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

Quantum field theory on noncommutative spaces

We consider the standard model on a non-commutative space and expand the action in the non-commutativity parameter $\theta^{\mu \nu}$
 . No new particles are introduced; the structure group is $SU(3)\times SU(2)\times U(1)$
 . We derive the leading order action. At zeroth order the action coincides with the ordinary standard model. At leading order in $\theta^{\mu\nu}$
 we find new vertices which are absent in the standard model on commutative space-time. The most striking features are couplings between quarks, gluons and electroweak bosons and many new vertices in the charged and neutral currents. We find that parity is violated in non-commutative QCD. The Higgs mechanism can be applied. QED is not deformed in the minimal version of the NCSM to the order considered.

The standard model on non-commutative space-time

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.

Construction of non-Abelian gauge theories on noncommutative spaces

Comments on perturbative dynamics of non-commutative Yang–Mills theory

We show that the photon self-energy in quantum electrodynamics on noncommutative 4 is renormalizable to all orders (both in θ and ) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the θ-deformed classical action which are necessary to compensate the divergences coming from loop integrations.

Renormalization of the noncommutative photon self-energy to all orders via Seiberg-Witten map

With the help of the Seiberg-Witten map for photons and fermions we define a theta-deformed QED at the classical level. Two possibilities of gauge-fixing are discussed. A possible non-Abelian extension for a pure theta-deformed Yang-Mills theory is also presented.

Deformed QED via Seiberg-Witten Map

We show that the non-commutative Yang–Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The non-commutative Yang–Mills action is invariant under combined conformal transformations of the Yang–Mills field and of the non-commutativity parameter $\theta$
 . The Seiberg–Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.

Non-commutative Lorentz symmetry and the origin of the Seiberg Witten map

We present a simple derivation of vector supersymmetry transformations for topological field theories of Schwarz- and Witten-type. Our method is similar to the derivation of BRST-transformations from the so-called horizontality conditions or russian formulae. We show that this procedure reproduces in a concise way the known vector supersymmetry transformations of various topological models (in particular of general relativity viewed as a topological-like model) and we use it to obtain some new transformations of this type for 4d topological YM-theories in different gauges.

Vector supersymmetry in topological field theories

A U(N) Chern-Simons theory on noncommutative $\mathbb{R}^{3}$ is constructed as a $\q$-deformed field theory. The model is characterized by two symmetries: the BRST-symmetry and the topological linear vector supersymmetry. It is shown that the theory is finite and $\q_{\m\n}$-independent at the one loop level and that the calculations respect the restriction of the topological supersymmetry. Thus the topological $\q$-deformed Chern-Simons theory is an example of a model which is non-singular in the limit $\q \to 0$.

J. M. Grimstrup

Papers

Renormalization of the noncommutative photon self-energy to all orders via Seiberg-Witten map

Deformed QED via Seiberg-Witten Map

Non-commutative Lorentz symmetry and the origin of the Seiberg Witten map

Vector supersymmetry in topological field theories

Perturbative Chern-Simons Theory on Noncommutative R^3