Journal Articles (Peer-Reviewed)
Much, Albert, Steffen Pottel and Klaus Sibold (2016): Preconjugate variables in quantum field theory and their applications, Physical Review D, 94 (6: ID 065008).
Abstract: Preconjugate variables X have commutation relations with the energy-momentum P of the respective system which are of a more general form than just the Hamiltonian one. Since they have been proven useful in their own right for finding new spacetimes, we present a study of them here. Interesting examples X can be found via geometry—motions on the mass shell for massive and massless systems—and via group theory—invariance under special conformal transformations of the mass shell and light cone, respectively. Both find representations on Fock space. We work mainly in ordinary four-dimensional Minkowski space and spin zero. The limit process from nonzero to vanishing mass turns out to be nontrivial and leads naturally to wedge variables. We point out some applications and extensions to more general spacetimes. In a companion paper, we discuss the transition to conjugate pairs.
Pottel, Steffen and Klaus Sibold (2016): Conjugate variables in quantum field theory and a refinement of Pauli’s theorem, Physical Review D, 94 (6: ID 065008).
Abstract: For the case of spin zero, we construct conjugate pairs of operators on Fock space. On states multiplied by polarization vectors, coordinate operators Q conjugate to the momentum operators P exist. In the massive case the notion of interest is derived from a geometrical quantity, the massless case is realized by taking the limit m2→0 on the one hand, on the other, starting with m2=0 directly, from conformal transformations. The norm problem of the states on which the Q’s act is crucial: the states determine eventually how many independent conjugate pairs exist. It is intriguing that (light-) wedge variables and, hence, the wedge-local case seem to be preferred.
Pottel, Steffen and Klaus Sibold (2010): Conformal Transformations of the S-Matrix; the β-Function Identifies the Change of Spacetime, Physical Review D, 82 (2: ID 025001).
Abstract: First conformal transformations of the S-matrix are derived in massless ϕ^4-theory. Then it is shown that the anomalous transformations can be rewritten as a symmetry once one has introduced a local coupling and interprets the charge of the symmetry accordingly. By introducing a suitable effective coupling on which the S-matrix depends one is able to identify via the β-function an underlying new spacetime with non-trivial conformal (flat) metric.
Journal Articles (Professional)
Pottel, Steffen (2018): BPHZ renormalization in configuration space for the A^4-model, Special Issue: In memoriam Wolfhart Zimmermann: work, life and heritage, Nuclear Physics B, 927: 274-293.
Abstract: Recent developments for BPHZ renormalization performed in configuration space are reviewed and applied to the model of a scalar quantum field with quartic self-interaction. An extension of the results regarding the short-distance expansion and the Zimmermann identity is shown for a normal product, which is quadratic in the field operator. The realization of the equation of motion is computed for the interacting field and the relation to parametric differential equations is indicated.
Pottel, Steffen (2018): Bogoliubov-Parasiuk-Hepp-Zimmermann Renormalization in Configuration Space (Dissertation), Universität Leipzig: Leipzig.
Otto, Felix, Steffen Pottel and Camilla Nobili (2017): Rigorous Bounds on Scaling Laws in Fluid Dynamics, in: Feireisl, Eduard and Elisabetta Rocca (ed.): Mathematical thermodynamics of complex fluids: Cetraro, Italy 2015, Springer International Publishing: Berlin, 101-145.
Pottel, Steffen (2017): A BPHZ Theorem in Configuration Space.
Abstract: The concept of BPHZ renormalization is translated into configuration space. A new version of the convergence theorem by means of Zimmermann's forest formula is proved and a sufficient condition for the existence of the constant coupling limit is derived in the new setting. https://arxiv.org/abs/1706.06762
Pottel, Steffen (2017): Configuration Space BPHZ Renormalization on Analytic Spacetimes.
Abstract: A configuration space version of BPHZ renormalization is proved in the realm of perturbative algebraic quantum field theory. All arguments are formulated entirely in configuration space so that the range of application is extended to analytic spacetimes. Further the relation to the momentum space method is established. In the course of that, it is necessary to study the limit of constant coupling. https://arxiv.org/abs/1708.04112
Pottel, Steffen (2017): Normal Products and Zimmermann Identities in Configuration Space BPHZ Renormalization.
Abstract: The notion of normal products, a generalization of Wick products, is derived with respect to BPHZ renormalization formulated entirely in configuration space. If inserted into time-ordered products, they admit the limit of coinciding field operators, which constitute the normal product. The derivation requires the introduction of Zimmermann identities, which relate field monomials or renormalization parts with differing subtraction degree. Furthermore, we calculate the action of wave operators on elementary fields inserted into time-ordered products using the properties of normal products. https://arxiv.org/abs/1708.04115