We show that every tempered distribution, which is a solution of the (homogenous) Klein-Gordon equation, admits a ``tame'' restriction to the characteristic (hyper)surface $\{x^0+x^n=0\}$ in $(1+n)$-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space $\cS'_{\partial_-}(\R^n)$ which we have introduced in \cite{PullJMP}. Moreover, we show that every element of $\cS'_{\partial_-}(\R^n)$ appears as the ``tame'' restriction of a solution of the (homogeneous) Klein-Gordon equation.

article Closed-form formulas for projecting constructible sets in the theory of algebraically closed fields Share on Author: Peter Ullrich Technische Universität München, Boltzmannstr, Garching, Germany Technische Universität München, Boltzmannstr, Garching, GermanyView Profile Authors Info & Claims ACM Communications in Computer AlgebraVolume 40Issue 2June 2006 pp 45–48https://doi.org/10.1145/1182553.1182560Online:01 June 2006Publication History 11citation78DownloadsMetricsTotal Citations11Total Downloads78Last 12 Months2Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

We investigate the generally assumed inconsistency in light cone quantum field theory that the restriction of a massive, real, scalar, free field to the nullplane $\Sigma=\{x^0+x^3=0\}$ is independent of mass \cite{LKS}, but the restriction of the two-point function depends on it (see, e.g., \cite{NakYam77, Yam97}). We resolve this inconsistency by showing that the two-point function has no canonical restriction to $\Sigma$ in the sense of distribution theory. Only the so-called tame restriction of the two-point function exists which we have introduced in \cite{Ull04sub}. Furthermore, we show that this tame restriction is indeed independent of mass. Hence the inconsistency appears only by the erroneous assumption that the two-point function would have a (canonical) restriction to $\Sigma$.

To treat the front-form Hamiltonian approach to quantum field theory, called light cone quantum field theory, in a mathematically rigorous way, the existence of a well-defined restriction of the corresponding free fields to the hypersurface {x0+x3=0} in Minkowski space is of an essential necessity. However, even in the situation of a real scalar free field such a restriction does canonically not exist; this is called the restriction problem. Furthermore, since the beginning of light cone quantum field theory there is the problem of nonexistence of a well-defined Fock space expansion of a free quantum field in terms of light cone momenta which is called the zero-mode problem. In this paper we present solutions to these long outstanding problems where the study of the zero-mode problem (of the corresponding classical field) will lead us to a solution of the restriction problem. We introduce a new function space of “squeezed” smooth functions which can canonically be embedded into the Schwartz space S(R3). The restriction of the free field to {x0+x3=0} is canonically definable on this function space and we show that the covariant field is uniquely determined by this “tame” restriction.

Abstract We analyze an only recently identified Riemann autograph that had been in the possession of Georg August Thieme (1831–1910). These are the notes which Bernhard Riemann (1826–1866) made during his conversation with Thieme in order to explain the central points of his theory of functions of a complex variable, in particular his version of the Dirichlet principle. Copyright 1999 Academic Press. Wir analysieren ein erst kurzlich identifiziertes Riemann-Autograph aus dem Besitz von Georg August Thieme (1831–1910). Es handelt sich um Notizen, mittels derer Bernhard Riemann (1826–1866) im Gesprach mit Thieme die zentralen Punkte seiner Theorie der Funktionen einer komplexen Veranderlichen erlautert hat, insbesondere seine Version des Dirichlet-Prinzips. Copyright 1999 Academic Press. MSC 1991 subject classification: 01A55; 30-03; 31-03; 01A72.

In [1, Kapitel I] Grauert and Remmert prove the Weierstraß division theorem for convergent power series by use of the Banach algebra of all power series converging absolutely on the closure \(\overline {P}\) of an open polycylinder P. An analogous argument is given by Hörmander in [3, Section 6.1] who, however, uses the Banach algebra of all functions holomorphic and bounded on P. The present paper gives an axiomatic approach to this Weierstraß division in Banach algebras of convergent power series which is based merely on a simple application of the geometric series and which works for both types of Banach algebras mentioned above and also, e.g, for the Banach algebra of all functions which are holomorphic on P and continuous on \(\overline {P}\).

A genuine continuum treatment of the massive phi(1+1)(4) theory in light-cone quantization is proposed. Fields are treated as operator-valued distributions, thereby leading to a mathematically well-defined handling of ultraviolet-and light-cone-induced infrared divergences and of their renormalization. Although nonperturbative, the continuum light-cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light-cone quantization, the critical coupling increases by 30% to a value r=1.5. Conventional perturbation theory at the corresponding order yields r(1) = 1, whereas the RG-improved fourth-order result is r(4) = 1.8 +/- 0.05.

A genuine continuum treatment of the massive ${\ensuremath{\varphi}}_{1+1}^{4}$ theory in light-cone quantization is proposed. Fields are treated as operator-valued distributions, thereby leading to a mathematically well-defined handling of ultraviolet- and light-cone-induced infrared divergences and of their renormalization. Although nonperturbative, the continuum light-cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light-cone quantization, the critical coupling increases by 30% to a value $r=1.5.$ Conventional perturbation theory at the corresponding order yields ${r}_{1}=1,$ whereas the RG-improved fourth-order result is ${r}_{4}=1.8\ifmmode\pm\else\textpm\fi{}0.05.$

A genuine continuum treatment of the massive φ41+1-theory in light-cone quantization is proposed. Fields are treated as operator valued distributions thereby leading to a mathematically well defined handling of ultraviolet and light cone induced infrared divergences and of their renormalization. Although non-perturbative the continuum light cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light cone quantization, the critical coupling increases by 30% to a value r = 1.5. Conventional perturbation theory at the corresponding order yields r1 = 1, whereas the RG improved fourth order result is r4 = 1.8± 0.05. PM 97/18, June 1997 PACS : 11.10.Ef, 11.10.St, 11.30.Rd

A genuine continuum treatment of the massive 4 1+1-theory in light-cone quantiza- tion is proposed. Fields are treated as operator valued distributions thereby leading to a mathematically well dened handling of ultraviolet and light cone induced infrared divergences and of their renormalization. Although non-perturbative the continuum light cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light cone quantization, the critical coupling increases by 30% to a value r = 1:5. Conventional perturbation theory at the corresponding order yields r1 = 1, whereas the RG improved fourth order result is r4 = 1:8 0:05.