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

We present in a manifestly gauge-invariant form the theory of classical linear gravitational perturbations in part I, and a quantum theory of cosmological perturbations in part II. Part I includes applications to several important examples arising in cosmology: a univese dominated by hydrodynamical matter, a universe filled with scalar-field matter, and higher-derivative theories of gravity. The growth rates of perturbations are calculated analytically in most interesting cases. The analysis is applied to study the evolution of fluctuations in inflationary universe models. Part II includes a unified description of the quantum generation and evolution of inhomogeneities about a classial Friedmann background. The method is based on standard canonical quantization of the action for cosmological perturbations which has been reduced to an expression in terms of a single gauge-invariant variable. The spectrum of density perturbations originating in quantum fluctuations is calculated in universe with hydrodynamical matter, in inflationary universe models with scalar-field matter, and in higher-derivative theories of gravity.

The gauge-invariant theory of classical and quantized cosmological perturbations developed in parts I and II is applied in part III to several interesting physical problems. It allows a simple derivation of the relation between temperature anistropes in the cosmic microwave background. radiation and the gauge-invariant potential for metric perturbations. The generation and evolution of gravitational waves is studied. As another example, a simple analysis of entropy perturbations and non-scale-invariant spectra in inflationary universe models is presented. The gauge-invariant theory of cosmological perturbations also allows a consistent and gauge-invariant definition of statistical fluctuations.

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Theory of Cosmological Perturbations

We review the present status of QCD corrections to weak decays beyond the leading-logarithmic approximation, including particle-antiparticle mixing and rare and $\mathrm{CP}$-violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective Hamiltonians of all decays for which the next-to-leading-order corrections are known. Subsequently, we present the phenomenological implications of these calculations. The values of various parameters are updated, in particular the mass of the newly discovered top quark. One of the central issues in this review are the theoretical uncertainties related to renormalization-scale ambiguities, which are substantially reduced by including next-to-leading-order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated. [S0034-6861(96)00304-2]

Weak decays beyond leading logarithms

We discuss the cosmological constant problem in the light of dilatation symmetry and its possible anomaly. For dilatation symmetric quantum theories realistic asymptotic cosmology is obtained provided the effective potential has a non-trivial minimum. For theories with dilatation anomaly one needs as a non-trivial "cosmon condition" that the energy-momentum tensor in the vacuum is purely anomalous. Such a condition is related to the short-distance renormalization group behavior of the fundamental theory. Observable deviations from the standard hot big bang cosmology are possible.

Cosmology and the Fate of Dilatation Symmetry

Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance

The form of the anomaly in the trace of the energy-momentum tensor in a general theory of interacting fermions and non-Abelian gauge bosons is dervied. The result is shown to involve precisely those gaugevariant operatos which are known to mix with the naive trace under renormalization. The trace is shown to be soft on the mass shell if and only if the theory is at an eigenvalue of the Callan-Symanzik $\ensuremath{\beta}$ function. The dilatation anomaly in the matrix element of $\ensuremath{\theta}_{}^{\ensuremath{\lambda}}{}_{\ensuremath{\lambda}}{}^{}$ with two electromagnetic currents (to lowest order in electromagnetism, but including all orders of the strong gauge interaction) is derived and shown to be infinitely renormalized in finite orders of strong perturbation theory. This anomaly is then shown to be canonical and given precisely by the lowest-order result provided the strong interactions are summed to all orders before going to the limit of physical space-time dimensions.

Trace and Dilatation Anomalies in Gauge Theories

We relate the energy-momentum-tensor trace anomaly in spin-1/2 quantum electrodynamics to the functions $\ensuremath{\beta}(\ensuremath{\alpha})$, $\ensuremath{\delta}(\ensuremath{\alpha})$ defined through the Callan-Symanzik equations, and prove finiteness of ${\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\nu}}$ when the anomaly is taken into account.

Energy-momentum-tensor trace anomaly in spin-1/2 quantum electrodynamics

A method for computing electromagnetic properties of hadrons in lattice QCD is described and preliminary numerical results are presented. The electromagnetic field is introduced dynamically, using a noncompact formulation. Employing enhanced electric charges, the dependence of the pseudoscalar meson mass on the (anti)quark charges and masses can be accurately calculated. At {beta}=5.7 with Wilson action, the {pi}{sup +}-{pi}{sup 0} splitting is found to be 4.9(3) MeV. Using the measured {ital K}{sup 0}-{ital K}{sup +} splitting, we also find {ital m}{sub {ital u}}/{ital m}{sub {ital d}}=0.512(6). Systematic errors are discussed. {copyright} {ital 1996 The American Physical Society.}

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Anthony Duncan

Papers

Trace and Dilatation Anomalies in Gauge Theories

Energy-momentum-tensor trace anomaly in spin-1/2 quantum electrodynamics

Electromagnetic splittings and light quark masses in lattice QCD

Regularization and renormalization of quantum field theory in curved space-time

Nonperturbative physics from interpolating actions