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Formal means

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, ... More

This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.Less

Gaussian expectation values. Steepest descent method

Jean Zinn-Justin

in Phase Transitions and Renormalization Group

This chapter presents a number of technical results concerning generating functions, Gaussian measures, and the steepest descent method. It begins by introducing the notion of generating function of ... More

This chapter presents a number of technical results concerning generating functions, Gaussian measures, and the steepest descent method. It begins by introducing the notion of generating function of the moments of a probability distribution. It then calculates Gaussian integrals and prove Wick′s theorem for Gaussian expectation values, a result that is simple but of major practical importance. The steepest descent method provides asymptotic evaluations, in some limits, of real or complex integrals. It leads to calculations of Gaussian expectation values, which explains its presence in this chapter. Moreover, the steepest descent method will be directly useful in this work. Exercises are provided at the end of the chapter.Less

Continuum limit and path integrals

Jean Zinn-Justin

in Phase Transitions and Renormalization Group

This chapter shows that, as in the case of the random walk, one can associate to the continuum limit a path integral, which generalizes the path integral of the Brownian motion. It first studies the ... More

This chapter shows that, as in the case of the random walk, one can associate to the continuum limit a path integral, which generalizes the path integral of the Brownian motion. It first studies the Gaussian example (which is simpler) and then the general case. Exercises are provided at the end of the chapter.Less

GAUSSIAN INTEGRALS

Jean Zinn-Justin

in Path Integrals in Quantum Mechanics

This chapter introduces, in the case of ordinary integrals, concepts and methods that can be generalized to path integrals. The first part is devoted to the calculation of ordinary Gaussian ... More

This chapter introduces, in the case of ordinary integrals, concepts and methods that can be generalized to path integrals. The first part is devoted to the calculation of ordinary Gaussian integrals, Gaussian expectation values, and the proof of the corresponding Wick's theorem. The notion of connected contributions is discussed, and it is shown that expectation values of monomials can conveniently be associated with graphs called Feynman diagrams. Expectation values corresponding to measures that deviate slightly from Gaussian measures can be reduced to sums of infinite series of Gaussian expectation values, a method known as perturbation theory. This chapter also contains a short presentation of the steepest descent method, a method that allows evaluating a class of integrals by reducing them, in some limit, to Gaussian integrals. To discuss properties of expectation values with respect to some measure or probability distribution, it is always convenient to introduce a generating function of the moments of the distribution.Less

PATH INTEGRALS: FERMIONS

Jean Zinn-Justin

in Path Integrals in Quantum Mechanics

The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric ... More

The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.Less

Gaussian integrals. Algebraic preliminaries

Jean Zinn-Justin

in Quantum Field Theory and Critical Phenomena: Fifth Edition

In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. ... More

In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. This physics textbook thus begins with a discussion of algebraic properties of Gaussian measures, and Gaussian expectation values for a finite number of variables. The important role of Gaussian measures is not unrelated to the central limit theorem of probabilities, although the interesting physics is generally hidden in essential deviations from Gaussian distributions. A few algebraic identities about Gaussian expectation values, in particular Wick's theorem are recalled. Integrals over some type of formally complex conjugate variables, directly relevant for boson systems are defined. Fermion systems require the introduction of Grassmann or exterior algebras, and the corresponding generalization of the notions of differentiation and integration. Both for complex and Grassmann integrals, Gaussian integrals, and Gaussian expectation values are calculated, and generalized Wick's theorems proven. The concepts of generating functions and Legendre transformation are recalled.Less

A formula for semisimple orbital integrals

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

This chapter states without proof the main result of this volume, which expresses the elliptic orbital integrals associated with the heat kernel as a Gaussian integral on t(γ‎). In addition, the ... More

This chapter states without proof the main result of this volume, which expresses the elliptic orbital integrals associated with the heat kernel as a Gaussian integral on t(γ‎). In addition, the chapter shows how to derive from this formula a corresponding formula for other kernels, which include the wave kernel. First, the chapter gives an explicit formula for the orbital integrals associated with the heat kernel of ℒAX in terms of a Gaussian integral on t(γ‎). From the formula for the heat kernel, this chapter derives a corresponding formula for the semisimple orbital integrals associated with the wave operator of ℒAX.Less

Quantum mechanics (QM): Path integrals in phase space

Jean Zinn-Justin

in Quantum Field Theory and Critical Phenomena: Fifth Edition

In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to ... More

In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.Less

The case where [k(γ),p0]=0

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

This chapter explicitly evaluates the Gaussian integral that appears in the right-hand side of a formula in Chapter 6 for the orbital integrals of the heat kernel, when γ‎ is nonelliptic and ... More

This chapter explicitly evaluates the Gaussian integral that appears in the right-hand side of a formula in Chapter 6 for the orbital integrals of the heat kernel, when γ‎ is nonelliptic and [t(γ‎),p₀] = 0. The computations here can be easily extended to the more general kernels considered in Chapter 6; the index formulas of Chapter 7 also play a key role here. This chapter begins by considering the case where G = K. It then computes explicitly the Gaussian integral when γ‎ is nonelliptic and [t(γ‎),p₀] = 0. Finally, the chapter works out the case where G = SL₂(R).Less

Fundamental Ideas

Nicholas Manton and Nicholas Mee

in The Physical World: An Inspirational Tour of Fundamental Physics

Chapter 1 offers a simple introduction to the use of variational principles in physics. This approach to physics plays a key role in the book. The chapter starts with a look at how we might minimize ... More

Chapter 1 offers a simple introduction to the use of variational principles in physics. This approach to physics plays a key role in the book. The chapter starts with a look at how we might minimize a journey by car, even if this means taking a longer route. Soap films are also discussed. It then turns to geometrical optics and uses Fermat’s principle to explain the reflection and refraction of light. There follows a discussion of the significance of variational principles throughout physics. The chapter also covers some introductory mathematical ideas and techniques that will be used in later chapters. These include the mathematical representation of space and time and the use of vectors; partial differentiation, which is necessary to express all the fundamental equations of physics; and Gaussian integrals, which arise in many physical contexts. These mathematical techniques are illustrated by their application to waves and radioactive decay.Less

Path integrals: I said to him, ‘You’re crazy’

Tom Lancaster and Stephen J. Blundell

in Quantum Field Theory for the Gifted Amateur

The chapter describes Feynman"s path integral approach and demonstrate how to evaluate certain Gaussian integrals, using the simple harmonic oscillator as an example.

Relativistic fermions: Introduction

Jean Zinn-Justin

in Quantum Field Theory and Critical Phenomena: Fifth Edition

Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the ... More

Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the statistical operator e<συπ>−βH</συπ> for the non-relativistic Fermi gas in the formalism of second quantization, and an expression for the evolution operator. Here, it is first recalled how relativistic fermions transform under the spin group. The free action for Dirac fermions is analysed, the relation between fields and particles explained, an expression for the scattering matrix obtained, and the non-relativistic limit of a model of self-coupled massive Dirac fermions derived. A formalism of Euclidean relativistic fermions is then introduced. In the Euclidean formalism: fermions transform under the fundamental representation of the spin group Spin(d) associated with the SO(d) rotation group (spin 1/2 fermions for d = 4). As for the scalar field theory, the Gaussian integral, which corresponds to a free field theory is calculated. Then the generating functional of correlation functions is obtained by adding a source term to the action. The field integral corresponding to a general action with an interaction expandable in powers of the field, can be expressed in terms of a series of Gaussian integrals, which can be calculated, for example, with the help of Wick's theorem. The connection between spin and statistics is verified by a simple perturbative calculation. The appendix describes a few additional properties of the spin group, the algebra of γ matrices, and the corresponding spinors for Euclidean fermions.Less