Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0026
- Subject:
- Physics, Nuclear and Plasma Physics
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
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.
Jean Zinn-Justin
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
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
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.
Jean Zinn-Justin
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only ...
More
A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only situations where the Hamiltonian eigenvalues are not degenerate are considered. The power of the path integral formalism is then illustrated beyond simple perturbation theory: it is used to derive a variational principle. It is evaluated, by applying the steepest descent method, in the case of O(N) symmetric Hamiltonians, in the large N limit. Steepest descent calculations of path integrals involve determinants of differential operators that are given a perturbative definition. This chapter shows how to calculate the spectrum of a Hamiltonian in the semi-classical limit or WKB approximation, starting from the semi-classical expansion of the partition function obtained earlier. It is demonstrated that the ground state of simple quantum systems is not degenerate.Less
A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only situations where the Hamiltonian eigenvalues are not degenerate are considered. The power of the path integral formalism is then illustrated beyond simple perturbation theory: it is used to derive a variational principle. It is evaluated, by applying the steepest descent method, in the case of O(N) symmetric Hamiltonians, in the large N limit. Steepest descent calculations of path integrals involve determinants of differential operators that are given a perturbative definition. This chapter shows how to calculate the spectrum of a Hamiltonian in the semi-classical limit or WKB approximation, starting from the semi-classical expansion of the partition function obtained earlier. It is demonstrated that the ground state of simple quantum systems is not degenerate.
JEAN ZINN-JUSTIN
- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent ...
More
This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent method, which reduces a certain type of integrals to gaussian expectation values. It then defines and discusses a few properties of dierentiation and integration in a Grassmann, that is, antisymmetric algebra, relevant for theories with fermion particles. In particular, gaussian integrals are calculated and general integrals are again reduced to gaussian expectation values. The chapter also recalls the concept of Legendre transformation, generating functional, functional dierentiation and the algebraic defiition of the determinant of an operator.Less
This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent method, which reduces a certain type of integrals to gaussian expectation values. It then defines and discusses a few properties of dierentiation and integration in a Grassmann, that is, antisymmetric algebra, relevant for theories with fermion particles. In particular, gaussian integrals are calculated and general integrals are again reduced to gaussian expectation values. The chapter also recalls the concept of Legendre transformation, generating functional, functional dierentiation and the algebraic defiition of the determinant of an operator.
Anatoly I. Ruban
- Published in print:
- 2015
- Published Online:
- October 2015
- ISBN:
- 9780199681747
- eISBN:
- 9780191761614
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199681747.003.0002
- Subject:
- Physics, Soft Matter / Biological Physics
Chapter 1 discusses the mathematical aspects of the asymptotic theory. It starts with basic definitions, using for this purpose the so-called coordinate asymptotic expansions. Then the chapter turns ...
More
Chapter 1 discusses the mathematical aspects of the asymptotic theory. It starts with basic definitions, using for this purpose the so-called coordinate asymptotic expansions. Then the chapter turns to asymptotic analysis of integrals and describe, among others, the method of steepest descent. However, the main attention is with parametric asymptotic expansions. These are used in a wide variety of fluid–dynamic problems, for which purpose a number of asymptotic techniques has been developed. We discuss in Chapter 1 the method of matched asymptotic expansions, the method of multiple scales, the method of strained coordinates, and the WKB method.Less
Chapter 1 discusses the mathematical aspects of the asymptotic theory. It starts with basic definitions, using for this purpose the so-called coordinate asymptotic expansions. Then the chapter turns to asymptotic analysis of integrals and describe, among others, the method of steepest descent. However, the main attention is with parametric asymptotic expansions. These are used in a wide variety of fluid–dynamic problems, for which purpose a number of asymptotic techniques has been developed. We discuss in Chapter 1 the method of matched asymptotic expansions, the method of multiple scales, the method of strained coordinates, and the WKB method.
