Scalar field quantization without divergences in all spacetime dimensions

A recently-proposed technique, called the dimensional expansion, uses the space-time dimension $D$ as an expansion parameter to extract nonperturbative results in quantum field theory. Here we apply dimensional-expansion methods to examine the Ising limit of a self-interacting scalar field theory. We compute the first few coefficients in the dimensional expansion for $\gamma_{2n}$, the renormalized $2n$-point Green's function at zero momentum, for $n\!=\!2$, 3, 4, and 5. Because the exact results for $\gamma_{2n}$ are known at $D\!=\!1$ we can compare the predictions of the dimensional expansion at this value of $D$. We find typical errors of less than $5\%$. The radius of convergence of the dimensional expansion for $\gamma_{2n}$ appears to be ${{2n}\over {n-1}}$. As a function of the space-time dimension $D$, $\gamma_{2n}$ appears to rise monotonically with increasing $D$ and we conjecture that it becomes infinite at $D\!=\!{{2n}\over {n-1}}$. We presume that for values of $D$ greater than this critical value, $\gamma_{2n}$ vanishes identically because the corresponding $\phi^{2n}$ scalar quantum field theory is free for $D\!>\!{{2n}\over{n-1}}$.

Renormalizability and UV/IR mixing in noncommutative theories with scalar fields

Loop corrections to unequal-time correlation functions in Minkowski spacetime exhibit secular growth due to a breakdown of time-dependent perturbation theory. This is analogous to secular growth in equal-time correlators on time-dependent backgrounds, except that in Minkowski the divergences must not signal a real IR issue. In this paper, we calculate the late-time limit of the two-point correlator for different massless self-interacting scalar quantum field theories on a Minkowski background. We first use a late-time version of the in-in path integral starting in the vacuum of the free theory; in this limit, the calculation, including UV renormalization, reduces to that in in-out. We find linear or logarithmic growth in time, depending on whether the interaction strength is dimension-one or dimensionless, respectively. We next develop the Weisskopf-Wigner resummation method, that proceeds by demanding unitarity within a truncated Hilbert space, to calculate the resummed correlator and find that it gives an exact exponentiation of the late-time perturbative result. The resummed (unequal-time) correlator thus decays with an exponential or polynomial time-dependence, which is suggestive of ‘universal’ behavior that depends on the dimensions of the interaction strength.

We consider scalar fields which are coupled to Einstein gravity with a negative cosmological constant, and construct periodic solutions perturbatively. In particular, we study tachyonic scalar fields whose mass is at or above the Breitenlohner–Freedman bound in four, five, and seven spacetime dimensions. The critical amplitude of the leading order perturbation, for which the perturbative expansion breaks down, increases as we consider less massive fields. We present various examples including a model with a self-interacting scalar field which is derived from a consistent truncation of IIB supergravity.

The geometric structure of S-matrix encapsulated by the “Amplituhedron program” has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan [1] it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions.In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with n − 4 massless poles and one massive pole at m2. The resulting amplitudes are associated to λ1 {phi}_1^3 + λ2 {phi}_1^2 ϕ2 potential where ϕ1 and ϕ2 are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the “decoupling limit” (where m → ∞, λ2 → ∞ such that g : frac{uplambda_2}{m} = finite) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by λ {phi}_1^3 + g {phi}_1^4 .

Applying recursive renormalization group transformations to a scalar field theory, we obtain an effective quantum gravity theory with an emergent extra dimension, described by a dual holographic Einstein-Klein-Gordon type action. Here, the dynamics of both the dual order-parameter field and the metric tensor field originate from density-density and energy-momentum tensor-tensor effective interactions, respectively, in the recursive renormalization group transformation, performed approximately in the Gaussian level. This linear approximation in the recursive renormalization group transformation for the gravity sector gives rise to a linearized quantum Einstein-scalar theory along the z-directional emergent space. In the large N limit, where N is the flavor number of the original scalar fields, quantum fluctuations of both dynamical metric and dual scalar fields are suppressed, leading to a classical field theory of the Einstein-scalar type in (D+1)-spacetime dimensions. We show that this emergent background gravity describes the renormalization group flows of coupling functions in the UV quantum field theory through the extra dimension. More precisely, the IR boundary conditions of the gravity equations correspond to the renormalization group β-functions of the quantum field theory, where the infinitesimal distance in the extra-dimensional space is identified with an energy scale for the renormalization group transformation. Finally, we also show that this dual holographic formulation describes quantum entanglement in a geometrical way, encoding the transfer of quantum entanglement from quantum matter to classical gravity in the large N limit. We claim that this entanglement transfer serves as a microscopic foundation for the emergent holographic duality description.

In this paper we compare the construction of equilibrium states at finite temperature for self-interacting massive scalar quantum field theories on Minkowski spacetime proposed by Fredenhagen and Lindner with results obtained in ordinary thermal field theory, by means of real time and Matsubara formalisms. In the construction of this state, even if the adiabatic limit is considered, the interaction Lagrangian is multiplied by a smooth time cut-off. In this way the interaction starts adiabatically and the correlation functions are free from divergences. The corresponding interaction Hamiltonian is a local interacting field smeared over the interval of time where the chosen cut-off is not constant. In order to cope with this smearing, the Matsubara propagator needs to be modified. We obtain an expansion of the correlation functions of the interacting equilibrium state as a sum over certain type of graphs with mixed edges, some of them correspond to modified Matsubara propagators and others to propagators of the real time formalism. An integration over the adiabatic time cut-off is present in every vertex. However, at every order in perturbation theory, the final result does not depend on the particular form of the cut-off function. The obtained graphical expansion contains in it both the real time formalism and the Matsubara formalism as particular cases. Finally, we show that a particular factorisation which is used to derive the ordinary real time formalism holds only in special cases and we present a counterexample. We conclude with the analysis of certain correlation functions and we notice that corrections to the self-energy in a $\lambda\phi^4$ at finite temperature theory are expected.

Kink dynamics in a system of two coupled scalar fields in two space–time dimensions

We construct a consistent quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-times free of infrared divergences. We show that in such a quantum field theory (i) a continuous symmetry of (pseudo)scalar field translations is spontaneously broken, (ii) Goldstone bosons appear as quanta of a free massless (pseudo)scalar field and (iii) there is a non-vanishing spontaneous {magnetization}. In spite of the existence of a spontaneous {magnetization} the main inequality between vacuum expectation values of certain operators which have been used for the derivation of the Mermin–Wagner–Hohenberg theorem (C. Itzykson and J.-M. Drouffe, {Statistical field theory}, Vol. I, 1989, pp. 219–224) is fulfilled.

An (n+1) factorization of an (n+1)-dimensional Riemann manifold is performed. For a space permitting a Killing vector, the (n+l)-dimensional Hubert variational principle reduces to the variational principle for the corresponding quantities in an n-dimensional space. Hence, setting n=4 and n=3, versions of a unified theory of gravitational, electromagnetic, and scalar fields and the steady-space theory of general relativity theory, respectively, are constructed. The five-dimensional variational principle for geodesics reduces to the four-dimensional leastaction principle for the test charged particle moving in gravitational, electromagnetic, and scalar fields.

We study rotating boson stars in five spacetime dimensions. The boson fields consist of a complex doublet scalar field. Considering boson stars rotating in two orthogonal planes with both angular momenta of equal magnitude, a special ansatz for the boson field and the metric allows for solutions with nontrivial dependence on the radial coordinate only. The charge of the scalar field equals the sum of the angular momenta. The rotating boson stars are globally regular and asymptotically flat. For our choice of a sextic potential, the rotating boson star solutions possess a flat spacetime limit. We study the solutions in flat and curved spacetime.

This paper describes an approach for interactive visualization of mixed scalar and vector fields, in which vector icons are generated from pre-voxelized icon templates and volume-rendered together with the volumetric scalar data. This approach displays simultaneously the global structure of the scalar field and the detailed features of the vector field. Interactive visualization is achieved with incremental image update, by re-rendering only a small portion of the image wherever and whenever a change occurs. This technique supports a set of interactive visualization tools, including change of vector field visualization parameters, real-time animation of vector icons advected within the scalar field, a zooming lens, and a local probe. Subject Terms: data visualisation; interactive systems; graphical user interfaces; rendering (computer graphics); interactive visualization; scalar fields; vector fields; vector icons; pre-voxelized icon templates; volume-rendered; volumetric scalar data; incremental image update; real-time animation; zooming lens; local probe

In [1, 2, 3] the Corolla Polynomial $ \mathcal C (\Gamma) \in \mathbb C [a_{h_1}, \ldots, a_{h_{\left \vert \Gamma^{[1/2]} \right \vert}}] $ was introduced as a graph polynomial in half-edge variables $ \left \{ a_h \right \} _{h \in \Gamma^{[1/2]}} $ over a 3-regular scalar quantum field theory (QFT) Feynman graph $ \Gamma $. It allows for a covariant quantization of pure Yang-Mills theory without the need for introducing ghost fields, clarifies the relation between quantum gauge theory and scalar QFT with cubic interaction and translates back the problem of renormalizing quantum gauge theory to the problem of renormalizing scalar QFT with cubic interaction (which is super renormalizable in 4 dimensions of spacetime). Furthermore, it is, as we believe, useful for computer calculations. In [4] on which this paper is based the formulation of [1, 2, 3] gets slightly altered in a fashion specialized in the case of the Feynman gauge. It is then formulated as a graph polynomial $ \mathcal C ( \Gamma ) \in \mathbb C [a_{h_{1 \pm}}, \ldots, a_{h_{\left \vert \Gamma^{[1/2]} \right \vert} \vphantom{h}_\pm}, b_{h_1}, \ldots, b_{h_{\left \vert \Gamma^{[1/2]} \right \vert}}] $ in three different types of half-edge variables $ \left \{ a_{h_+} , a_{h_-} , b_h \right \} _{h \in \Gamma^{[1/2]}} $. This formulation is also suitable for the generalization to the case of spontaneously broken gauge theories (in particular all bosons from the Standard Model), as was first worked out in [4] and gets reviewed here.

In this paper the quantum vacuum energies induced by massive fluctuations of\none real scalar field on a configuration of two partially transparent plates\nare analysed. The physical properties of the infinitely thin plates are\ncharacterized by two Dirac-$\\delta$ potentials. We find that an\nattractive/repulsive Casimir force arises between the plates when the weights\nof the $\\delta$'s have equal/different sign. If some of the plates absorbs\nfluctuations below some threshold of energy (the corresponding weight is\nnegative) there is the need to set a minimum mass to the scalar field\nfluctuations to preserve unitarity in the corresponding quantum field theory.\nTwo repulsive $\\delta$-interactions are compatible with massless fluctuations.\nThe effect of Dirichlet boundary conditions at the endpoints of the interval\n$(-a,a)$ on a massless scalar quantum field theory defined on this interval is\ntantamount letting the weights of the repulsive $\\delta$-interactions to\n$+\\infty$.\n

A solution of the renormalised Thirring model constructed only from two scalar massless fields is studied. The representations of the conformal group for the solution as well as those for the scalar fields are obtained. It is proved that the renormalised Thirring equation is covariant with respect to the above representations.