A new perturbative approximation applied to supersymmetric quantum field theory

We show that a recently proposed graphical perturbative calculational scheme in quantum field theory is consistent with global supersymmetry invariance. We examine a two-dimensional supersymmetric quantum field theory in which we do not know of any other means for doing analytical calculations. We illustrate the power of this new technique by computing the ground-state energy density E to second order in this new perturbation theory. We show that there is a beautiful and delicate cancellation between infinite classes of graphs which leads to the result that E=0.     

Chaotic Quantization of Classical Gauge Fields

We argue that the quantized non-Abelian gauge theory can be obtained as the infrared limit of the corresponding classical gauge theory in a higher dimension. We show how the transformation from classical to quantum field theory emerges, and calculate Planck's constant from quantities defined in the underlying classical gauge theory.     

Essay: Cosmic Censorship: The Role of Quantum Gravity

The cosmic censorship hypothesis introduced by Penrose thirty years ago is still one of the most important open questions in classical general relativity. In this essay we put forward the idea that cosmic censorhip is intrinsically a quantum gravity phenomena. To that end, we construct a gedanken experiment in which cosmic censorship is violated within the purely classical framework of general relativity. We prove, however, that quantum effects restore the validity of the conjecture. This suggests that classical general relativity is inconsistent and that cosmic censorship might be enforced only by a quantum theory of gravity.     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000     

Simplex equations and their solutions

We investigate ann-simplex generalization of the classical and quantum Yang-Baxter equation. For the case ofsl(2) we find the most general solution of the classicaln-simplex equation for alln. These classical solutions can be quantized (in the sense of quantum group theory) forn=2,3 and we exhibit a quantum solution to the tetrahedron equations (n=3). The classical nondegenerate solutions cannot be quantized forn=4.     

Aspects of supersymmetric quantum mechanics

We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E₀ = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P_c(y) is identical to |Ψ₀(y)|² in the quantum theory.     

Supersymmetry breaking for the observable sector in superstring models

We discuss the supersymmetry breaking in four-dimensional supergravity models from superstrings including perturbative or non-perturbative string σ-model corrections. It is found that various soft supersymmetry breaking terms can appear with values of the order ofm _3/2 even at the tree level of effective field theory and of string theory.     

Quantum virial coefficients of molecular nitrogen

We report virial coefficients up to third order in density for molecular nitrogen, investigating 103 temperatures in the range (15 K, 3000 K). All calculations are based on an ab initio-based potential taken from the literature. Path-integral Monte Carlo (PIMC) is applied to account for nuclear quantum effects, and these results are compared to a more approximate but faster semiclassical treatment. Additionally, we examine a PIMC approach that employs semiclassical beads for the path-integral images, but find that it offers marginal advantage. A recently developed orientation sampling algorithm is used in conjunction with Mayer sampling to compute precise virial coefficients. We find that, within the precision of our calculations of the second-order coefficient (B₂), semiclassical methods are adequate for temperatures greater than 250 K, and are needed to correct classical behaviour for temperatures as high as 800 K. For the third-order coefficient (B₃), the semiclassical methods are adequate above 150 K, and are required up to the highest temperature examined (3000 K) in order to correct the classical treatment within the precision of the calculations. However, three-body contributions to the potential are much more significant than nuclear quantum effects for the evaluation of B₃.     

Stochastic and quantum space-time metrics and the weak-field limit

In this review we present a simple method of introducing stochastic and quantum metrics into gravitational theory at short distances in terms of small fluctuations around a classical background space-time. We consider only residual effects due to the stochastic (or quantum) theory of gravity and use a perturbative stochastization (or quantization) method. By using the general covariance and correspondence principles, we reconstruct the theory of gravitational, mechanical, electromagnetic, and quantum mechanical processes and tensor algebra in the space-time with stochastic and quantum metrics. Some consequences of the theory are also considered, in particular, it indicates that the value of the fundamental lengthl lies in the interval 10^–23l10^–22 cm.     

Quantum Instantons with Classical Moduli Spaces

 We introduce a quantum Minkowski space-time based on the quantum group SU(2) _q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective space ℙ³. Received: 10 May 2002 / Accepted: 10 January 2003 Published online: 5 May 2003 Communicated by L. Takhtajan     

Low Temperature Phase Diagrams¶of Fermionic Lattice Systems

We consider fermionic lattice systems with Hamiltonian H=H ^{(0)}+λH _Q , where H ^{(0)} is diagonal in the occupation number basis, while H _Q is a suitable “quantum perturbation”. We assume that H ^{(0)} is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitations, while H _Q is a finite range or exponentially decaying Hamiltonian that can be written as a sum of even monomials in the fermionic creation and annihilation operators. Mapping the d dimensional quantum system onto a classical contour system on a d+1 dimensional lattice, we use standard Pirogov–Sinai theory to show that the low temperature phase diagram of the quantum system is a small perturbation of the zero temperature phase diagram of the classical system, provided λ is sufficiently small. Particular attention is paid to the sign problems arising from the fermionic nature of the quantum particles. As a simple application of our methods, we consider the Hubbard model with an additional nearest neighbor repulsion. For this model, we rigorously establish the existence of a paramagnetic phase with commensurate staggered charge order for the narrow band case at sufficiently low temperatures. Received: 23 December 1996/ Accepted: 7 April 1999     

Multi-loop Feynman integrals and conformal quantum mechanics

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless φ³ theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the φ³ scalar field theory are given by the Green function for the conformal quantum mechanics.     

FRW minisuperspace with local N=4 supersymmetry and self‐interacting scalar field

A supersymmetric FRW model with a scalar supermultiplet and generic superpotential is analysed from a quantum cosmological perspective. The corresponding Lorentz and supersymmetry constraints allow to establish a system of first order partial differential equations from which solutions can be obtained. We show that this is possible when the superpotential is expanded in powers of a parameter λ?1. At order λ⁰ we find the general class of solutions, which include in particular quantum states reported in the current literature. New solutions are partially obtained at order λ¹, where the dependence on the superpotential is manifest. These classes of solutions can be employed to find states for higher orders in λ. Our analysis further points to the following: (i) supersymmetric wave functions can only be found when the superpotential has either an exponential behaviour, an effective cosmological constant form or is zero; (ii) If the superpotential behaves differently during other periods, the wave function is trivial ( = 0, i.e., no supersymmetric states). We conclude this paper discussing how our FRW minisuperspace (with N = 4 supersymmetry and invariance under time‐reparametrization) can be relevant concerning the issue of supersymmetry breaking.     

Vaidya Space-Time in Black-Hole Evaporation

We take a boundary-value approach to quantum amplitudes arising in gravitational collapse to a black hole. Pose boundary data on initial and final space-like hypersurfaces Σ _F,I , separated at spatial infinity by a Lorentzian proper-time interval T. Quantum amplitudes are calculated following Feynman's approach; rotate: T→|T|exp (−iθ) into the complex, where 0< θ≤π/2, and solve the corresponding well-posed complex classical boundary-value problem. We compute the classical Lorentzian action S _class and corresponding semi-classical quantum amplitude, proportional to exp (iS _class). To recover the Lorentzian amplitude, take the limit θ→ 0₊ of the semi-classical amplitude. For the classical boundary-value problem with given perturbative boundary data, we compute an effective spherically-symmetric energy-momentum tensor 〉 T ^μν〈 _EFF , averaged over several wavelengths of the radiation, describing the averaged extra energy-momentum contribution in the Einstein field equations, due to the perturbations. This takes the form of a null fluid, describing the radiation (of quantum origin) streaming radially outwards. The classical space-time metric, in this region of the space time, is of Vaidya form, justifying the adiabatic radial mode equations, for spins s = 0 and s = 2.     

The Elliptic Genus and Hidden Symmetry

We study the elliptic genus (a partition function) in certain interacting, twist quantum field theories. Without twists, these theories have N=2 supersymmetry. The twists provide a regularization, and also partially break the supersymmetry. In spite of the regularization, one can establish a homotopy of the elliptic genus in a coupling parameter. Our construction relies on a priori estimates and other methods from constructive quantum field theory; this mathematical underpinning allows us to justify evaluating the elliptic genus at one endpoint of the homotopy. We obtain a version of Witten's proposed formula for the elliptic genus in terms of classical theta functions. As a consequence, the elliptic genus has a hidden SL(2, ℤ) symmetry characteristic of conformal theory, even though the underlying theory is not conformal. Received: 7 January 2000 / Accepted: 10 April 2000     

Classical and path-integral molecular-dynamics study on liquid water and ice melting using non-empirical TTM2.1-F model

The TTMF2.1-F model is a non-empirical intermolecular water potential parametrised from ab-initio calculations of the water dimer with a complete basis set limit including dispersion correction from second-order Moller-Plesset perturbation theory. In this work, using two-phase ice-water NVT molecular-dynamics (MD) simulations, we found the ice melting temperature using the TTM2.1-F potential is close to 273?K when the nuclear quantum effects (NQEs) were included using path-integral centroid MD. Detailed analysis of the radial distribution functions, angle distribution functions, and associated joint probability for both liquid water and the two-phase cases showed that the melting-point-temperature drop when using path-integral simulation is due to the weakening of hydrogen bonds vis-à-vis classically-propagated MD.     

A supersymmetric extension of the standard model with bilinear R-parity violation

The minimum supersymmetric standard model with bilinear R-parity violation is studied systematically. As we consider low-energy supersymmetry, we examine the structure of the bilinear R-parity violating model carefully. We analyze the mixing of, e.g., Higgs bosons with sleptons, neutralinos with neutrinos and charginos with charged leptons in the model. Possible and some important physics results, e.g. that the lightest Higgs may be heavier than the weak Z-boson at tree level, are obtained. The Feynman rules for the model are derived in the 't Hooft–Feynman gauge, which is convenient if perturbative calculations are needed beyond the tree level. Received: 26 April 1999 / Revised version: 28 June 1999 / Published online: 16 November 1999     

A new technique for the evaluation of Feynman diagrams in the high energy,large momentum transfer limit

The problem of finding scattering amplitudes in the high energy, large momentum transfer limit is reconsidered. We propose a novel technique of evaluating Feynman diagrams in this regime with higher order perturbative corrections in mind. The relation to other methods is discussed and the ladder diagrams in φ³ theory, recalculated up to sixth order, serve as examples.     

Classical supersymmetric matter and cosmology

We show that supersymmetry may play an important role in giving a reasonable explanation of relevant cosmological features. We work in a classicalN=1 supergravity scenario showing that supersymmetric matter leads, in the classical limit, to a fluid model acting as the source of the gravitational field. The equation of state arises, in a natural way, as a consequence of the field equations. We study conditions for the matter dominated era and the existence of an inflationary solution. Using upper bounds of several red-shift experimental results we are able to give an estimate of the corresponding Hubble constant.