Wick rotation

Summary

In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. This transformation is also used to find solutions to problems in quantum mechanics and other areas. Overview Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature (−1, +1, +1, +1) convention) :ds^2 = -\left(dt^2\right) + dx^2 + dy^2 + dz^2 and the four-dimensional Euclidean metric :ds^2 = d\tau^2 + dx^2 + dy^2 + dz^2 are equivalent if one permits the coordinate t to take on imaginary values. The Minkowski metric becomes Euclidean when t is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates

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https://en.wikipedia.org/wiki/Wick_rotation

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The course builds on the two previous courses on the subject. The main subject is the study of quantum field theories at the loop level. The course introduces the concept of loop divergences and renormalization. Non abelian gauge theories are also discussed in depth.

Introduction to the path integral formulation of quantum mechanics. Derivation of the perturbation expansion of Green's functions in terms of Feynman diagrams. Several applications will be presented, including non-perturbative effects, such as tunneling and instantons.

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The present dissertation considers the capabilities, limitations and possible extensions of modelling the hysteresis that is exhibited by type-II superconductors, especially those with high critical temperature. Superconductors of type-II, including high temperature superconductors, are partially penetrated by magnetic flux. The tubes, through which the flux passes the superconductor, are 'pinned' to certain locations due to impurities in the crystal structure of the material, and they must be forced by an external magnetic field or a transport current in order to move. Thus, the pinning of flux tubes constitutes a memory that gives rise to a hysteresis with corresponding losses. The critical state model is a well-known, macroscopic model that describes well this partial flux penetration. Furthermore, the flux tubes can start to flow due to the Lorenz-force when a large transport current flows in the superconductor. This produces an additional resistive voltage. The concept of hysteresis and its main properties are discussed, and a number of various models describing this phenomenon are presented. An emphasis is made on the classical Preisach model of hysteresis, which is a weighted superposition of relay operators. A hysteretic system produces higher harmonics, just as most nonlinear systems. An investigation reveals under what conditions the Preisach model generates only odd harmonics, and also when all odd harmonics are present, as well as how this knowledge can be utilised. A parameterised Preisach model is proposed, which always applies when the critical state model is an acceptable approximation of the superconductor hysteresis. Its capabilities and limitations as a hysteresis model for superconductors are investigated. It is demonstrated how the parameters can be estimated from different electric measurements on high temperature superconductors and that the output and the losses can quickly and accurately be computed for an arbitrary signal, once the parameter identification is made. Moreover, the structure of the parameterised model is such that an inverse model can easily be obtained. It is also shown that the hysteresis saturation, which occurs when the flux flow starts, can be modelled by introducing limiting functions in the model. A generalised equivalent circuit has been proposed that takes into account both the hysteretic and resistive behaviour, the latter being due to flux flow. This extended model describes the global electric behaviour of a superconducting device, which can be applied either when the superconductor is part of a larger system or when it stands alone. A few examples of its application are given.

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