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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Quantum Evolution: From Particles To Fields

Quantum Evolution: From Particles To Fields

Chapter:
(p.110) 6 QUANTUM EVOLUTION: FROM PARTICLES TO FIELDS
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0006

To calculate scattering S-matrix elements, quantities relevant to Particle Physics, it is necessary to consider instead the quantum evolution operator in real time. This chapter begins by deriving the path integral representation of the evolution operator and the S-matrix in simple quantum mechanics. To illustrate the power of the formalism, it shows how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Section 5.1 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas the evolution operator is then given by a holomorphic functional integral. Using the parallel formalism of Section 5.6, the chapter derives an analogous representation for the evolution operator of a system of non-relativistic fermions. It then begins the study of relativistic quantum field theory with the example of the self-coupled neutral scalar boson. It shows that the holomorphic formalism, in a form that extends the construction of Section 5.5 to relativistic real time evolution, leads to various representations of the S-matrix in terms of functional integrals. The chapter relates S-matrix elements to the continuation to real time of various kinds of euclidean correlation functions.

Keywords:   S-matrix, quantum evolution operator, functional integrals, quantum field theory, holomorphic formalism, quantum mechanics

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