Singular Field Theories Research Articles

Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe mechanical systems), as well as certain cases of multisymplectic manifolds, while extends the Darboux theorem in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories, i.e. with locally non-invertible Legendre maps. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.

More Insights into Symmetries in Multisymplectic Field Theories

This work provides a general overview for the treatment of symmetries in classical field theories and (pre)multisymplectic geometry. The geometric characteristics of the relation between how symmetries are interpreted in theoretical physics and in the geometric formulation of these theories are clarified. Finally, a general discussion is given on the structure of symmetries in the presence of constraints appearing in singular field theories. Symmetries of some typical theories in theoretical physics are analyzed through the construction of the relevant multimomentum maps which are the conserved quantities (by Noether’s theorem) on the (pre)multisymplectic phase spaces.

Multisymplectic constraint analysis of scalar field theories, Chern-Simons gravity, and bosonic string theory

The (pre)multisymplectic geometry of the De Donder–Weyl formalism for field theories is further developed for a variety of field theories including a scalar field theory from the canonical Klein-Gordon action, the electric and magnetic Carrollian scalar field theories, bosonic string theory from the Nambu-Goto action, and 2+1 gravity as a Chern-Simons theory. The Lagrangians for the scalar field theories and for 2+1 Chern-Simons gravity are found to be singular in the De Donder–Weyl sense while the Nambu-Goto Lagrangian is found to be regular. Furthermore, the constraint structure of the premultisymplectic phase spaces of singular field theories is explained and applied to these theories. Finally, it is studied how symmetries are developed on the (pre)multisymplectic phase spaces in the presence of constraints.

Constraint algorithm for singular field theories in the <i>k</i>-cosymplectic framework

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of $k$-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of $k$-precosymplectic structure, which is a generalization of the $k$-cosymplectic structure. Next $k$-precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.

Lagrangian presymplectic constraint analysis of mechanical models of field theories coupled to external fields

The Lecanda-Roman-Roy geometric constraint algorithm for presymplectic Lagrangian systems is applied to a mechanical model of singular field theories coupled to time independent external fields. The simple, yet intrinsic structure of the algorithm allows the influence of the external field to be traced through the constraint analysis, showing clearly where pathologies arise-namely in the second generation nondynamical constraints arising from the stability of the first generation compatibility constraints. Using a coordinate independent geometric algorithm provides a more systematic tool for investigating singular field theories than the usual ad hoc manipulation of the field equations; where the essential structure is often obscured by the details of the representation of the model and the complexity of the algebra.

Dynamics of unrenormalizable interactions in Minkowski and Euclidean spaces

A new approach to unrenormalizable and marginally singular field theories is developed using generalized-function techniques and an unsubtracted Lehmann representation for increasing spectral functions. Solution methods for a class of unrenormalizable equations of time-ordered and nontime-ordered functions are devised in Minkowski and Euclidean spaces. It is shown that within ladder and string approximations unrenormalizable theories possess perturbative solutions in Minkowski space which are finite in every order and do not suffer from an infinite set of subtraction constants, the axioms of relativistic field theory being satisfied. Renormalization factorsZ 2,3 −1 turn out to be differential operators (depending on a scaling parameter to a change of which observable quantities are insensitive) and may be set equal to one without the theory becoming a free one. By complex extension, solutions nonanalytic in the coupling constant can be generated, representing interactions not associable with particle exchange. A mathematical formulation of peratization is given and confronted with the Minkowski-space perturbation approach. The formalism is applied to propagators, Wightman functions and Bethe-Salpeter amplitudes of relativistic four-fermion theories in string and ladder approximations and it is shown that the locality and positive-definiteness axioms are satisfied.