*Jean Zinn-Justin*

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of ...
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Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths. Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties, or barrier penetration effects. Even though the formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulation is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. It allows simple construction of a many-body theory both for bosons and fermions.Less

Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths. Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties, or barrier penetration effects. Even though the formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulation is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. It allows simple construction of a many-body theory both for bosons and fermions.

*G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691142173
- eISBN:
- 9781400842650
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142173.003.0010
- Subject:
- Mathematics, Applied Mathematics

This chapter indicates how scattering theories can be developed when wave motions in chiral media are studied. To begin, this chapter remarks that a scattering process describes the effects of a ...
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This chapter indicates how scattering theories can be developed when wave motions in chiral media are studied. To begin, this chapter remarks that a scattering process describes the effects of a perturbation on a system about which everything is known in the absence of the perturbation. It then presents a general formulation of the class of problems thus described in terms of evolution operators, and outlines an approach to scattering theory in the time domain. The chapter also shows how the use of spectral theory allows the explicit construction of solutions to abstract initial boundary value problems in terms of generalised integral transforms, and how these generalised integral transforms can be used for the construction of the wave operators and the scattering operator. Furthermore, this chapter explores the extension of these ideas to the study of electromagnetics of complex media.Less

This chapter indicates how scattering theories can be developed when wave motions in chiral media are studied. To begin, this chapter remarks that a scattering process describes the effects of a perturbation on a system about which everything is known in the absence of the perturbation. It then presents a general formulation of the class of problems thus described in terms of evolution operators, and outlines an approach to scattering theory in the time domain. The chapter also shows how the use of spectral theory allows the explicit construction of solutions to abstract initial boundary value problems in terms of generalised integral transforms, and how these generalised integral transforms can be used for the construction of the wave operators and the scattering operator. Furthermore, this chapter explores the extension of these ideas to the study of electromagnetics of complex media.