In a previous paper we investigate a Lagrangian field theory for the gravitational field (which is there represented by a section g^a of the orthonormal coframe bundle over Minkowski spacetime. Such theory, under appropriate conditions, has been proved to be equivalent to a Lorentzian spacetime structure, where the metric tensor satisfies Einstein field equations. Here, we first recall that according to quantum field theory ideas gravitation is described by a Lagrangian theory of a possible massive graviton field (generated by matter fields and coupling also to itself) living in Minkowski spacetime. The graviton field is moreover supposed to be represented by a symmetric tensor field h carrying the representations of spin two and zero of the Lorentz group. Such a field, then (as it is well known), must necessarily satisfy the gauge condition given by Eq.(3) below. Next, we introduce an ansatz relating h to the 1-form fields g^a. Then, using the Clifford bundle formalism we derive, from our Lagrangian theory, the exact wave equation for the graviton and investigate the role of the gauge condition given by Eq.(3) in obtaining a reliable conservation law for the energy-momentum tensor of the gravitational plus the matter fields in Minkowski spacetime. Finally we ask the question: does Eq.(3) fix any gauge condition for the field g of the effective Lorentzian spacetime structure that represents the field h in our theory? We show that no gauge condition is fixed a priory, as is the case in General Relativity. Moreover we investigate under which conditions we may fix Logunov gauge condition.
In this note, we consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator $\square_g$ is essentially self-adjoint. We review a recent result which gives the meromorphic continuation of the Lorentzian spectral zeta function density, i.e. of the trace density of complex powers $α\mapsto (\square_g-i \varepsilon)^{-α}$. In even dimension $n\geq 4$, the residue at $\frac{n}{2}-1$ is shown to be a multiple of the scalar curvature in the limit $\varepsilon\to 0^+$. This yields a spectral action for gravity in Lorentzian signature.
A sketch of an approach towards Lorentzian spectral geometry (based on joint work with Mario Paschke) is described, together with a general way to define abstractly the quantized Dirac field on such Lorentzian spectral geometries. Moyal-Minkowski spacetime serves as an example. The scattering of the quantized Dirac field by a non-commutative (Moyal-deformed) action of an external scalar potential is investigated. It is shown that differentiating the S-matrix with respect to the strength of the scattering potential gives rise to quantum field operators depending on elements of the non-commutati…
In this paper using the Clifford bundle formalism a Lagrangian theory of the Yang-Mills type (with a gauge fixing term and an auto interacting term) for the gravitational field in Minkowski spacetime is presented. It is shown how two simple hypothesis permit the interpretation of the formalism in terms of effective Lorentzian or teleparallel geometries. In the case of a Lorentzian geometry interpretation of the theory the field equations are shown to be equivalent to Einstein's equations.
CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We …
We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-…
We characterize those spacetimes which admit a isometric (or conformal) embedding in some Lorentz-Minkowski space L^N. In particular, any globally hyperbolic spacetime can be isometrically embedded in L^N. This is proven by a result of its own interest: the construction of a smooth time function whose gradient is bounded away from zero -and, thus, an orthogonal global splitting of the spacetime with bounded lapse. The role of the so-called "folk problems on smoothability" is stressed.
We investigate a link between the energy-momentum dispersion relation and the spectral distance in the context of a Lorentzian almost-commutative spectral geometry, defined by the product of Minkowski spacetime and an internal discrete noncommutative space. Using the causal structure, the almost-commutative manifold can be identified with a pair of four-dimensional Minkowski spacetimes embedded in a five-dimensional Minkowski geometry. Considering fermions travelling within the light cone of the ambient five-dimensional spacetime, we then derive the energy-momentum dispersion relation.
Recent results obtained in $κ$-Poincaré invariant gauge theories on $κ$-Minkowski space are reviewed and commented. A Weyl quantization procedure can be applied to convolution algebras to derive a convenient star product. For such a star product, gauge invariant polynomial action functional depending on the curvature exists only in 5 dimensions. The corresponding noncommutative differential calculus and the related connection are twisted together with the BRST structure linked to the gauge invariance. Phenomenological consequences stemming from the existence of one extra dimension are commente…
We consider the sharp interface limit $ε\to 0$ of the semilinear wave equation $u_{tt} - Δu + \nabla W(u)/ ε^2 = 0$ in $\mathbf R^{1+n}$, where $u$ takes values in $\mathbf R^k$, $k = 1,2$, and $W$ is a double-well potential if $k = 1$ and vanishes on the unit circle and is positive elsewhere if $k = 2$. For fixed $ε> 0$ we find some special solutions, constructed around minimal surfaces in $\mathbf R^n$. In the general case, under some additional assumptions, we show that the solutions converge to a Radon measure supported on a time-like $k$-codimensional minimal submanifold of the Minkowski …
Lattice field theory is a very powerful tool to study Feynman's path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator $ζ$-functions can treat Feynman's path integral non-pertubatively in Lorentzian background metrics. In this article, we formally $ζ$-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator $ζ$-function regularization to be applicable. Furthermore, we show that the clas…
We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with $N$ faces. Starting with the Schwinger representation of the $\mathfrak{su}(1,1)$ Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the three-dimensional Minkowski space $\mathbb{R}^{1,2}$. Considering $N$ copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with $N$ faces whose normal vectors are space-like, up to Lorentz transformation…
We give a geometrically intrinsic construction of a global time function for relatively compact diamond-shaped regions in arbitrary spacetimes. In the case of Minkowski spacetime, the flow of diffeomorphisms associated to a suitably normalized gradient of this time function becomes the conformal isotropy subgroup of the diamond. In full generality, this time function is elegantly expressed in terms of the Lorentzian distance function, and it has an asymptotic behavior at large absolute times similar to the one in Minkowski spacetime.
Minkowski space can be sliced, outside the lightcone, in terms of Euclidean Anti-de Sitter and Lorentzian de Sitter slices. In this paper we investigate what happens when we apply holography to each slice separately. This yields a dual description living on two spheres, which can be interpreted as the boundary of the light cone. The infinite number of slices gives rise to a continuum family of operators on the two spheres for each separate bulk field. For a free field we explain how the Green's function and (trivial) S-matrix in Minkowski space can be reconstructed in terms of two-point functi…
The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzi…
We consider a gauge theory on the 5-d $κ$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_μ$ is necessarily twisted. Performing a BRST gauge-fixing with a Lorentz-type gauge, we carry out a first exploration of the one loop quantum properties of this gauge theory. We find that the gauge-fixed theory gives rise to a non-vanishing tadpole for the time component of the gauge potential, while there is no non-vanishing tadpole 1-point function for the spatial components of $A_μ$. This signal…
Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of descripting flat spacetimes, creating stronger links with hyperbolic geometry and compactifying spacetimes. In particular massive particles and extreme BTZ singular lines arise naturally. This paper is three-fold. First, prove background local properties which will be useful for futu…
Spacetimes obtained by dimensional reduction along lattices containing a lightlike direction can admit semigroup extensions of their isometry groups. We show by concrete examples that such a semigroup can exhibit a natural order, which in turn implies the existence of preferred coordinate charts on the underlying space. Specifically, for spacetimes which are products of an external Minkowski space with an internal two-dimensional Lorentzian space, where one of the lightlike directions has a compact size, the preferred charts consist of "infinite-momentum" frames on the internal space. This imp…
We extend our previous study of Hopf-algebraic $κ$-deformations of all inhomogeneous orthogonal Lie algebras ${\rm iso}(g)$ as written in a tensorial and unified form. Such deformations are determined by a vector $τ$ which for Lorentzian signature can be taken time-, light- or space-like. We focus on some mathematical aspects related to this subject. Firstly, we describe real forms with connection to the metric's signatures and their compatibility with the reality condition for the corresponding $κ$-Minkowski (Hopf) module algebras. Secondly, $h$-adic vs $q$-analog (polynomial) versions of def…
The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this funct…
The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $Λ$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $κ$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $κ$-deformation, while curved momentum spaces are defined as orbits gener…
We work on a 4-manifold equipped with Lorentzian metric $g$ and consider a volume-preserving diffeomorphism which is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric $h$, the pullback of $g$. Motivated by elasticity theory, we introduce a Lagrangian expressed algebraically (without differentiations) via our pair of metrics. Analysis of the resulting nonlinear field equations produces three main results. Firstly, we show that for Ricci-flat manifolds our linearised field equations are Maxwell's equations in the Lorenz gauge with exact current…
