Starting from a mathematical basis where one analyses and developing different techniques in how to solve and represent different kinds of integrals with diagrams. Representing the integrals as n-valent vertices and introducing propagators is a great tool that helps with the book-keeping of the solutions and will sometimes do the calculations redundant. The symmetries of diagrams are analysed and how one extracts the symmetry factors from looking at a diagram by using some fairly simple combinatorics and cleverness. Introducing the probability amplitude and do some analysis of the path integral formulation is the step into physics. Discussing experiments as the double-slit experiment and deriving the Schrödinger equation from the generating functional. Looking at diagrams in Quantum Mechanics and Quantum Field Theory will explore the use of the crucial understanding of our generators for the diagrams. This thesis makes use of the so called generating functional almost throughout and to connect the first discussed mathematics to real physical theories is the aim.
Starting from a mathematical basis where one analyses and developing different techniques in how to solve and represent different kinds of integrals with diagrams. Representing the integrals as n-valent vertices and introducing propagators is a great tool that helps with the book-keeping of the solutions and will sometimes do the calculations redundant. The symmetries of diagrams are analysed and how one extracts the symmetry factors from looking at a diagram by using some fairly simple combinatorics and cleverness. Introducing the probability amplitude and do some analysis of the path integral formulation is the step into physics. Discussing experiments as the double-slit experiment and deriving the Schrödinger equation from the generating functional. Looking at diagrams in Quantum Mechanics and Quantum Field Theory will explore the use of the crucial understanding of our generators for the diagrams. This thesis makes use of the so called generating functional almost throughout and to connect the first discussed mathematics to real physical theories is the aim.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
DiagrammaticsThe diagrammatic method is one of the most powerful tools of theoretical physics.It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory.The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams (graphs).Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs.There exist several kinds of diagrams.We will try to present them in a systematic way.In Sect.20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure.This formalism is used extensively in classical statistical physics.It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables.We use the term "Gaussian integration" in a rather broad sense.Beside commuting "bosonic" variables, we also consider anti-commuting "fermionic" variables, where we use the Berezin integral with respect to a Gaussian weight.Even in the case of commuting variables, the "Gaussian integral" is not necessarily meant in the sense of measure theory.It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight.However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.We distinguish two kinds of spaces on which we perform the integrals: real and complex.As in many other places in our work, we treat these two cases in parallel.Of course, the difference between the real (i.e.neutral), and the complex (i.e.charged) formalism is mainly that of a different notation.In particular, charged lines need to be equipped with an arrow, whereas neutral lines need not.The terminology that we use is inspired by quantum field theory.Therefore, the variables that enter the integral are associated with "particles"; they are divided into "bosons" and "fermions", each subdivided into "neutral" and "charged" particles.In the main part of the chapter we describe the diagram formalism used in quantum many-body physics and quantum field theory.As a preparation, we include a brief Sect.20.2 devoted to the basic terminology of perturbation theory
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.