There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

Euclid is a European Space Agency medium-class mission selected for launch in 2020 within the cosmic vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

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Cosmology and Fundamental Physics with the Euclid Satellite

One century after its formulation, Einstein's general relativity (GR) has made remarkable predictions and turned out to be compatible with all experimental tests. Most of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that GR should be modified when gravitational fields are strong and spacetime curvature is large. The best astrophysical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of GR. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einstein's theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime.

/pdf/testing-general-relativity-with-present-and-future-3ik25z2kf9.pdf

Testing general relativity with present and future astrophysical observations

We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali-Gabadadze-Porrati model, cascading gravity and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware-Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solutions in these models. Finally we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

Massive Gravity

We construct a fully covariant theory of massive gravity which does not require the introduction of an external reference metric, and overcomes the usual problems of massive gravity theories (fatal ghosts instabilities, acausality and/or van Dam-Veltman-Zakharov discontinuity). The equations of motion of the theory are nonlocal but respect causality. The starting point is the quadratic action proposed in the context of the degravitation idea. We show that it is possible to extend it to a fully nonlinear covariant theory. This theory describes the 5 degrees of freedom of a massive graviton plus a scalar ghost. However, contrary to generic nonlinear extensions of Fierz-Pauli massive gravity, the ghost has the same mass m as the massive graviton, independently of the background, and smoothly goes into a nonradiative degree of freedom for m→0. As a consequence, for m∼H0 the vacuum instability induced by the ghost is irrelevant even over cosmological time scales. We finally show that an extension of the model degravitates a vacuum energy density of order M4Pl down to a value of order M2Plm2, which for m=O(H0) is of order of the observed value of the vacuum energy density.

Nonlocal theory of massive gravity

We study the cosmological dynamics of a recently proposed infrared modification of the Einstein equations, based on the introduction of a nonlocal term constructed with m2gμν□-1R, where m is a mass parameter. The theory generates automatically a dynamical dark energy component, that can reproduce the observed value of the dark energy density without introducing a cosmological constant. Fixing m so to reproduce the observed value ΩDE≃0.68, and writing wDE(a) = w0+(1-a)wa, the model provides a neat prediction for the equation of state parameters of dark energy, w0≃-1.042 and wa≃-0.020, and more generally provides a pure prediction for wDE as a function of redshift. We show that, because of some freedom in the definition of □-1, one can extend the construction so to define a more general family of nonlocal models. However, in a first approximation this turns out to be equivalent to adding an explicit cosmological constant term on top of the dynamical dark energy component. This leads to an extended model with two parameters, ΩΛ and m. Even in this extension the EOS parameter w0 is always on the phantom side, in the range -1.33 ≲w0≤-1, and there is a prediction for the relation between w0 and wa.

Cosmological dynamics and dark energy from nonlocal infrared modifications of gravity

We perform a detailed study of the cosmological dynamics of a recently proposed infrared modification of the Einstein equations, based on the introduction of a non-local term constructed with $m^2g_{\mu\nu}\Box^{-1} R$, where $m$ is a mass parameter. The theory generates automatically a dynamical dark energy component, that can reproduce the observed value of the dark energy density without introducing a cosmological constant. Fixing $m$ so to reproduce the observed value $\Omega_{\rm DE}\simeq 0.68$, and writing $w(a)=w_0+(1-a) w_a$, the model provides a neat prediction for the equation of state parameters of dark energy, $w_0\simeq -1.042$ and $w_a\simeq -0.020$. We show that, because of some freedom in the definition of $\Box^{-1}$, one can extend the construction so to define a more general family of non-local models. However, in a first approximation this turns out to be equivalent to adding an explicit cosmological constant term on top of the dynamical dark energy component. This leads to an extended model with two parameters, $\Omega_{\Lambda}$ and $m$. Even in this extension the EOS parameter $w_0$ is always on the phantom side, in the range $-1.33 < w_0\leq -1$, and there is a prediction for the relation between $w_0$ and $w_a$.

Ermis Mitsou

Papers

Nonlocal theory of massive gravity

Cosmological dynamics and dark energy from nonlocal infrared modifications of gravity

Cosmological dynamics and dark energy from non-local infrared modifications of gravity

Apparent ghosts and spurious degrees of freedom in non-local theories

Early dark energy from zero-point quantum fluctuations