The author has tried to include several examples to illustrate the general methods being treated and ease the reader through the journey. Felipe Zaldivar, MAA Reviews, July, This is a complete guide to the mathematics underlying the laws of physics in general and quantum physics in particular. Frederic Paugam has given us a work of enormous scope and shows students a road to a coordinate-free presentation of mathematical objects underlying classical and quantum field theories. The book will be useful to graduate student, but also to pure and applied mathematicians interested in modern mathematical developments related to quantum theory.

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## Renormalization of Quantum Field Theories with Non-linear Field Transformations

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### Quantum field theory

Anderen bekeken ook. The construction of local , relativistic-covariant quantum non-Abelian gauge theories, involves the so-called Faddeev-Popov determinant and the introduction of ghost fields and relies on manipulations of field integrals Faddeev and Popov BRST symmetry emerges from this formalism. In what follows, as a slight simplification, we discuss only gauge theories without matter, the modifications due to the addition of matter fields being simple even though, in the case of fermions and chiral gauge invariance it may lead to obstruction to quantization in the form of gauge anomalies.

However, as a consequence of gauge invariance , the integrand is constant along gauge orbits the trajectories obtained by starting from one gauge field and acting on it with all gauge transformations and thus the integral is not defined. The idea is then to introduce a surface section that cuts all gauge orbits once and to restrict the integral to this section with a measure that ensures that all choices of sections are equivalent. It follows from the rules of integration in Grassmann algebras that determinants can be represented by integrals over generators of Grassmann algebras , which are anti-commuting variables see the section The origin of BRST symmetry and equation 30 in particular.

Another way of expressing the symmetry is based on the introduction of a functional differential operator.

In the field integral implementation of BRST symmetry, one must also verify the invariance of the integration measure in 8. Note, however, that in presence of matter fields, this condition may apply to generators in other representations and is then only satisfied for semi-simple Lie algebras.

Gauge independence, in particular, is essential to prove that physical observables satisfy the unitarity requirement, a property that is not obvious for a quantized gauge theory. Without entering into too many details, let us point out that one observation facilitates the construction of general BRST invariant polynomials in the fields.

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## Noncommutative Supersymmetric Field Theories

The initial field integral 8 of the quantized gauge theory is not defined, even in the sense of a perturbative expansion: the perturbative expansion is generated by keeping in the exponential the part of the classical action that is quadratic in the fields free action and expanding in a formal power series the exponential of the remaining part.

This expansion is usefully described in terms of Feynman diagrams , each one representing a Feynman integral contribution to the perturbative series. A first necessary step is called regularization , in which one modifies in some unphysical way the classical action to render the expansion finite. The construction of quantum gauge theories and practical calculations are much simplified if the regularization preserves the BRST symmetry, even if, strictly speaking, this requirement is not mandatory. Different methods are available: one can modify the theory at short distance, in the continuum at a scale specified by a cut-off or by introducing a space-time lattice lattice gauge theories.

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When the number of different field monomials, required to render finite the perturbative expansion, is bounded at all orders by some fixed number, one calls the quantum field theory perturbatively renormalizable by power counting. Gauge theories with properly chosen gauge sections satisfy this criterion, like, for example, Landau's gauge in 7.

### Proceedings of a Workshop, Held at Ringberg Castle Tegernsee, FRG, February 16-20, 1987

However, in gauge theories one still has to prove that renormalization can be achieved without spoiling the geometric structure that ensures that physical results do not depend on the choice of the gauge section. Initial proofs of gauge independence of the renormalized gauge theory were based on the Slavnov-Taylor identities, suitably extended to the situation of spontaneous symmetry breaking by Lee-Zinn-Justin. Note that all these proofs apply to compact Lie groups with semi-simple Lie algebras; the simpler Abelian case has to be dealt with by different methods.

Let us point out that if renormalization would preserve the BRST symmetry in the explicit form 12 , gauge independence could be proved easily. Other fields are similarly renormalized. Interesting links of methodological nature have emerged with Mould calculus and multizeta values, inso- far as Hopf algebras prove to be a very useful tool to deal with divergences in general.

This can be illustrated by the implementation of Abel type theorems for the non com- mutative generating series of multizeta values to obtain the counter-terms eliminating divergence of multizeta values. Hence multiple zeta functions should be of motivic origin as indicated by Bloch, Esnault and Kreimer and also Connes and Marcolli in their joint book Kontsevich has also recently presented his own view of real space renormalization techniques using vertex algebras.