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CRC Press eBooks

Further Development of Functional Integrals

John W. Negele, Henri Orland

Published Mar 5, 2018• 0 citations
Publisher

Abstract

This chapter addresses a range of problems for which functional integrals provide unique insights and approximations. It shows how to generate a variety of alternative functional integrals for the evolution operator which gives rise to different physical approximations in the stationary-phase approximation. The chapter aims to perform the general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental vibrational excitations of the system. It also shows how time-dependent mean field solutions may be used to calculate transition amplitudes between specific states, approximate quantum eigenstates of large amplitude collective motion, study barrier penetration and spontaneous fission. The chapter evaluates the asymptotic behavior of large orders of perturbation theory and explores the freedom available in formulating functional integrals for the evolution operator and how to exploit this freedom in making physical approximations.

DOI
10.1201/9780429497926-7

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This chapter addresses a range of problems for which functional integrals provide unique insights and approximations. It shows how to generate a variety of alternative functional integrals for the evolution operator which gives rise to different physical approximations in the stationary-phase approximation. The chapter aims to perform the general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental vibrational excitations of the system. It also shows how time-dependent mean field solutions may be used to calculate transition amplitudes between specific states, approximate quantum eigenstates of large amplitude collective motion, study barrier penetration and spontaneous fission. The chapter evaluates the asymptotic behavior of large orders of perturbation theory and explores the freedom available in formulating functional integrals for the evolution operator and how to exploit this freedom in making physical approximations.
A theory is presented in which approximations to quantum observables are obtained by applying the stationary-phase approximation to an exact functional integral representation of the many-body evolution operator. The requirement that the leading correction to the stationary-phase approximation vanish leads to a time-dependent Hartree-Fock mean field. Application of the theory to the Fourier transform of the trace of the evolution operator and the study of its poles yields quantized bound states with large amplitude. The theory is shown to reproduce the familiar static Hartree-Fock and random-phase approximations in the appropriate limits and yields an excellent approximation to the entire spectrum of the exactly solvable Lipkin model.NUCLEAR STRUCTURE Derivation of time-dependent mean-field theory from functional integral representation. Quantum theory of large amplitude collective motion. Application to Lipkin model.
Previously, we have seen how the field integral method can be deployed to formulate perturbative approximation schemes to explore weakly interacting theories. In this chapter, we will learn how elements of the perturbative approach can be formulated more efficiently by staying firmly within the framework of the field integral. More importantly, in doing so, we will see how the field integral provides a method for identifying and exploring non-trivial reference ground states – “mean-fields.” A fusion of perturbative and mean-field methods will provide us with analytical machinery powerful enough to address a spectrum of rich applications ranging from superfluidity and superconductivity to metallic magnetism and the interacting electron gas. As mentioned in Chapter 5, the perturbative machinery is but one part of a larger framework. In fact, the diagrammatic series already contained hints indicating that a straightforward expansion of a theory in the interaction operator might not always be an optimal strategy: all previous examples that contained a large parameter “ N ” – and usually it is only problems of this type that are amenable to controlled analytical analysis – shared the property that the diagrammatic analysis bore structures of “higher complexity.” (For example, series of polarization operators appeared rather than series of the elementary Green functions, etc.) This phenomenon suggests that large- N problems should qualify for a more efficient and, indeed, a more physical formulation.
Previously, we have seen how the field integral method can be deployed to formulate perturbative approximation schemes to explore weakly interacting theories. In this chapter, we will learn how elements of the perturbative approach can be formulated more efficiently by staying firmly within the framework of the field integral. More importantly, in doing so, we will see how the field integral provides a method for identifying and exploring non-trivial reference ground states – “mean-fields.” A fusion of perturbative and mean-field methods will provide us with analytical machinery powerful enough to address a spectrum of rich applications ranging from superfluidity and superconductivity to metallic magnetism and the interacting electron gas. As mentioned in Chapter 5, the perturbative machinery is but one part of a larger framework. In fact, the diagrammatic series already contained hints indicating that a straightforward expansion of a theory in the interaction operator might not always be an optimal strategy: all previous examples that contained a large parameter “ N ” – and usually it is only problems of this type that are amenable to controlled analytical analysis – shared the property that the diagrammatic analysis bore structures of “higher complexity.” (For example, series of polarization operators appeared rather than series of the elementary Green functions, etc.) This phenomenon suggests that large- N problems should qualify for a more efficient and, indeed, a more physical formulation. While these remarks appear to be largely methodological, the rationale behind searching for an improved theoretical formulation is, in fact, much deeper.
A nonperturbative method is developed for calculating the excitation of a many-body system by a time-dependent Hamiltonian. The stationary-phase approximation to a functional-integral representation of the interaction-picture many-body evolution operator results in a mean-field approximation to the SSS matrix which is asymptotically time independent. A one-body temporally nonlocal evolution equation defines the stationary mean-field configurations. The general method and character of the stationary solutions are illustrated by application to the forced harmonic oscillator and forced Lipkin model. Potential applications to realistic nuclear and atomic scattering situations are discussed.NUCLEAR REACTIONS Mean-field approximation, many-body SSS matrix, time-dependent Hartree Fock.
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