We construct a Lagrangian for general nonlinear electrodynamics that features electric and magnetic potentials on equal footing. In the language of this Lagrangian, discrete and continuous electric-magnetic duality symmetries can be straightforwardly imposed, leading to a simple formulation for theories with the SO(2) duality invariance. When specialized to the conformally invariant case, our construction provides a manifestly duality-symmetric formulation of the recently discovered ModMax theory. We briefly comment on a natural generalization of this approach to $p$-forms in $2p+2$ dimensions.

We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case, the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differential operator and gauge transformations of such operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case.

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions. Supplementary Information: The online version contains the Armenian language version of the article available at https://doi.org/10.1007/s10958-023-06674-w.