We apply the linear delta expansion to the quantum mechanical version of the slow rollover transition which is the principal feature of inflationary models of the early Universe. The method, which goes beyond the Gaussian approximation, gives results which stay close to the exact solution for longer than previous methods. It provides a promising basis for extension to a full field theoretic treatment.
Abstract. Urine sediments of dogs with experimentally induced ethylene glycol poisoning were examined by light microscopy and X-ray diffraction. Massive calcium oxalate crystalluria was observed in all poisoned dogs. By light microscopy, the frequency with which six-sided hippurate-like prisms and envelope forms of calcium oxalate dihydrate occurred was approximately equal. The hippurate-like crystals were shown to be calcium oxalate monohydrate by X-ray diffractometry.Ethylene glycol (antifreeze) toxicosis is commonly observed in dogs and cats.23 Ethylene glycol is metabolized to glycoaldehyde, glycolic acid, glyoxylic acid, oxalic acid, and hippuric acid, which result in acidosis, cytotoxicosis, and subsequent renal failure. I The diagnosis often depends on laboratory findings such as metabolic acidosis, a large anion and osmolal gap, and ~rystalluria.~ The presence of calcium oxalate crystalluria is frequently cited as a useful diagnostic aid in ethylene glycol poisoning in people and animals, although crystals are not seen in every ~a s e .~,~~ The type of calcium oxalate crystal referred to most commonly in standard texts on urinalysis is the octahedral calcium oxalate dihydrate ("envelope" or "Maltese cross" form).2 Less frequently reported is the "hempseed"-shaped calcium oxalate monohydrate.".22 In addition, crystals very similar to published photographs and diagrams of six-sided prismatic hippurate crystals have been reported in p e~p l e ' .~.~. '~,~' and a r~i m a l s . '~.~~ In a retrospective study of ethylene glycol toxicosis in 50 dogs and cats, calcium oxalate dihydrate crystals (envelope forms) were observed in the urine of only 18% of the dogs and cats. Hippurate-like crystalluria was more frequently observed (44% of dogs and 18% of cats).23 Although hippuric acid is excreted in the urine of ethylene glycol-poisoned dogs," it is thought to be the least significant metabolite q~antitatively,'~ and probably could not account for the massive hippurate-like crystalluria which is commonly observed. In order to accurately identify these hippurate-like crystals, urine 6 sediments of eight dogs with experimentally induced ethylene glycol poisoning were examined by light microscopy and X-ray diffraction. X-ray diffraction was used because of its ability to absolutely identify crystals with a certainty, since the X-ray diffraction pattern of a compound is unique. lo, l9 X-ray diffraction involves the passage of a monochromatic beam of X-rays through the crystalline substance. Diffraction of the Xray beam by the planes of atoms is observed as a line or peak. The angular spacings of the lines are then used to calculate the distance (d-spacing) between the planes of atoms in the crystal. To identify crystals, the unknown pattern (d-spacing) is then compared with standard X-ray patterns of known substances." Materials and MethodsEight mature, healthy mixed-breed dogs were fasted for 24 hours, pretreated with chlorpromazine ( 1 mg/kg body weight) to prevent vomiting, and fed 10 ml/kg body weight of 95% ethyle...
We use an optimised hopping parameter expansion for the free energy (linear δ-expansion) to study the phase transitions at finite temperature and finite charge density in a global U(1) scalar Higgs sector on the lattice at large lattice couplings. We are able to plot out phase diagrams in lattice parameter space and find that the standard second-order phase transition with temperature at zero chemical potential becomes first order as the chemical potential increases. * email: T.Evans@ic.ac.uk , WWW: http://theory.ic.ac.uk/~time † Having ruled out a perturbative approach we turn to the various available nonperturbative techniques. One's first thought is to use Monte Carlo (MC) techniques to tackle the problem. However, MC methods usually fail when considering models at finite densities. This is because the Euclidean action, which is used as a statistical weight for the system, becomes complex, making a simple statistical integration technique imposible.To replace the MC approach we need an analytical non-perturbative method. The method chosen in this paper is an example of a general family sometimes called linear δ-expansions [5]. Examples of these methods have appeared under many names, including optimised pertubation theory[4], action-variational approach [17], variational perturbation theory [21], method of self-similar approximations [22], screened perturbation theory [20],and the variational cumulant expansion [23]. The method has been applied successfully to 1 the evaluation of simple integrals [5,14,15], solving non-linear differential equations [16], quantum mechanics [5,13,24,25,26], cosmological slow roll transitions [7] and to quantum field theory, both in the continuum [26,21] and on the lattice [5,8,9,10,11,12,17,18,19,23,28,29]. Since the LDE approach is analytical, we do not have to worry about the presence of a complex action. The expectation value of all physical observables will turn out to be real.The work in this paper with the U(1) or O(2) model builds on that set out in [4] for the case of zero temperature and zero chemical potential. However, here we phrase the model in terms of the field and its conjugate (Φ, Φ * ) rather than working with the real components of the field. This change is made because the charge operator is diagonal in this representation and so much easier to deal with [32].This paper is also complementary to the work done on the U(1) model using LDE methods in the continuum at finite temperature and finite chemical potential [6]. The Linear Delta expansionThe general format of the LDE method is to take a given expansion, whether it be a perturbative expansion in the continuum or a cumulant expansion on the lattice, and to provide an order by order optimisation of this expansion. It is in this process of optimisation that non-perturbative information emerges. It is straightforward, in principle, to expand beyond leading order, unlike other non-perturbative methods like large-N expansions or mean-field approximations.The first step in the method is to replace the physical ac...
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