Book contents
- Frontmatter
- Contents
- Preface
- 1 Quantum fields
- 2 Operators on the multi-particle state space
- 3 Quantum dynamics and Green's functions
- 4 Non-equilibrium theory
- 5 Real-time formalism
- 6 Linear response theory
- 7 Quantum kinetic equations
- 8 Non-equilibrium superconductivity
- 9 Diagrammatics and generating functionals
- 10 Effective action
- 11 Disordered conductors
- 12 Classical statistical dynamics
- Appendices
- Bibliography
- Index
10 - Effective action
Published online by Cambridge University Press: 24 December 2009
- Frontmatter
- Contents
- Preface
- 1 Quantum fields
- 2 Operators on the multi-particle state space
- 3 Quantum dynamics and Green's functions
- 4 Non-equilibrium theory
- 5 Real-time formalism
- 6 Linear response theory
- 7 Quantum kinetic equations
- 8 Non-equilibrium superconductivity
- 9 Diagrammatics and generating functionals
- 10 Effective action
- 11 Disordered conductors
- 12 Classical statistical dynamics
- Appendices
- Bibliography
- Index
Summary
In the previous chapter we introduced the one-particle irreducible effective action by collecting the one-particle irreducible vertex functions into a generator whose argument is the field, the one-state amplitude in the presence of the source. The effective action thus generates the one-particle irreducible amputated Green's functions. We shall now enhance the usability of the non-equilibrium effective action by establishing its relationship to the sum of all one-particle irreducible vacuum diagrams. To facilitate this it is convenient to add the final mathematical tool to the arsenal of functional methods, viz. functional integration or path integrals over field configurations. We are then following Feynman and instead of describing the field theory in terms of differential equations, we get its corresponding representation in terms of functional or path integrals. This analytical condensed technique shall prove powerful when unraveling the content of a field theory. The loop expansion of the non-equilibrium effective action is developed, and taken one step further as we introduce the two-particle irreducible effective action valid for non-equilibrium states. As an application of the effective action approach, we consider a dilute Bose gas and a trapped Bose–Einstein condensate.
Functional integration
Functional differentiation has its integral counterpart in functional integration. We shall construct an integration over functions and not just numbers as in elementary integration of a function. We approach this infinite-dimensional kind of integration with care (or, from a mathematical point of view, carelessly), i.e. we base it on our usual integration with respect to a single variable and take it to a limit.
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- Chapter
- Information
- Quantum Field Theory of Non-equilibrium States , pp. 313 - 372Publisher: Cambridge University PressPrint publication year: 2007