX µ/δxv = const., hence under global space‐time transformations such as the Poincar´e transformation. The corresponding amended system that is form‐invariant under local space‐time transformations δX µ/δxv ∼ const. then describes the coupling of the fields to the space‐time geometry and thus yields the dynamics of space‐time that is associated with the given physical system. Non‐zero spin matter determines thereby the space‐time curvature via a well‐defined source term in a covariant Poisson‐type equation for the Riemann tensor. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

General relativity as an extended canonical gauge theory

It is widely accepted that the fundamental geometrical law of nature should follow from an action principle. The particular subset of transformations of a system's dynamical variables that maintain the form of the action principle comprises the group of canonical transformations. In the context of canonical field theory, the adjective ``extended'' signifies that not only the fields but also the space-time geometry is subject to transformation. Thus, in order to be physical, the transition to another, possibly noninertial frame of reference must necessarily constitute an extended canonical transformation that defines the general mapping of the connection coefficients, hence the quantities that determine the space-time curvature and torsion of the respective reference frame. The canonical transformation formalism defines simultaneously the transformation rules for the conjugates of the connection coefficients and for the Hamiltonian. As will be shown, this yields unambiguously a particular Hamiltonian that is form-invariant under the canonical transformation of the connection coefficients and thus satisfies the general principle of relativity. This Hamiltonian turns out to be a quadratic function of the curvature tensor. Its Legendre-transformed counterpart then establishes a unique Lagrangian description of the dynamics of space-time that is not postulated but derived from basic principles, namely the action principle and the general principle of relativity. Moreover, the resulting theory satisfies the principle of scale invariance and is renormalizable.

Relativistic non-Hermitian quantum mechanics

We develop relativistic wave equations in the framework of the new non-Hermitian $\mathcal{P}\mathcal{T}$ quantum mechanics. The familiar Hermitian Dirac equation emerges as an exact result of imposing the Dirac algebra, the criteria of $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics, and relativistic invariance. However, relaxing the constraint that, in particular, the mass matrix be Hermitian also allows for models that have no counterpart in conventional quantum mechanics. For example it is well known that a quartet of Weyl spinors coupled by a Hermitian mass matrix reduces to two independent Dirac fermions; here, we show that the same quartet of Weyl spinors, when coupled by a non-Hermitian but $\mathcal{P}\mathcal{T}$-symmetric mass matrix, describes a single relativistic particle that can have massless dispersion relation even though the mass matrix is nonzero. The $\mathcal{P}\mathcal{T}$-generalized Dirac equation is also Lorentz invariant, unitary in time, and CPT respecting, even though as a noninteracting theory it violates $\mathcal{P}$ and $\mathcal{T}$ individually. The relativistic wave equations are reformulated as canonical fermionic field theories to facilitate the study of interactions and are shown to maintain many of the canonical structures from Hermitian field theory, but with new and interesting possibilities permitted by the non-Hermiticity parameter ${m}_{2}$.

Quartic canonical force field in curvilinear internal coordinates for XY3 (D3h) molecules. The case of the BH3 molecule.

Using the canonical force field theory, expressions of quadratic, cubic, and quartic canonical force constants are obtained for XY3 (D3h) molecules in curvilinear redundant coordinates, i.e., simple valence internal coordinates (VICs), in terms of force constants in normal coordinates and in independent symmetry coordinates. To carry out this task, it was previously necessary to obtain for the first time the non-linear redundancy relation and the corresponding orthogonal projection onto the pure vibrational manifold for XY3 (D3h) molecules corresponding to a set of seven VICs. As an application, the quartic canonical force field in curvilinear redundant internal coordinates of BH3 is determined from ab initio force fields in normal coordinates calculated at the coupled-cluster singles and doubles level with perturbative treatment of the triples in conjunction with a triple- and quadruple-ζ size basis set. This anharmonic force field so obtained for the borane molecule, and in general for XY3 (D3h) molecules, is uniquely defined (therefore in an unambiguous form) and depending on the same number of parameters, i.e., force constants, when independent coordinates (natural or symmetry) are used in its description.

Distinguishingk-defects from their canonical twins

We study k-defects - topological defects in theories with more than two derivatives and second-order equations of motion - and describe some striking ways in which these defects both resemble and differ from their analogues in canonical scalar field theories. We show that, for some models, the homotopy structure of the vacuum manifold is insufficient to establish the existence of k-defects, in contrast to the canonical case. These results also constrain certain families of DBI instanton solutions in the 4-dimensional effective theory. We then describe a class of k-defect solutions, which we dub doppelgangers, that precisely match the field profile and energy density of their canonical scalar field theory counterparts. We give a complete characterization of Lagrangians which admit doppelganger domain walls. By numerically computing the fluctuation eigenmodes about domain wall solutions, we find different spectra for doppelgangers and canonical walls, allowing us to distinguish between k-defects and the canonical walls they mimic. We search for doppelgangers for cosmic strings by numerically constructing solutions of DBI and canonical scalar field theories. Despite investigating several examples, we are unable to find doppelganger cosmic strings, hence the existence of doppelgangers for defects with codimension >1 remains an open question.

Canonical force field theory in terms of curvilinear internal coordinates: Application to the methane molecule

This Letter generalizes the Kuczera’s theory of the vibrational canonical force field in order to obtain an unambiguous and uniquely defined expression for the anharmonic force field in terms of curvilinear internal coordinates. By using this generalization it has been shown mathematically that the canonical force constants could be transferred between molecules with the same geometrical structure. As illustration, the theory is used to obtain the expressions for the quadratic and cubic canonical force constants of the methane molecule.

On general properties of Lorentz-invariant formulation of noncommutative quantum field theory

We study general properties of certain Lorentz-invariant noncommutative quantum field theories proposed in the literature. We show that causality in those theories does not hold, in contrast to the canonical noncommutative field theory with the light-wedge causality condition. This is the consequence of the infinite nonlocality of the theory getting spread in all spacetime directions. We also show that the time-ordered perturbation theory arising from the Hamiltonian formulation of noncommutative quantum field theories remains inequivalent to the covariant perturbation theory with usual Feynman rules even after restoration of Lorentz symmetry.