Publications of Andreas Wipf

 106. Daniel August, Bjoern Wellegehausen and Andreas Wipf Wipf Mass spectrum of 2-dimensional N=(2,2) super Yang-Mills theory on the lattice In the present work we analyse N=(2,2) supersymmetric Yang-Mills (SYM) theory in two dimensions by means of lattice simulations. The theory arises as dimensional reduction of N=1 SYM theory in four dimensions. As in other gauge theories with extended supersymmetry, the classical scalar potential has flat directions which may destabilize numerical simulations. In addition, the fermion determinant need not be positive and this sign-problem may cause further problems in a stochastic treatment. We demonstrate that N=(2,2) super Yang-Mills theory has actually no sign problem and that the flat directions are lifted and thus stabilized by quantum corrections. Only the bare mass of the scalars experience a finite additive renormalization in this finite theory. On various lattices with different lattice constants we determine the scalar masses and hopping parameters for which the supersymmetry violating terms are minimal. By studying four Ward identities and by monitoring the $\pi$-mass we show that supersymmetry is indeed restored in the continuum limit. In the second part we calculate the masses of the low-lying bound states. We find that in the infinite-volume and supersymmetric continuum limit the Veneziano-Yankielowicz super-multiplet becomes massless and the Farrar-Gabadadze-Schwetz super-multiplet decouples from the theory. In addition, we estimate the masses of the excited mesons in the Veneziano-Yankielowicz multiplet. We observe that the gluino-glueballs have comparable masses to the excited mesons. preprint ArXiv:1802.07797 pdf file (1795kb) Polina Feldmann, Andreas Wipf and Luca Zambelli Critical Wess-Zumino models with four supercharges from the functional renormalization group We analyze the N=1 supersymmetric Wess-Zumino model dimensionally reduced to the N= supersymmetric model in three Euclidean dimensions. As in the original model in four dimensions and the N=(2,2) model in two dimensions the superpotential is not renormalized. This property puts severe constraints on the non-trivial fixed-point solutions, which are studied in detail. We admit a field-dependent wave function renormalization that in a geometric language relates to a Kähler metric. The Kähler metric is not protected by supersymmetry and we calculate its explicit form at the fixed point. In addition we determine the exact quantum dimension of the chiral superfield and several critical exponents of interest, including the correction-to-scaling exponent ω, within the functional renormalization group approach. We compare the results obtained at different levels of truncation, exploring also a momentum-dependent wave function renormalization. Finally we briefly describe a tower of multicritical models in continuous dimensions. preprint ArXiv:1712.03910 pdf file (496kb) Boris Merzlikin, Ilya Shapiro, Andreas Wipf and Omar Zanusso Renormalization group flows and fixed points for a scalar field in curved space with nonminimal F(φ)R coupling Using covariant methods, we construct and explore the Wetterich equation for a non-minimal coupling F(φ)R of a quantized scalar field to the Ricci scalar of a prescribed curved space. This includes the often considered non-minimal coupling ξRφ2 as a special case. We consider the truncations without and with scale- and field-dependent wave function renormalization in dimensions between four and two. Thereby the main emphasis is on analytic and numerical solutions of the fixed point equations and the behavior in the vicinity of the corresponding fixed points. We determine the non-minimal coupling in the symmetric and spontaneously broken phases with vanishing and non-vanishing average fields, respectively. Using functional perturbative renormalization group methods, we discuss the leading universal contributions to the RG flow below the upper critical dimension d=4. Phys. Rev. D96 (2017) 125007 pdf file (440kb) Bjoern Wellegehasuen, Daniel Schmidt and Andreas Wipf Critical flavour number of the Thirring model in three dimensions The Thirring model is a four-fermion theory with a current-current interaction and U(2N) chiral symmetry. It is closely related to three-dimensional QED and other models used to describe properties of graphene. In addition it serves as a toy model to study chiral symmetry breaking. In the limit of flavour number N → 1/2 it is equivalent to the Gross-Neveu model, which shows a parity-breaking discrete phase transition. The model was already studied with different methods, including Dyson-Schwinger equations, functional renormalisation group methods and lattice simulations. Most studies agree that there is a phase transition from a symmetric phase to a spontaneously broken phase for a small number of fermion flavours, but no symmetry breaking for large N. But there is no consensus on the critical flavour number Ncrit above which there is no phase transition anymore and on further details of the critical behaviour. Values of N found in the literature vary between 2 and 7. All earlier lattice studies were performed with staggered fermions. Thus it is questionable if in the continuum limit the lattice model recovers the internal symmetries of the continuum model. We present new results from lattice Monte Carlo simulations of the Thirring model with SLAC fermions which exactly implement all internal symmetries of the continuum model even at finite lattice spacing. If we reformulate the model in an irreducible representation of the Clifford algebra, we find, in contradiction to earlier results, that the behaviour for even and odd flavour numbers is very different: For even flavour numbers, chiral and parity symmetry are always unbroken. For odd flavour numbers parity symmetry is spontaneously broken below the critical flavour number Nir,crit=9 while chiral symmetry is still unbroken. Phys. Rev. D96 (2017) 094504 (Editor's choice) pdf file (525kb) Holger Gies, Tobias Hellwig, Andreas Wipf and Omar Zanusso A functional perspective on emergent supersymmetry We investigate the emergence of N=1 supersymmetry in the long-range behavior of three-dimensional parity-symmetric Yukawa systems. We discuss a renormalization approach that manifestly preserves supersymmetry whenever such symmetry is realized, and use it to prove that supersymmetry-breaking operators are irrelevant, thus proving that such operators are suppressed in the infrared. All our findings are illustrated with the aid of the ε-expansion and a functional variant of perturbation theory, but we provide numerical estimates of critical exponents that are based on the non-perturbative functional renormalization group. JHEP 1712 (2017) 132 pdf file (347kb) Tobias Hellwig, Andreas Wipf and Omar Zanusso Scaling and superscaling solutions from the functional renormalization group We study the renormalization group flow of Z2-invariant supersymmetric and non-supersymmetric scalar models in the local potential approximation using functional renormalization group methods. We focus our attention to the fixed points of the renormalization group flow of these models, which emerge as scaling solutions. In two dimensions these solutions are interpreted as the minimal (supersymmetric) models of conformal field theory, while in three dimension they are manifestations of the Wilson-Fisher universality class and its supersymmetric counterpart. We also study the analytically continued flow in fractal dimensions between 2 and 4 and determine the critical dimensions for which irrelevant operators become relevant and change the universality class of the scaling solution. We also include novel analytic and numerical investigations of the properties that determine the occurrence of the scaling solutions within the method. For each solution we offer new techniques to compute the spectrum of the deformations and obtain the corresponding critical exponents. Phys. Rev. D92 (2015) 085027 pdf file (1036kb) Ilya Shapiro, Poliane de Morais Teixeira and Andreas Wipf On the functional renormalization group for the scalar file on curved background with non-minimal interaction. The running of the non-minimal parameter ξ of the interaction of the real scalar field and scalar curvature is explored within the non-perturbative setting of the func- tional renormalization group (RG). We establish the RG flow in curved space-time in the scalar field sector, in particular derive an equation for the non-minimal parameter. The RG trajectory is numerically explored for different sets of initial data. Eur. Phys. J. C 75 C (2015) 262 pdf file (468kb) Marianne Heilmann, Tobian Hellwig, Benjamin Knorr, Marcus Ansorg and Andreas Wipf Convergence of Derivative Expansion in Supersymmetric Functional RG Flows. We confirm the convergence of the derivative expansion in two supersymmetric models via the functional renormalization group method. Using pseudo-spectral methods, high-accuracy results for the lowest energies in supersymmetric quantum mechanics and a detailed description of the supersymmetric analogue of the Wilson-Fisher fixed point of the three-dimensional Wess-Zumino model are obtained. The superscaling relation proposed earlier, relating the relevant critical exponent to the anomalous dimension, is shown to be valid to all orders in the supercovariant derivative expansion and for all d≥2. JHEP 1502 (2015) 109 pdf file (1430kb) Björn Wellegehausen, Daniel Körner and Andreas Wipf Asymptotic safety on the lattice: The Nonlinear O(N) Sigma Model. We study the non-perturbative renormalization group flow of the nonlinear O(N) sigma model in two and three spacetime dimensions using a scheme that combines an effective local Hybrid Monte Carlo update routine, blockspin transformations and a Monte Carlo demon method. In two dimensions our results verify perturbative renormalizability. In three dimensions, we determine the flow diagram of the theory for various N and different truncations and find a non-trivial fixed point, which indicates non-perturbative renormalizability. It is related to the well-studied phase transition of the O(N) universality class and characterizes the continuum physics of the model. We compare the obtained renormalization group flows with recent investigations by means of the Functional Renormalization Group. Annals of Physics 349 (2014) 374 pdf file (575kb) Björn Wellegehausen, Axel Maas, Andreas Wipf and Lorenz von Smekal Hadron masses and baryonic scales in G2-QCD at finite density. The QCD phase diagram at densities relevant to neutron stars remains elusive, mainly due to the fermion-sign problem. At the same time, a plethora of possible phases has been predicted in models. Meanwhile G2-QCD, for which the SU(3) gauge group of QCD is replaced by the exceptional Lie group G2, does not have a sign problem and can be simulated at such densities using standard lattice techniques. It thus provides benchmarks to models and functional continuum methods, and it serves to unravel the nature of possible phases of strongly interacting matter at high densities. Instrumental in understanding these phases is that G2-QCD has fermionic baryons, and that it can therefore sustain a baryonic Fermi surface. Because the baryon spectrum of G2-QCD also contains bosonic diquark and probably other more exotic states, it is important to understand this spectrum before one can disentangle the corresponding contributions to the baryon density. Here we present the first systematic study of this spectrum from lattice simulations at different quark masses. This allows us to relate the mass hierarchy, ranging from scalar would-be-Goldstone bosons and intermediate vector bosons to the G2-nucleons and deltas, to individual structures observed in the total baryon density at finite chemical potential. Phys. Rev. D89 (2014) 056007 pdf file (233kb) Mikhail Plyushchay and Andreas Wipf Particle in a self-dual dyon background: hidden free nature, and exotic superconformal symmetry. We show that a non-relativistic particle in a combined field of a magnetic monopole and 1/r2 potential reveals a hidden, partially free dynamics when the strength of the central potential and the charge-monopole coupling constant are mutually fitted to each other. In this case the system admits both a conserved Laplace-Runge-Lenz vector and a dynamical conformal symmetry. The supersymmetrically extended system corresponds then to a background of a self-dual or anti-self-dual dyon. It is described by a quadratically extended Lie superalgebra D(2,1;α) with α=1/2, in which the bosonic set of generators is enlarged by a generalized Laplace-Runge-Lenz vector and its dynamical integral counterpart related to Galilei symmetry, as well as by the chiral Z2-grading operator. The odd part of the nonlinear superalgebra comprises a complete set of 24=2×3×4 fermionic generators. Here a usual duplication comes from the Z2-grading structure, the second factor can be associated with a triad of scalar integrals --- the Hamiltonian, the generator of special conformal transformations and the squared total angular momentum vector, while the quadruplication is generated by a chiral spin vector integral which exits due to the (anti)-self-dual nature of the electromagnetic background. Phys. Rev. D89 (2014) 045017 pdf file (361kb) Raphael Flore, Andreas Wipf and Omar Zanusso Functional renormalization group of the nonlinear sigma model and the O(N) universality class. We study the renormalization group flow of the O(N) nonlinear sigma model in arbitrary dimensions. The effective action of the model is truncated to fourth order in the derivative expansion and the flow is obtained by combining the non-perturbative renormalization group and the background field method. We investigate the flow in three dimensions and analyze the phase structure for arbitrary N. While a nontrivial fixed point is present in a reduced truncation of the effective action and has critical properties which can be related to the well-known features of the O(N) universality class, one of the fourth order operators destabilizes this fixed point and has to be discussed carefully. Phys.Rev. D87 (2013) 065019 pdf file (584kb) Marianne Mastaler, Franziska Synatschke-Czerwonka and Andreas Wipf Supersymmetric renormalization group flows. The functional renormalization group equation for the quantum effective action is a powerful tool to investigate non-perturbative phenomena in quantum field theories. We discuss the application of manifest supersymmetric flow equations to the N = 1 Wess-Zumino model in two and three dimensions and the linear O(N) sigma model in three dimensions in the large-N limit. The former is a toy model for dynamical supersymmetry breaking, the latter for an exactly solvable field theory. Phys. Part. Nucl. 43 (2012) 593 pdf file (769kb) Marianne Heilman, Daniel Litim, Franziska Synatschke-Czerwonka and Andreas Wipf Phases of supersymmetric O(N) theories. We perform a global renormalization group study of O(N) symmetric Wess-Zumino theories and their phases in three euclidean dimensions. At infinite N the theory is solved exactly. The phases and phase transitions are worked out for finite and infinite short-distance cutoffs. A distinctive new feature arises at strong coupling, where the effective superfield potential becomes multi-valued, signalled by divergences in the fermion-boson interaction. Our findings resolve the long-standing puzzle about the occurrence of degenerate O(N) symmetric phases. At finite N, we find a strongly-coupled fixed point in the local potential approximation and explain its impact on the phase transition. We also examine the possibility for a supersymmetric Bardeen-Moshe-Bander phenomenon, and relate our findings with the spontaneous breaking of supersymmetry in other models. Phys. Rev. D86 (2012) 105006 pdf file (6722kb) Raphael Flore, Daniel Körner, Andreas Wipf and Christian Wozar Supersymmetric Nonlinear O(3) Sigma Model on the Lattice. A supersymmetric extension of the nonlinear O(3) sigma model in two spacetime dimensions is investigated by means of Monte Carlo simulations. We argue that it is impossible to construct a lattice action that implements both the O(3) symmetry as well as at least one supersymmetry exactly at finite lattice spacing. It is shown by explicit calculations that previously proposed discretizations fail to reproduce the exact symmetries of the target manifold in the continuum limit. We provide an alternative lattice action with exact O(3) symmetry and compare two approaches based on different derivative operators. Using the nonlocal SLAC derivative for the quenched model on moderately sized lattices we extract the value sigma(2,u0) = 1.2604(13) for the step scaling function at u0 = 1.0595, to be compared with the exact value 1.261210. For the supersymmetric model with SLAC derivative the discrete chiral symmetry is maintained but we encounter strong sign fluctuations, rendering large lattice simulations ineffective. By applying the Wilson prescription, supersymmetry and chiral symmetry are broken explicitly at finite lattice spacing, though there is clear evidence that both are restored in the continuum limit by fine tuning of a single mass parameter. JHEP 1211 (2012) 159 pdf file (2075kb) Axel Maas, Lorenz von Smekal, Bjoern Wellegehausen and Andreas Wipf The phase diagram of a gauge theory with fermionic baryons. The fermion-sign problem at finite density is a persisting challenge for Monte-Carlo simulations. Theories that do not have a sign problem can provide valuable guidance and insight for physically more relevant ones that do. Replacing the gauge group SU(3) of QCD by the exceptional group G2, for example, leads to such a theory. It has mesons as well as bosonic and fermionic baryons, and shares many features with QCD. This makes the G2 gauge theory ideally suited to study general properties of dense, strongly-interacting matter, including baryonic and nuclear Fermi pressure effects as relevant in compact stars and heavy-ion collisions. We present the first lattice simulations of the phase diagram of this theory at finite temperature and baryon chemical potential. Phys. Rev. D86 (2012) 111901 pdf file (349kb) Kurt Langfeld and Andreas Wipf Fermi-Einstein condensation in dense QCD-like theories. While pure Yang-Mills theory feature the centre symmetry, this symmetry is explicitly broken by the presence of dynamical matter. We study the impact of the centre symmetry in such QCD-like theories. In the analytically solvable Schwinger model, centre transitions take place even under extreme conditions, temperature and/or density, and we show that they are key to the solution of the Silver-Blaze problem. We then develop an effective SU(3) quark model which confines quarks by virtue of centre sector transitions. The phase diagram by confinement is obtained as a function of the temperature and the chemical potential. We show that at low temperatures and intermediate values for the chemical potential the centre dressed quarks undergo condensation due to Bose like statistics. This is the Fermi Einstein condensation. To corroborate the existence of centre sector transitions in gauge theories with matter, we study (at vanishing chemical potential) the interface tension in the three-dimensional Z2 gauge theory with Ising matter, the distribution of the Polyakov line in the four-dimensional SU(2)-Higgs model and devise a new type of order parameter which is designed to detect centre sector transitions. Our analytical and numerical findings lead us to conjecture a new state of cold, but dense matter in the hadronic phase for which Fermi Einstein condensation is realised. Annals of Physics 327 (2012) 994 pdf file (694kb) Christian Wozar and Andreas Wipf Supersymmetry Breaking in Low Dimensional Models. We analyse supersymmetric models that show supersymmetry breaking in one and two dimensions using lattice methods. Starting from supersymmetric quantum mechanics we explain the fundamental principles and problems that arise in putting supersymmetric models onto the lattice. We compare our lattice results (built upon the non-local SLAC derivative) with numerically exact results obtained within the Hamiltonian approach. A particular emphasis is put on the discussion of boundary conditions. We investigate the ground state structure, mass spectrum, effective potential and Ward identities and conclude that lattice methods are suitable to derive the properties of supersymmetric quantum mechanics, even with broken supersymmetry. Based on this result we analyse the two dimensional N=1 Wess-Zumino model with spontaneous supersymmetry breaking. First we show that (in agreement with earlier analytical and numerical studies) the SLAC derivative is a sensible choice in the quenched model, which is nothing but the two dimensional φ4 model. Then, we present the very first computation of a renormalised critical coupling for the complete supersymmetric model. This calculation makes use of Binder cumulants and is supported by a direct comparison to Ward identity results, both in the continuum and infinite volume limit. The physical picture is completed by masses at two selected couplings, one in the supersymmetric phase and one in the supersymmetry broken phase. Signatures of the Goldstino in the fermionic correlator are clearly visible in the broken case. Annals of Physics 327 (2012) 774 pdf file (913kb) Daniel Litim, Marianne Mastaler, Franzsika Synatschke-Czerwonka and Andreas Wipf Critical behavior of supersymmetric O(N) models in the large-N limit. We derive a supersymmetric renormalization group (RG) equation for the scale-dependent superpotential of the supersymmetric O(N) model in three dimensions. For a supersymmetric optimized regulator function, we solve the RG equation for the superpotential exactly in the large-N limit. The fixed-point solutions are classified by an exactly marginal coupling. In the weakly coupled regime there exists a unique fixed-point solution, for intermediate couplings we find two separate fixed-point solutions and in the strong coupling regime no globally defined fixed-point potentials exist. We determine the exact critical exponents both for the superpotential and the associated scalar potential. Finally, we relate the high-temperature limit of the four-dimensional theory to the Wilson-Fisher fixed point of the purely scalar theory. Phys. Rev. D 84 125009 (2011) pdf file (1671kb) Björn Wellegehausen, Andreas Wipf and Christian Wozar Phase diagram of the lattice G2 Higgs Model. We study the phases and phase transition lines of the finite temperature G2 Higgs model. Our work is based on an efficient local hybrid Monte-Carlo algorithm which allows for accurate measurements of expectation values, histograms and susceptibilities. On smaller lattices we calculate the phase diagram in terms of the inverse gauge coupling β and the hopping parameter κ. For κ > 0 the model reduces to G2 gluodynamics and for κ → ∞ to SU(3) gluodynamics. In both limits the system shows a first order confinement-deconfinement transition. We show that the first order transitions at asymptotic values of the hopping parameter are almost joined by a line of first order transitions. A careful analysis reveals that there exists a small gap in the line where the first order transitions turn into continuous transitions or a cross-over region. For β → ∞ the gauge degrees of freedom are frozen and one finds a nonlinear O(7) sigma model which exhibits a second order transition from a massive O(7)-symmetric to a massless O(6)-symmetric phase. The corresponding second order line for large β remains second order for intermediate β until it comes close to the gap between the two first order lines. Besides this second order line and the first order confinement-deconfinement transitions we find a line of monopole-driven bulk transitions which do not interfer with the confinement-deconfinment transitions. Phys. Rev. D 83 114502 (2011) pdf file (559kb) Björn Wellegehausen, Andreas Wipf and Christian Wozar Casimir Scaling and String Breaking in G2 Gluodynamics. We study the potential energy between static charges in G2 gluodynamics in three and four dimensions. Our work is based on an efficient local hybrid Monte-Carlo algorithm and a multi-level Lüscher-Weisz algorithm with exponential error reduction to accurately measure expectation values of Wilson and Polyakov loops. Both in three and four dimensions we show that at intermediate scales the string tensions for charges in various G2 representations scale with the second order Casimir. In three dimensions Casimir scaling is confirmed within four percent for charges in representations of dimensions 7,14,27,64,77,77',182 and 189 and in four dimensions within five percent for charges in representations of dimensions 7,14,27 and 64. In three dimensions we detect string breaking for charges in the two fundamental representations. The scale for string breaking agrees very well with the mass of the created pair of glue-lumps. Close to the string breaking distance Casimir scaling between adjoint and defining representation is violated by 2.5 percent. The analytical prediction for the continuum string tension is confirmed for the defining representation in three dimensions. Phys. Rev. D 83 016001 (2011) pdf file (288kb) Franziska Synatschke, Jens Braun and Andreas Wipf N=1 Wess Zumino Model in d=3 at zero and finite temperature. Supersymmetric renormalization group (RG) flow equations for the effective superpotential of the three-dimensional Wess-Zumino model are derived at zero and non-zero temperature. This model with fermions and bosons interacting via a Yukawa term possesses a supersymmetric analogue of the Wilson-Fisher fixed-point. At zero temperature we determine the phase-transition line in coupling-constant space separating the supersymmetric from the non-supersymmetric phase. At finite temperature we encounter dimensional reduction from 3 to 2 dimensions in the infrared regime. We determine the finite-temperature phase diagram for the restoration of the global Z2-symmetry and show that for temperatures above the Z2 phase transition the pressure obeys the Stefan-Boltzmann law of a gas of massless bosons in 2+1 dimensions. Phys. Rev. D 81 125001 (2010) pdf file (333kb) Kurt Langfeld, Bjoern Wellegehausen and Andreas Wipf Confinement and the quark Fermi-surface in SU(2N) QCD-like theories. Yang-Mills theories with a gauge group SU(Nc≠3) and quark matter in the fundamental representation share many properties with the theory of strong interactions, QCD with Nc=3. We show that, for Nc even and in the confinement phase, the quark determinant is independent of the boundary conditions, periodic or anti-periodic ones. We then argue that a Fermi sphere of quarks can only exist under extreme conditions when the centre symmetry is spontaneously broken and colour is liberated. Our findings are supported by lattice gauge simulations for Nc=2… 5 and are illustrated by means of a simple quark model. Phys. Rev. D 81 114502 (2010) pdf file (120kb) Franziska Synatschke, Holger Gies and Andreas Wipf Phase Diagram and Fixed-Point Structure of two dimensional N=1 Wess-Zumino Model. We study the phases and fixed-point structure of two-dimensional supersymmetric Wess-Zumino models with one supercharge. Our work is based on the functional renormalization group formulated in terms of a manifestly off-shell supersymmetric flow equation for the effective action. Within the derivative expansion, we solve the flow of the superpotential also including the anomalous dimension of the superfield. The models exhibit a surprisingly rich fixed-point structure with a discrete number of fixed-point superpotentials. Each fixed-point superpotential is characterized by its number of nodes and by the number of RG relevant directions. In limiting cases, we find periodic superpotentials and potentials which confine the fields to a compact target space. The maximally IR-attractive fixed point has one relevant direction, the tuning of which distinguishes between supersymmetric and broken phases. For the Wess-Zumino model defined near the Gau\ss ian fixed point, we determine the phase diagram and compute the corresponding ground-state masses. Phys. Rev. D 80 085007 (2009) pdf file (478kb) Bjoern Wellegehausen, Christian Wozar and Andreas Wipf Effective Polyakov Loop Dynamics for Finite Temperature G2 Gluodynamics. Based on the strong coupling expansion we obtain effective 3-dimensional models for the Polyakov loop in finite temperature G2 gluodynamics. The Svetitsky-Jaffe conjecture relates the resulting continuous spin models with G2 gluodynamics near phase transition points. In the present work we analyse the effective theory in leading order with the help of a generalised mean field approximation and with detailed Monte Carlo simulations. In addition we derive a Potts-type discrete spin model by restricting the characters of the Polyakov loops to the three extremal points of the fundamental domain of G2. Both the continuous and discrete effective models show a rich phase structure with a ferromagnetic, symmetric and several anti-ferromagnetic phases. The phase diagram contains first and second order transition lines and tricritical points. The modified mean field predictions compare very well with the results of our simulations. Phys. Rev. D 80 065028 (2009) pdf file (1009kb) Holger Gies, Franziska Synatschke and Andreas Wipf Supersymmetry breaking as a quantum phase transition. We explore supersymmetry breaking in the light of a rich fixed-point structure of two-dimensional supersymmetric Wess-Zumino models with one supercharge using the functional renormalization group (RG). We relate the dynamical breaking of supersymmetry to an RG relevant control parameter of the superpotential which is a common relevant direction of all fixed points of the system. Supersymmetry breaking can thus be understood as a quantum phase transition analogously to similar transitions in correlated fermion systems. Supersymmetry gives rise to a new superscaling relation between the critical exponent associated with the control parameter and the anomalous dimension of the field -- a scaling relation which is not known in standard spin systems. Phys. Rev. D 80 101701 (2009) pdf file (120kb) Vladimir Kassandrov, Nina Markova, Gerhard Schäfer and Andreas Wipf On a model of a classical relativistic particle of constant and universal mass and spin. The deformation of the classical action for a point-like particle recently suggested by A. Staruszkiewicz gives rise to a spin structure which constrains the values of the invariant mass and the invariant spin to be the same for any solution of the equations of motion. Both these Casimir invariants, the square of the four-momentum vector and the square of the Pauli-Lubanski vector, are shown to preserve the same fixed values also in the presence of an arbitrary external electromagnetic field. In the "free" case, in the centre-of-mass reference frame, the particle moves along a circle of fixed radius with arbitrary varying frequency. In a homogeneous magnetic field, a number of rotational "states" is possible with frequencies slightly different from the cyclotron frequency, and "phase-like" transitions with spin flops occure at some critical value of the particle's three-momentum. J. Phys. A: Math. Theor. 42 315204 (2009) pdf file (213kb) Wieland Brendel, Falk Bruckmann, Lukas Janssen, Andreas Wipf and Christian Wozar Instanton constituents and fermionic zero modes in twisted CP(n) models. We construct twisted instanton solutions of CP(n) models. Generically a charge-k instanton splits into k(n+1) well-separated and almost static constituents carrying fractional topological charges and being ordered along the noncompact direction. The locations, sizes and charges of the constituents are related to the moduli parameters of the instantons. We sketch how solutions with fractional total charge can be obtained. We also calculate the fermionic zero modes with quasi-periodic boundary conditions in the background of twisted instantons for minimally and supersymmetrically coupled fermions. The zero modes are tracers for the constituents and show a characteristic hopping. The analytical findings are compared to results extracted from Monte-Carlo generated and cooled configurations of the corresponding lattice models. Analytical and numerical results are in full agreement and it is demonstrated that the fermionic zero modes are excellent filters for constituents hidden in fluctuating lattice configurations. Phys. Lett. B 676 116-125 (2009) pdf file (766kb) Franziska Synatschke, Georg Bergner, Holger Gies, Andreas Wipf Flow Equation for Supersymmetric Quantum Mechanics. We study supersymmetric quantum mechanics with the functional RG formulated in terms of an exact and manifestly off-shell supersymmetric flow equation for the effective action. We solve the flow equation nonperturbatively in a systematic super-covariant derivative expansion and concentrate on systems with unbroken supersymmetry. Already at next-to-leading order, the energy of the first excited state for convex potentials is accurately determined within a 1% error for a wide range of couplings including deeply nonperturbative regimes. JHEP 03 028 (2009) pdf file (362kb) Tobias Kaestner, Georg Bergner, Sebastian Uhlmann, Andreas Wipf and Christan Wozar Two-Dimensional Wess-Zumino Models at Intermediate Couplings. We consider the two-dimensional N=(2,2) Wess-Zumino model with a cubic superpotential at weak and intermediate couplings. Refined algorithms allow for the extraction of reliable masses in a region where perturbation theory no longer applies. We scrutinize the Nicolai improvement program which is supposed to guarantee lattice supersymmetry and compare the results for ordinary and non-standard Wilson fermions with those for SLAC derivatives. It turns out that this improvement completely fails to enhance simulations for Wilson fermions and only leads to better results for SLAC fermions. Furthermore, even without improvement terms the models with all three fermion species reproduce the correct values for the fermion masses in the continuum limit. Phys. Rev. D 78 095001 (2008) pdf file (2200kb) Franziska Synatschke, Andreas Wipf and Kurt Langfeld Relation between chiral symmetry breaking and confinement in YM-theories. Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop are thoroughly investigated. The approach by Gattringer is generalized to mode sums which reconstruct the Polyakov loop locally. This opens the possibility to study the mode sum approximation to the Polyakov loop correlator. The approach is re-derived for the ab initio continuum formulation of Yang-Mills theories, and the convergence of the mode sum is studied in detail. The mode sums are then explicitly calculated for the Schwinger model and SU(2) gauge theory in a homogeneous background field. Using SU(2) lattice gauge theory, the IR dominated mode sums are considered and the mode sum approximation to the static quark anti-quark potential is obtained numerically. We find a good agreement between the mode sum approximation and the static potential at large distances for the confinement and the high temperature plasma phase. Phys. Rev. D 77 114018 (2008) pdf file (479kb) Tobias Kaestner, Georg Bergner, Sebastian Uhlmann, Andreas Wipf, and Christian Wozar Supersymmetric lattice models in one and two dimensions. We study and simulate N=2 supersymmetric Wess-Zumino models in one and two dimensions. For any choice of the lattice derivative, the theories can be made manifestly supersymmetric by adding appropriate improvement terms corresponding to discretizations of surface integrals. In particular, we check that fermionic and bosonic masses coincide and the unbroken Ward identities are fulfilled to high accuracy. Equally good results for the effective masses can be obtained in a model with the SLAC derivative (even without improvement terms). In two dimensions we introduce a non-standard Wilson term in such a way that the discretization errors of the kinetic terms are only of order a squared. Masses extracted from the corresponding manifestly supersymmetric model prove to approach their continuum values much quicker than those from a model containing the standard Wilson term. Again, a comparable enhancement can be achieved in a theory using the SLAC derivative. arXiv:0709.0822v2 [hep-lat] pdf file (134kb) Christian Wozar, Tobias Kaestner, Sebastian Uhlmann, Andreas Wipf and Thomas Heinzl Z(3) Polyakov loop models and inverse Monte-Carlo methods. We study effective Polyakov loop models for SU(3) Yang-Mills theory at finite temperature. A comprehensive mean field analysis of the phase diagram is carried out and compared to the results obtained from Monte-Carlo simulations. We find a rich phase structure including ferromagnetic and antiferromagnetic phases. Due to the presence of a tricritical point the mean field approximation agrees very well with the numerical data. Critical exponents associated with second-order transitions coincide with those of the Z3 Potts model. Finally, we employ inverse Monte-Carlo methods to determine the effective couplings in order to match the effective models to Yang-Mills theory. arXiv:0708.4146v1 [hep-lat] pdf file (127kb) Georg Bergner, Tobias Kaestner, Sebastian Uhlmann and Andreas Wipf Low-dimensional Supersymmetric Lattice Models. We study and simulate N=2 supersymmetric Wess-Zumino models in one and two dimensions. For any choice of the lattice derivative, the theories can be made manifestly supersymmetric by adding appropriate improvement terms corresponding to discretizations of surface integrals. In one dimension, our simulations show that a model with the Wilson derivative and the Stratonovitch prescription for this discretization leads to far better results at finite lattice spacing than other models with Wilson fermions considered in the literature. In particular, we check that fermionic and bosonic masses coincide and the unbroken Ward identities are fulfilled to high accuracy. Equally good results for the effective masses can be obtained in a model with the SLAC derivative (even without improvement terms). In two dimensions we introduce a non-standard Wilson term in such a way that the discretization errors of the kinetic terms are only of order O(a4). Masses extracted from the corresponding manifestly supersymmetric model prove to approach their continuum values much quicker than those from a model containing the standard Wilson term. Again, a comparable enhancement can be achieved in a theory using the SLAC derivative. Annals Phys. 323 946-988 (2008) pdf file (774kb) Christian Wozar, Tobias Kaestner, Andreas Wipf and Tom Heinzl Inverse Monte-Carlo determination of effective lattice models for SU(3) Yang-Mills theory at finite temperature. This paper concludes our efforts in describing SU(3)-Yang-Mills theories at different couplings/temperatures in terms of effective Polyakov-loop models. The associated effective couplings are determined through an inverse Monte Carlo procedure based on novel Schwinger-Dyson equations that employ the symmetries of the Haar measure. Due to the first-order nature of the phase transition we encounter a fine-tuning problem in reproducing the correct behavior of the Polyakov-loop from the effective models. The problem remains under control as long as the number of effective couplings is sufficiently small. Phys. Rev. D 76 085004 (2007) pdf file (179kb) Franziska Synatschke, Christian Wozar and Andreas Wipf Spectral sums of the Dirac-Wilson Operator and their relation to the Polyakov loop. We investigate and compute spectral sums of the Wilson lattice Dirac operator for quenched SU(3) gauge theory. It is demonstrated that there exist sums which serve as order parameters for the confinement-deconfinement phase transition and get their main contribution from the IR end of the spectrum. They are approximately proportional to the Polyakov loop. In contrast to earlier studied spectral sums some of them are expected to have a well-defined continuum limit. Phys. Rev. D75 114003 (2007) pdf file (281kb) Sebastian Uhlmann, Reinhard Meinel and Andreas Wipf Ward Identities for Invariant Group Integrals. We derive two types of Ward identities for the generating functions for invariant integrals of monomials of the fundamental characters for arbitrary simple compact Lie groups. The results are applied to the groups SU(3), Spin(5) and G2 of rank 2 as well as SU(4). J. Phys. A 40 4367-4389 (2007) pdf file (330kb) Alexei Abrikosov and Andreas Wipf The integral form of APS boundary conditions in the Bag Model. We propose the integral form of Atiah-Patodi-Singer spectral boundary conditions (SBC) and find explicitly the integral projector onto SBC for the 3-dimensional spherical cavity. After discussion of a simple example we suggest that the relation between the projector and fermion propagator is universal and stays valid independently of the bag form and space dimension. J. Phys. A 40 5163-5172 (2007) pdf file (134kb) Andreas Wipf, Tobias Kaestner, Christian Wozar and Tom Heinzl Generalized Potts-Models and their Relevance for Gauge Theories We study the Polyakov loop dynamics originating from finite-temperature Yang-Mills theory. The effective actions contain center-symmetric terms involving powers of the Polyakov loop, each with its own coupling. For a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and anti-ferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents nu and gamma at the continuous transition between symmetric and anti-ferromagnetic phases are the same as for the 3-state spin Potts model. SIGMA 3 006 (2007) 14 pages pdf file (378kb) Andreas Kirchberg, Klaus Kirsten, Mariel Santangelo and Andreas Wipf Spectral asymmetry on the ball and asymptotics of the asymmetry kernel Let iD be the Dirac operator on a D=2d dimensional ball B with radius R. We calculate the spectral asymmetry eta(0,iD) for D=2 and D=4, when local chiral bag boundary conditions are imposed. With these boundary conditions, we also analyze the small-t asymptotics of the heat trace Tr(F P exp(-t P22)) where P is an operator of Dirac type and F is an auxiliary smooth smearing function. J. Phys. A 39 9573-9589 (2006) pdf file (178kb) Christian Wozar, Tobias Kaestner, Andreas Wipf, Tom Heinzl and Balazs Pozsgay Phase Structure of Z3-Polyakov-Loop Models We study effective lattice actions describing the Polyakov loop dynamics originating from finite-temperature Yang-Mills theory. Starting with a strong-coupling expansion the effective action is obtained as a series of Z3-invariant operators involving higher and higher powers of the Polyakov loop, each with its own coupling. Truncating to a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches concerning the phase structure of the theories. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and anti-ferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents (nu) and (gamma) at the continuous transition between symmetric and anti-ferromagnetic phases are the same as for the 3-state Potts model. Phys. Rev. D 74 114501 (2006) pdf file (714kb) Tom Heinzl, Ben Liesfeld, Kay-Uwe Amthor, Heinrich Schwoerer, Roland Sauerbrey and Andreas Wipf On the observation of vacuum birefringence We suggest an experiment to observe vacuum birefringence induced by intense laser fields. A high-intensity laser pulse is focused to ultra-relativistic intensity and polarizes the vacuum which then acts like a birefringent medium. The latter is probed by a linearly polarized x-ray pulse. We calculate the resulting ellipticity signal within strong-field QED assuming Gaussian beams. The laser technology required for detecting the signal will be available within the next three years. Optics Communications 267 318-321 (2006) pdf file (146kb) Andreas Wipf, Andreas Kirchberg and Dominique Laenge Algebraic Solution of the Supersymmetric Hydrogen Atom The N=2 supersymmetric extension of the Schrödinger-Hamiltonian with 1/r-potential in d dimension is constructed. The system admits a supersymmetrized Laplace-Runge-Lenz vector which extends the rotational SO(d) symmetry to a hidden SO(d+1) symmetry. It is used to determine the discrete eigenvalues with their degeneracies and the corresponding bound state wave functions. (Proceedings of the 4th International Symposium "Quantum Theory and Symmetries" in Varna, 2005) in Proceedings 'Quantum Theory and Symmetries', Heron Press (2006), pages 887-897 pdf-file (91kb) Tom Heinzl, Tobias Kaestner and Andreas Wipf Effective Lattice Actions for Finite-Temperature Yang-Mills Theory We determine effective lattice actions for the Polyakov loop using inverse Monte Carlo techniques. (Proceedings of the XCQC-Meeting in Swansea, 2005) hep-th/0504180 pdf-file (243kb) Andreas Wipf Non-Perturbative Methods in Supersymmetric Theories The aim of these notes is to provide a short introduction to supersymmetric theories: supersymmetric quantum mechanics, Wess-Zumino models and supersymmetric gauge theories. A particular emphasis is put on the underlying structures and non-perturbative effects in N=1, N=2 and N=4 Yang-Mills theories. (Extended version of lectures given at the TROISIEME CYCLE DE LA PHYSIQUE EN SUISSE ROMANDE) hep-th/0504180 pdf-file (394kb) Tom Heinzl, Tobias Kaestner and Andreas Wipf Effective Actions for the SU(2) Confinement--Deconfinement Phase Transition We compare different Polyakov loop actions yielding effective descriptions of finite--temperature SU(2) Yang--Mills theory on the lattice. The actions are motivated by a simultaneous strong--coupling and character expansion obeying center symmetry and include both Ising and Ginzburg--Landau type models. To keep things simple we limit ourselves to nearest--neighbor interactions. Some truncations involving the most relevant characters are studied within a novel mean--field approximation. Using inverse Monte--Carlo techniques based on exact geometrical Schwinger--Dyson equations we determine the effective couplings of the Polyakov loop actions. Monte--Carlo simulations of these actions reveal that the mean--field analysis is a fairly good guide to the physics involved. Our Polyakov loop actions reproduce standard Yang--Mills observables well up to limitations due to the nearest--neighbor approximation. Phys. Rev. D 72 065005 (2005) pdf file (362kb) Andreas Kirchberg, Dominique Laenge and Andreas Wipf From the Dirac Operator to Wess-Zumino Models on Spatial Lattices We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the nonlocal and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling. Annals Phys. 316 357-392 (2005) pdf file (169kb) Andreas Kirchberg, Dominique Laenge and Andreas Wipf Extended Supersymmetries and the Dirac Operator We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges that exist and restrictions on the geometry of the underlying spaces as well as the admissible gauge field configurations. From the superalgebra with two or more real supercharges we infer the existence of integrability conditions and obtain a corresponding superpotential. This potential can be used to deform the supercharges and to determine zero modes of the Dirac operator. The general results are applied to the Kähler spaces CPN. Annals Phys. 315 467-487 (2005) pdf file (112kb) Leander Dittmann, Tom Heinzl and Andreas Wipf An effective lattice theory for Polyakov loops We derive effective actions for SU(2) Polyakov loops using inverse Monte Carlo techniques. In a first approach, we determine the effective couplings by requiring that the effective ensemble reproduces the single--site distribution of the Polyakov loops. The latter is flat below the critical temperature implying that the (untraced) Polyakov loop is distributed uniformly over its target space, the SU(2) group manifold. This allows for an analytic determination of the Binder cumulant and the distribution of the mean--field, which turns out to be approximately Gaussian. In a second approach, we employ novel lattice Schwinger--Dyson equations which reflect the SU(2) x SU(2) invariance of the functional Haar measure. Expanding the effective action in terms of SU(2) group characters makes the numerics sufficiently stable so that we are able to extract a total number of 14 couplings. The resulting action is short--ranged and reproduces the Yang--Mills correlators very well. JHEP 0406:005,2004 pdf file (317kb) Leander Dittmann, Tom Heinzl and Andreas Wipf Effective sigma models and lattice Ward identities We perform a lattice analysis of the Faddeev--Niemi effective action conjectured to describe the low--energy sector of SU(2) Yang--Mills theory. To this end we generate an ensemble of unit vector fields (color spins') n from the Wilson action. The ensemble does not show long--range order but exhibits a mass gap of the order of 1 GeV. From the distribution of color spins we reconstruct approximate effective actions by means of exact lattice Schwinger--Dyson and Ward identities (inverse Monte Carlo'). We show that the generated ensemble cannot be recovered from a Faddeev--Niemi action, modified in a minimal way by adding an explicit symmetry--breaking term to avoid the appearance of Goldstone modes. JHEP 0212:014 (2002) pdf file (461kb) Andreas Kirchberg, Dominique Laenge, Pablo Pisani and Andreas Wipf Algebraic Solution of the Supersymmetric Hydrogen Atom in d Dimension In this paper the N=2 supersymmetric extension of the Schrödinger Hamiltonian with 1/r-potential in arbitrary space-dimensions is constructed. The supersymmetric hydrogen atom admits a conserved Laplace-Runge-Lenz vector which extends the rotational symmetry SO(d) to a hidden SO(d+1) symmetry. This symmetry of the system is used to determine the discrete eigenvalues with their degeneracies and the corresponding bound state wave functions. Annals Phys. 303 359-388 (2003) pdf file (150kb) Gabriela Beneventano, Mariel Santangelo and Andreas Wipf Spectral asymmetry for bag boundary conditions We give an expression, in terms of boundary spectral functions, for the spectral asymmetry of the Euclidean Dirac operator in two dimensions, when its domain is determined by local boundary conditions, and the manifold is of product type. As an application, we explicitly evaluate the asymmetry in the case of a finite-length cylinder, and check that the outcome is consistent with our general result. Finally, we study the asymmetry in a disk, which is a non-product case, and propose an interpretation. J. Phys. A 35 9343-9354 (2002) pdf file (81kb) Horacio Falomir, Pablo Pisani and Andreas Wipf Pole structure of the Hamiltonian zeta-function for a singular potential We study the pole structure of the zeta-function associated to the Hamiltonian H of a quantum mechanical particle living in the half-line R +, subject to the singular potential g/x 2 +x 2. We show that H admits nontrivial self-adjoint extensions (SAE) in a given range of values of the parameter g. The zeta-functions of these operators present poles which depend on g and, in general, do not coincide with half an integer (they can even be irrational). The corresponding residues depend on the SAE considered. J. Phys. A 35 5427-5444 (2002) pdf file (110kb) Miguel Aguado, Manuel Asorey and Andreas Wipf Nahm transform and moduli spaces of CPN-models on the torus There is a Nahm transform for two-dimensional gauge fields which establishes a one-to-one correspondence between the orbit space of U(N) gauge fields with topological charge k defined on a torus and that of U(k) gauge fields with charge N on the dual torus. The main result of this paper is to show that a similar duality transform cannot exist for CPN instantons. This fact establishes a significative difference between 4-D gauge theories and CPN models. The result follows from the global analysis of the moduli space of instantons based on a complete and explicit parametrization of all self-dual solutions on the two-dimensional torus. The boundary of the space of regular instantons is shown to coincide with the space of singular instantons. This identification provides a new approach to analyzing the role of overlapping instantons in the infrared sector of CPN sigma models. Annals Phys. 298 2-23 (2002) pdf file (330kb) Pierre van Baal and Andreas Wipf Classical gauge vacua as knots The four dimensional O(3) non-linear sigma model introduced by Faddeev and Niemi, with a Skyrme-like higher order term to stabilise static knot solutions classified by the Hopf invariant, can be rewritten in terms of the complex two-component CP variables. A further rewriting of these variables in terms of SU(2) curvature free gauge fields is performed. This leads us to interpret SU(2) pure gauge vacuum configurations, in a particular maximal abelian gauge, in terms of knots with the Hopf invariant equal to the winding number of the gauge configuration. Phys. Lett. B 515 181-184 (2001) pdf file (45kb) Falk Bruckmann, Tom Heinzl, Temo Vekua and Andreas Wipf Magnetic monopoles vs. Hopf defects in the Laplacian (Abelian) gauge We investigate the Laplacian Abelian gauge on the sphere S 4 in the background of a single t Hooft instanton. To this end we solve the eigenvalue problem of the covariant Laplace operator in the adjoint representation. The ground state wave function serves as an auxiliary Higgs field. We find that the ground state is always degenerate and has nodes. Upon diagonalisation, these zeros induce topological defects in the gauge potentials. The nature of the defects crucially depends on the order of the zeros. For first-order zeros one obtains magnetic monopoles. The generic defects, however, arise from zeros of second order and are pointlike. Their topological invariant is the Hopf index S 3 --> S 2. These findings are corroborated by an analysis of the Laplacian gauge in the fundamental representation where similar defects occur. Possible implications for the confinement scenario are discussed. Nucl. Phys. B 593 545-561 (2001) pdf file (111kb) Chris Ford, Jan Pawlowski, Torsten Tok and Andreas Wipf ADHM construction of instantons on the torus We apply the ADHM instanton construction to SU(2) gauge theory on T 2 n x R 4-n for n=1,2,3,4. To do this we regard instantons on T n x R 4-n as periodic (modulo gauge transformations) instantons on R 4 Since the R 4 topological charge of such instantons is infinite the ADHM algebra takes place on an infinite dimensional linear space. The ADHM matrix M is related to a Weyl operator (with a self-dual background) on the dual torus T' n. We construct the Weyl operator corresponding to the one-instantons on T n x R 4-n. In order to derive the self-dual potential on T n x R 4-n it is necessary to solve a specific Weyl equation. This is a variant of the Nahm transformation. In the case n=2 (i.e. T x R) we essentially have an Aharonov Bohm problem on T' 2 In the one-instanton sector we find that the scale parameter, lambda, is bounded above, lambda 2 V'<4 pi, V' being the volume of the dual torus T' 2. Nucl. Phys. B 596 387-414 (2001) pdf file (155kb) Mikhail Volkov and Andreas Wipf Black hole pair creation in de Sitter space: A complete one-loop analysis We present an exact one-loop calculation of the tunneling process in Euclidean quantum gravity describing creation of black hole pairs in a de Sitter universe. Such processes are mediated by S 2 x S 2 gravitational instantons giving an imaginary contribution to the partition function. The required energy is provided by the expansion of the universe. We utilize the thermal properties of de Sitter space to describe the process as the decay of a metastable thermal state. Within the Euclidean path integral approach to gravity, we explicitly determine the spectra of the fluctuation operators, exactly calculate the one-loop fluctuation determinants in the zeta-function regularization scheme, and check the agreement with the expected scaling behaviour. Our results show a constant volume density of created black holes at late times, and a very strong suppression of the nucleation rate for small values of Lambda. Nucl. Phys. B 582 313-362 (2000) pdf file (228kb) Siyavush Azakov, Hans Joos and Andreas Wipf Witten-Veneziano relation for the Schwinger model The Witten-Veneziano relation between the topological susceptibility of pure gauge theories without fermions and the main contribution of the complete theory and the corresponding formula of Seiler and Stamatescu with the so-called contact term are discussed for the Schwinger model on a circle. Using the (Euclidean) path integral and the canonical (Hamiltonian) approaches at finite temperatures we demonstrate that both formulae give the same result in the limit of infinite volume and (or) zero temperature. Phys. Lett. B 479 245-258 (2000) pdf file (62kb) Falk Bruckmann, Tom Heinzl, Andreas Wipf and Torsten Tok Instantons and Gribov copies in the maximally Abelian Gauge We calculate the Faddeev-Popov operator corresponding to the maximally Abelian gauge for gauge group SU(N). Specializing to SU(2) we look for explicit zero modes of this operator. Within an illuminating toy model (Yang-Mills mechanics) the problem can be completely solved and understood. In the field theory case we are able to find an analytic expression for a normalizable zero mode in the background of a single t Hooft instanton. Accordingly, such an instanton corresponds to a horizon configuration in the maximally Abelian gauge. Possible physical implications are discussed. Nucl. Phys. B 584 589-614 (2000) pdf file (161kb) Chris Ford, Torsten Tok and Andreas Wipf SU(N)-gauge theories in Polyakov gauge on the torus We investigate the Abelian projection with respect to the Polyakov loop operator for SU(N) gauge theories on the four torus. The gauge fixed A 0 is time-independent and diagonal. We construct fundamental domains for A 0. In sectors with non-vanishing instanton number such gauge fixings are always singular. The singularities define the positions of magnetically charged monopoles, strings or walls. These magnetic defects sit on the Gribov horizon and have quantized magnetic charges. We relate their magnetic charges to the instanton number. Phys. Lett. B 456 155-161 (1999) pdf file (79kb) Chris Ford, Torsten Tok and Andreas Wipf Abelian projection on the torus for general gauge groups We consider Yang-Mills theories with general gauge groups G and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where A 0 is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed A 0 are constructed for arbitrary gauge groups. In the sectors with non-vanishing instanton number such gauge fixings are necessarily singular. The singularities can be restricted to Dirac strings joining magnetically charged defects. The magnetic charges of these monopoles take their values in the co-root lattice of the gauge group. We relate the magnetic charges of the defects and the windings of suitable Higgs fields about these defects to the instanton number. Nucl. Phys. B 548 585-612 (1999) pdf file (167kb) Chris Ford, Ulrich Mitreuter, Torsten Tok, Andreas Wipf and Jan Pawlowski Monopoles, Polyakov loops, and gauge fixing on the torus We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where A 0 is independent of time and in the Cartan subalgebra. In the non-perturbative sectors such gauge fixings are necessarily singular. These singularities can be restricted to Dirac strings joining monopole and anti-monopole like `defects'. Annals Phys. 269 26-50 (1998) pdf file (144kb) Ulrich Mitreuter, Jan Pawlowski and Andreas Wipf Polyakov loops and fermionic zero-modes in QCD2 on the torus A direct derivation of the free energy and expectation values of Polyakov-loops in QCD 2 via path integral methods is given. The chosen gauge fixing has no Gribov-copies and has a natural extension to four dimensions.The Faddeev-Popov determinant and the integration over the space component of the gauge field cancel exactly. It only remains an integration over the zero components of the gauge field in the Cartan sub-algebra. This way the Polyakov-loop operators become Vertex-operators in a simple quantum mechanical model. The number of fermionic zero modes is related to the winding-numbers of A 0 in this gauge. Nucl. Phys. B 514 381-398 (1998) pdf file (112k) Stefan Durr and Andreas Wipf Finite temperature Schwinger model with chirality breaking boundary conditions The N f-flavour Schwinger Model on a finite space [ 0 , L ] and subject to bag-type boundary-conditions at x 1= 0 und x 1= L is solved at finite temperature T=1/β. The boundary conditions depend on a real parameter theta and break the axial flavour symmetry. We argue that this approach is more appropriate to study the broken phases than introducing small quark masses, since all calculations can be performed analytically. In the imaginary time formalism we determine the thermal correlators for the fermion-fields and the determinant of the Dirac-operator in arbitrary background gauge-fields. We show that the boundary conditions induce a CP-odd theta-term in the effective action. The chiral condensate, and in particular its T- and L- dependence, is calculated for N f fermions. It is seen to depend on the order in which the two lengths β=1/T and L are sent to infinity. Annals Phys. 255 333-361 (1997) pdf file (136kb) Ivo Sachs and Andreas Wipf Generalized Thirring models The Thirring model and various generalizations of it are analyzed in detail. The four-Fermi interaction modifies the equation of state. Chemical potentials and twisted boundary conditions both result in complex fermionic determinants which are analyzed. The non-minimal coupling to gravity does deform the conformal algebra which in particular contains the minimal models. We compute the central charges, conformal weights and finite size effects. For the gauged model we derive the partition functions and the explicit expression for the chiral condensate at finite temperature and curvature. The Bosonization in compact curved space-times is also investigated. Annals Phys. 249 380-429 (1996) pdf file (178kb) Sergei Odintsov and Andreas Wipf Running surface couplings We discuss the renormalization group improved effective action and running surface couplings in curved spacetime with boundary. Using scalar self-interacting theory as an example, we study the influence of boundary effects to effective equations of motion in spherical cap and the relevance of surface running couplings to quantum cosmology and symmetry breaking phenomenon. Running surface couplings in the asymptotically free SU(2) gauge theory are found. Phys. Lett. B 356 26-31 (1995) pdf file (54kb) Andreas Wipf and Stefan Durr Gauge theories in a bag We investigate multi-flavour gauge theories confined in d=2n-dimensional Euclidean bags. The boundary conditions for the 'quarks' break the axial flavour symmetry and depend on a parameter theta We determine the theta-dependence of the fermionic correlators and determinants and find that a CP-breaking theta-term is generated dynamically. As an application we calculate the chiral condensate in multi-flavour QED2 and the abelian projection of QCD2. In the second model a condensate is generated in the limit where the number of colours, Nc, tends to infinity. We prove that the condensate in QCD2 decreases with increasing bag radius R at least as R -1/NcNf. Finally we determine the correlators of mesonic currents in QCD2. Nucl. Phys. B 443 201-232 (1995) pdf file (149kb) Viatcheslav Mukhanov, Andreas Wipf and Andrei Zelnikov On 4-D Hawking Radiation From Effective Action We determine the s-waves contribution of a scalar field to the four dimensional effective action for arbitrary spherically symmetric external gravitational fields. The result is applied to 4d-black holes and it is shown that the energy momentum tensor derived from the (nonlocal) effective action contains the Hawking radiation. The luminosity is close to the expected one in the s-channel. The energy momentum tensor may be used as starting point to study the backreaction problem. Phys. Lett. B 332 283-291 (1994) pdf file (92kb) Viatcheslav Mukhanov and Andreas Wipf On the symmetries of Hamiltonian systems In this paper we show how the well-known local symmetries of Lagrangean systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian- and Lagrangean formalisms is found. The possible applications of our results are discussed. Int. J. Mod. Phys. A 10 579-610 (1995) pdf file (127kb) Ivo Sachs and Andreas Wipf Temperature and curvature dependence of the chiral symmetry breaking in 2-D gauge theories The partition function and the order parameter for the chiral symmetry breaking are computed for a family of 2-dimensional interacting theories containing the gauged Thirring model. In particular we derive non-perturbative expressions for the dependence of the chiral condensate on the temperature and the curvature. Both, high temperature and high curvature suppress the condensate exponentially and we can associate an effective temperature to the curvature. Phys. Lett. B 326 105-110 (1994) pdf file (68kb) Laszlo Feher, Lochlainn O'Raifeartaigh, Philippe Ruelle, Izumi Tsutsui and Andreas Wipf On Hamiltonian Reductions Of The Wess-Zumino-Novikov-Witten Theories The structure of Hamiltonian reductions of the Wess-Zumino-Novikov-Witten (WZNW) theory by first class Kac-Moody constraints is analyzed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability, conformal invariance and W-symmetry in the reduced theories. A Lagrangean, gauged WZNW implementation of the reduction is established in the general case and thereby the path integral as well as the BRST formalism are set up for studying the quantum version of the reduction. The general results are applied to a number of examples. In particular, a W-algebra is associated to each embedding of sl(2) into the simple Lie algebras by using purely first class constraints. The importance of these sl(2) systems is demonstrated by showing that they underlie the Wn l-algebras as well. New generalized Toda theories are found whose chiral algebras are the W-algebras belonging to the half-integral sl(2) embeddings, and the W-symmetry of the effective action of those generalized Toda theories associated with the integral gradings is exhibited explicitly. Phys. Rept. 222 1-64 (1992) pdf file (228kb) Ivo Sachs, Andreas Wipf and Arne Dettki Conformal And Thermodynamic Properties Of A Family Of Thirring Like Model We investigate Thirring-like models containing fermionic and scalar fields propagating in 2-dimensional space time. The corresponding conformal algebra is studied and we disprove a conjecture relating the finite size effects to the central charge. Some new results concerning the fermionic determinant on the torus with chirally twisted boundary conditions and a chemical potential are presented. In particular we show how the thermodynamics of the Thirring model depends on the current-current interaction. Phys. Lett. B 317 545-549 (1993) pdf file (66kb) Claus Kiefer and Andreas Wipf Functional Schrödinger equation for fermions in external gauge fields We discuss the functional Schrödinger picture for fermionic fields in external gauge fields for both stationary and time - dependent problems. We give formal results for the ground state and the solution of the time - dependent Schrödinger equation for QED in arbitrary dimensions, while more explicit results are obtained in two dimensions. For both the massless and massive Schwinger model we give an explicit expression for the ground state functional as well as for the expectation values of energy, electric and axial charge. We also give the corresponding results for non-abelian fields. We solve the functional Schrödinger equation for a constant external field in four dimensions and obtain the amount of particle creation. We solve the Schrödinger equation for arbitrary external fields for massless QED in two dimensions and make a careful discussion of the anomalous particle creation rate. Finally, we discuss some subtleties connected with the interpretation of the quantized Gauss constraint. Annals Phys. 236 241-285 (1994) pdf file (148kb) Christian Wiesendanger and Andreas Wipf Running Coupling Constants From Finite Size Effects The dependence of effective actions on the finite size of the space-time region M is investigated in detail. It is shown explicitly that the one-loop effective actions on M and lambda x M are the same if the volume and surface coupling constants and fields scale according to the renormalization flow. An efficient algorithm for calculating the beta-functions and anomalous dimensions is derived. The general results are applied to a number of examples, in particular scalar field theories in two, four and six dimensions, O(N)-sigma models in two dimensions and gauge field theories with fermions in two and four dimensions. Annals Phys. 233 125-161 (1994) pdf file (131kb) Lochlainn O'Raifeartaigh, Norbert Straumann and Andreas Wipf Aharonov-Bohm Effect In Presence Of Superconductors The analysis of a previous paper, in which it was shown that the energy for the Aharonov-Bohm effect could be traced to the interaction energy between the magnetic field of the electron and the background magnetic field, is extended to cover the case in which the magnetic field of the electron is shielded from the background magnetic field by superconducting material. The paradox that arises from the fact that such a shielding would apparently preclude the possibility of an interaction energy is resolved and, within the limits of the ideal situation considered, the observed experimental result is derived. Found. Phys. 23 703-709 (1992) pdf file (52kb) Lochlainn O'Raifeartaigh, Norbert Straumann and Andreas Wipf On The Origin Of The Aharonov-Bohm Effect It is now generally accepted that the Aharanov-Bohm effect originates in the interaction between an electron and an external gauge-potential A whose B-field vanishes locally. Here it is shown that the effect can equally well be regarded as originating in the interaction of the magnetic field of the electron with the distant B field. From this point of view the effect is seen to have a natural classical origin and loses much of its mystery. Comments Nucl. Part. Phys. 20 15-22 (1991) pdf file (461kb) pdf file (44kb) Ivo Sachs and Andreas Wipf Finite Temperature Schwinger Model The temperature dependence of the order parameter of the Schwinger model is calculated in the euclidean functional integral approach. For that we solve the model on a finite torus and let the spatial extension tend to infinity at the end of the computations. The induced actions, fermionic zero-modes, relevant Green functions and Wilson loop correlators on the torus are derived. We find the analytic form of the chiral condensate for any temperature and in particular show that it behaves like -2e -pi sqrt(pi)T/e for temperatures large compared to the induced photon mass. Helv. Phys. Acta 65 652-678 (1992) pdf file (225kb) Arne Dettki and Andreas Wipf Finite Size Effects From General Covariance And Weyl Anomaly By exploiting the diffeomorphism invariance we relate the finite size effects of massless theories to their Weyl anomaly. We show that the universal contributions to the finite size effects are determined by certain coefficient functions in the heat kernel expansion of the related wave operators. For massless scalars confined in a 4-dimensional curved spacetime with boundary the relevant coefficients are given confirming the results of Moss and Dowker and also of Branson and Gilkey. We apply the general results to theories on bounded regions in two- and four-dimensional flat space-times and determine the change of the effective action under arbitrary conformal deformations of the regions. Nucl. Phys. B 377 252-280 (1992) pdf file (108kb) Laszlo Feher, Lochlainn O'Raifeartaigh, Philippe Ruelle, Izumi Tsutsui and Andreas Wipf Generalized Toda Theories And W Algebras Associated With Integral Gradings A general class of conformal Toda theories associated with integral gradings of the simple Lie algebras is investigated. These generalized Toda theories are obtained by reducing the Wess-Zumino-Novikov-Witten (WZNW) theory by first class constraints, and thus they inherit extended conformal symmetry algebras, generalized W-algebras, and current dependent Kac-Moody (KM) symmetries from the WZNW theory, which are analysed in detail in a non-degenerate case. WE uncover the sl(2) structure underlying the generalized W-algebras, which allows for identifying the primary fields, and give a simple algorithm for implementing the W-symmetries by current dependent KM transformations, which can be used to compute the action of the W-algebra on any quantity. We establish how the Lax pair of Toda theory arises in the WZNW framework and show that a recent result of Mansfield and Spence, which interprets the W-symmetry of the Toda theory by means of non-Abelian form preserving gauge transformations of the Lax pair, arises immediately as a consequence of the KM interpretation. Annals Phys. 213 1-20 (1992) pdf file (88kb) Lochlainn O'Raifeartaigh, Philippe Ruelle, Izumi Tsutsui and Andreas Wipf W algebras for generalized Toda theories The generalized Toda theories obtained in a previous paper by the conformal reduction of WZNW theories possess a new class of W-algebras, namely the algebras of gauge-invariant polynomials of the reduced theories. An algorithm for the construction of base-elements for the W-algebras of all such generalized Toda theories is found, and the W-algebras for the maximal SL(N,R) generalized Toda theories are constructed explicitly, the primary field basis being identified. Commun. Math. Phys. 143 333-354 (1992) pdf file /93kb) Lochlainn O'Raifeartaigh and Andreas Wipf Conformally Reduced WZNW Theories And Two-Dimensional Gravity The WZNW theories (for a non-compact form of the gauge groups) are reduced to a series of integrable theories that interpolate between WZNW theories and the corresponding Toda theories. They describe a set of WZNW fields in interaction with each other and with a two-dimensional gravitational field. An algorithm for constructing the general solutions, and a formula that relates the Virasoro and Kac-Moody centres of the reduced theories is given, together with a (conformally non-invariant) extension of the reduction to obtain affine Toda theories. Phys. Lett. B 251 361-368 (1990) pdf file (74kb) Janos Balog, Laszlo Feher, Peter Forgacs, Lochlainn O'Raifeartaigh and Andreas Wipf Kac-Moody Realization Of W Algebra By realizing the W-algebras of Toda field-theories as the algebras of gauge-invariant polynomials of constrained Kac-Moody systems we obtain a simple algorithm for constructing W-algebras without computing the W-generators themselves. In particular this realization yields an identification of a primary field basis for all the W-algebras, quadratic bases for the A,B,C-algebras, and the relation of W-algebras to Casimir algebras. At the quantum level it yields the general formula for the Virasoro centre in terms of the KM- level. Phys. Lett. B 244 435-441 (1990) pdf file (66kb) Janos Balog, Laszlo Feher, Lochlainn O'Raifeartaigh, Peter Forgacs and Andreas Wipf Toda Theory And W Algebra From A Gauged WZNW Point Of View Annals Phys. 203 76-136 (1990) A new formulation of Toda theories is proposed by showing that they can be regarded as certain gauge Wess-Zumino-Novikov-Witten (WZNW) models. It is argued that the WZNW variables are the proper ones for Toda theory, since all the physically permitted Toda solutions are regular when expressed in these variables. A detailed study of classical Toda theories and their W-algebras is carried out from this unified WZNW point of view. WE construct a primary field basis for the W-algebra for any group, we obtain a new method for calculating the W-algebra and it action on the Toda fields by constructing its Kac-Moody implementation, and we analyse the relationship between W-algebras and Casimir algebras. The W-algebra of G 2 and the Casimir algebras for the classical groups are exhibited explicitly. Annals Phys. 203 76-136 (1990) pdf file (182kb) Peter Forgacs, Andreas Wipf, Janos Balog, Laszlo Feher and Lochlainn O'Raifeartaigh Liouville and Toda Theories as Conformally Reduced WZNW Theories It is shown that the Liouville theory can be regarded as an SL(2,R) Wess-Zumino-Novikov-Witten theory with conformal invariant constraints and that Polyakov's SL(2,R) Kac-Moody symmetries of induced two-dimensional gravity is just one side of the WZNW current algebra. Analogously, Toda field theories can be regarded as conformal-invariantly constrained WZNW theories for appropriate (maximally non-compact) groups. Phys. Lett. B 227 214-220 (1989) pdf file (69kb) Emil Mottola and Andreas Wipf Unsuppressed Fermion Number Violation at High Temperature: An O(3) Model The O(3) nonlinear sigma-model in 1+1 dimensions, modified by an explicit symmetry-breaking term, is presented as a model for baryon- and lepton-number violation in the standard electroweak theory. Although arguments bases on the Atiyah-Singer index theorem and instanton physics apply to the model, we show by explicit calculations that the rate of chiral fermion-number violation due to the axial anomaly is entirely unsuppressed at sufficiently high temperatures. Our results apply to unbroken gauge theories as well and may require reevaluation of the role of instantons in high-temperature QCD. Phys. Rev. D 39 588-602 (1989) pdf file (2698kb) Janos Balog, Lochlainn O'Raifeartaigh, Peter Forgacs and Andreas Wipf Consistency of String Propagation on Curved Spacetimes: An SU(1,1) based Counterexample String propagation on non-compact group manifolds is studies as an exactly solvable example of propagation on more general curved spacetimes. It is shown that for the only viable group SU(1,1) x G c string propagation is consistent classically but not quantum mechanically (unitarity is violated). This shows that conformal invariance of the corresponding sigma-model (vanishing of the beta-functions) is not sufficient to guarantee unitarity. Nucl. Phys. B 325 225-241 (1989) pdf file (528kb) Steven Blau, Matt Visser and Andreas Wipf Analytical Results for the Effective Action We study the coupling of scalar fields and Dirac spinors to classical background gauge potentials. For the zero-field and constant-field configurations we derive some analytic expressions for the one-loop effective action. We discuss both massless and massive scalars and spinors. These results extend toe work of Schwinger and others Int. Journ. Mod. Phys. A 6 5408-5434 (1992) pdf file (1062kb) Steven Blau, Matt Visser and Andreas Wipf Zeta Functions and the Casimir Energy We use zeta functions techniques to give a finite definition for the Casimir energy of an arbitrary ultrastatic spacetime with or without boundaries. We find that the Casimir energy is intimately related to, but not identical to, the one-loop effective energy. We show that in general the Casimir energy depends on a normalization scale. This phenomenon has relevance to applications of the Casimir energy in bag models of QCD Within the framework of Kaluza-Klein models we discuss the one-loop corrections to the induced cosmological and Newton constants in terms of a Casimir like effect.We can calculate the dependence of these constants on the radius of the compact dimensions, without having to resort to detailed calculations. Nucl. Phys. B 310 163-180 (1988) pdf file (98kb) Steven Blau, Matt Visser and Andreas Wipf Determinants of Conformal Wave Operators in Four Dimensions We consider conformally coupled wave operators in four dimensions. Such operators are associated with conformally coupled massless scalars, massless spin 1/2 particles and Abelian gauge bosons. We explicitly calculate the change in the determinant of these wave operators as a function of conformal deformations of the background metric. This variation is given in terms of a geometrical object, the second Seeley-de Witt coefficient. Phys. Lett. B 209 209-213 (1988) pdf file (252kb) Steven Blau, Matt Visser and Andreas Wipf Determinants, Dirac Operators and One-Loops Physics We consider the Dirac operator. Its determinant is examined and in two Euclidean dimensions is explicitly evaluated in terms of geometrical quantities. This leads us to consider a generalization of the Wess-Zumino action that is applicable to arbitrary genus. Our analysis is relevant to a number of interesting systems: Schwinger models on curved two-manifolds; string theories with world-sheet vectors; and as an exploration of possible directions in evaluating determinants in four dimensions. Int. J. Mod. Phys. A 4 1467-1484 (1989) pdf file (540 kb) Fred Cooper, Avinash Khare, Renato Musto and Andreas Wipf Supersymmetry and the Dirac Equation We discuss in detail two supersymmetries of the 4-dimensional Dirac operator D2 where D=d-ieA, namely the usual chiral supersymmetry and a separate complex supersymmetry. Using SUSY methods developed to categorize solvable potentials in 1-dimensional quantum mechanics we systematically study the cases where the spectrum, eigenfunctions, and S-matrix of D2 can be obtained analytically. We relate these solutions to the solutions of the ordinary massive Dirac equation in external gauge fields. We show that whenever a Schrödinger equation for a potential V(x) is exactly solvable, then there always exists a corresponding static scalar field phi(x) for which the Jackiw-Rebbi type (1+1)-dimensional Dirac equation is exactly solvable with V(x) and φ(x) being related by V(x)=φ2(x)+φ'(x). We also discuss and exploit the supersymmetry of the path integral representation for the fermion propagator in an external field. Annals of Phys. 187 1-28 (1988) pdf file (1113kb) Fred Cooper, Joseph Ginnochio and Andreas Wipf Supersymmetry, Operator Transformations and Exactly Solvable Potentials A large class of potentials can be solved by using supersymmetry and shape invariance. In this paper we apply operator transformations (f transformations) to these algebraically solvable problems to obtain a larger class of solvable potentials - the Natanzon class of potentials which are not shape invariant. The important condition (which is independent of supersymmetry) for finding new potentials from old ones using operator transformations is that the resulting Schödinger equation has a potential which does not depend on the state. As a special case of the f transformation we rederive the previously known connection between the 2d harmonic oscillator, the hydrogen atom and the Morse potential. We also discuss the lack of commutivity of SUSY and the f transformations. J. Phys. A 22 3707-3716 (1989) pdf file (438kb) Fred Cooper, Joseph Ginnochio and Andreas Wipf Derivation of the S-Matrix using Supersymmetry Using supersymmetry and shape invariance the reflection and transmission coefficients for a large class of solvable potentials can be obtained algebraically. Phys. Lett. A 129 147-147 (1988) pdf file (167kb) Lochlainn O'Raifeartaigh and Andreas Wipf WKB Properties of the Time Dependent Schrödinger System It is shown that the time-dependent WKB expansion highlights some of the hidden properties of the Schrödinger equation and forms a natural bridge between that equation and the functional integral formulation of quantum mechanics. In particular it is shown that the leading (zero- and first-order in h) terms in the WKB expansion are essentially classical, and the relationship of this result to the classical nature of the WKB partition function, and of the anomalies in quantum field theory, is discussed. Found. Phys. 18 307-329 (1988) pdf file (986kb) Peter Forgacs, Lochlainn O'Raifeartaigh and Andreas Wipf Scattering Theory, U(1)-Anomaly and Index Theorems for Compact and Non-Compact Manifolds The L2 index theorem on even dimensional non-compact manifolds is related to the corresponding APS result for compact manifolds with boundaries. We show that generally the two index theorems are slightly different. For the two-dimensional Dirac operator on the disk we formulate (modified) nonlocal boundary conditions such that the two index theorems coincide. Exploiting the supersymmetric structure of D2 operator we explicitly evaluate the supersymmetric partition function in this case. Nucl. Phys. B 293 559-592 (1987) pdf file (1190kb) Renato Musto, Lochlainn O'Raifeartaigh and Andreas Wipf The U(1)-Anomaly, the Non-Compact Index Theorem, and the (Supersymmetric) BA-Effect The fractional discrepancy between the global U(1) chiral anomaly (described by a flux-integral of gauge-fields and not necessarily an integer on non-compact, euclidean space-times) and the index of the Dirac operator D is shown to be just (δ+(0) - δ-(0))/π where δ+(0) and δ-(0) are the left- and right-handed zero energy phase shifts. Phys. Lett. B 175 433-438 (1986) pdf file (684kb) Yasushi Fujimoto, Hiroshi Yoneyama and Andreas Wipf Symmetry Restoration of Scalar Models at Finite Temperature The symmetry restoration of scalar models at finite temperature and in less than four dimensions is investigated. For that purpose a series of approximations to the constraint effective lattice-potential is introduced. The continuum limit of these mean-field-like potentials is discussed and it is shown that the symmetry is always restored at finite temperature. As an application we derive an estimate for the critical temperature. Phys. Rev. D 38 2625-2634 (1988) pdf file (1678kb) Yasushi Fujimoto, Hiroshi Yoneyama and Andreas Wipf Finite Temperature lambda φ 4-Theory in Two and Three Dimensions and Symmetry Restoration lambda φ4 theory is studied in 2 and 3 dimensions to examine the validity of the finite temperature perturbation theory. We find that in some cases it is good even at high temperature in contrast to the case in 4 dimensions. We also discuss the problem of symmetry restoration and show an example of symmetry restoration within a save perturbation at high temperature. Z. Phys. C 35 351-354 (1987) pdf file (246kb) Andreas Wipf Some Results on Magnetic Monopoles and Vacuum Decay PhD-Thesis 1. Introduction 2. Magnetic monopoles 3. Explicit monopole solutions 4. The WKB-exponent in field theory 5. The primed determinant Helv. Phys. Acta 58 531-596 (1985) pdf.gz file (32 Mb) Andreas Wipf Non-Relativistic Yang-Mills Particles in a Spherically Symmetric Monopole Field We discuss the equations of motion and the conservation laws for a non-relativistic isospin carrying particle in a spherically symmetric monopole field and for a vanishing Yukawa coupling. In the 't Hooft-Polyakov monopole field we find no classical counterpart of the Rubakov effect. In the Prasad-Sommerfield limit we can solve the equations of motion analytically. J. Phys. A 18 2379-2384 (1985) pdf file (226kb) Lochlainn O'Raifeartaigh, Andreas Wipf and Hiroshi Yoneyama The Constraint Effective Potential Because of the non-perturbative nature of the conventional effective potential Γ(Ω,φ) (for classical Higgs potentials and volume Omega) and because of the inconvenience of a Legendre transform for numerical computations, it is proposed to replace Γ(Ω,φ) by a "constraint" effective potential U(Ω,φ), which has a direct intuitive meaning, which is very convenient for lattice computations, and from which Γ(Ω,φ) can immediately be recovered (as the convex hull). In particular, Γ(∞,φ)=U(∞,φ). Various properties of U(Ω,φ), such as convexity properties, upper and lower bounds and volume dependence are established. It is computed directly for zero dimensions and by Monte Carlo simulations in one and four dimensions, with up to 160 and 84 lattice sites, respectively. Nucl. Phys. B 271 653-680 (1986) pdf file (1420kb) Andreas Wipf Tunnel Determinants Methods for computing the regularized determinants of fluctuation operators are being developed. The results follow from the fact that these determinants can be expressed by eigenmodes of the fluctuation operator. As an application the tunnel determinants of some one- and higher-dimensional models are computed. It is shown that every fluctuation operator defines a supersymmetric quantum mechanical system. Nucl. Phys. B 269 24-44 (1986) pdf file (703kb) Andreas Wipf Upper and Lower Bounds For The Bounce Action We show how to bound the WKB exponent in field theory from above and below. The results follow from an inf-max characterisation of the Euclidean action of the bounce solution. For the lower bounds our method involves only some simple inequalities while for the upper bounds it leds to a finite dimensional extremising problem. We apply this method to approximate the WKB exponent for the effective potential in 1-loop approximation which is used in the inflationary cosmological models. J. Phys. A 18 2521-2529 (1985) pdf file (310kb) Martin Schweizer, Norbert Straumann and Andreas Wipf Postnewtonian Generation Of Gravitational Waves In A Theory Of Gravity With Torsion We adapt the post-Newtonian gravitational-radiation methods developed within general relativity by Epstein and Wagoner to the gravitation theory with torsion, recently proposed by Hehl et al., and show that the two theories predict in this approximation the same gravitational radiation losses. Since they agree also on the first-Newtonian level, they are at the present time - observationally - indistinguishable. Gen. Rel. and Grav. 12 951-961 (1980) pdf file (446kb)