Quantum field theory (QFT) provides us with one and almost only suitable language (or mathematical tool) for describing not only the motion and interaction of particles but also their “annihilation” and “creation” out of a field considered a priori in a sophisticated way, whose view seems to be suited for describing dislocations, as a particle or a string embedded within a crystalline ordered field. This chapter concisely overviews the method of QFT, emphasizing distinction from the quantum mechanics, conventionally used for a single and/or many particle problems, and its equivalence to the statistical mechanics. The alternative formalism based on Feynman path integral and its imaginary time representation are reviewed, as the foundation for our use in Chapter 10.

As we have seen in Chapter 3, much of the microscopic “specificities” are renormalized into a limited number of degrees of freedom at dislocation substructure scale (Scale A), especially into those with “cellular” morphology, essentially extending over 3D crystalline space. Therefore, as a critical step toward the successful multiscale plasticity, we are required to be ready to answer the following questions about the 3D cell structure; “why do they need the 3D ‘cellular’ morphology?,” “what is the substantial role, especially against the mechanical properties?,” why does the well-documented ‘universality’ manifested as a similitude law, hold?, and “how the microscopic degrees of freedom (information) are stored and when will they be released?” The first goal of this chapter is to derive an effective theory governing the dislocation substructure evolutions, particularly, cellular patterning, from a dislocation theory-based microscopic description of Hamiltonian through a rational “coarse-graining” procedure provided by the method of quantum field theory (QFT) (see Chapter 8). Secondly, after presenting some representative simulation results, an extensive series of discussions on the cell formation mechanisms and the mechanical roles are discussed and identified.

This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.