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This chapter provides a detailed description of the numerical algorithms used to study open and closed quantum systems out of equilibrium. In a first step, the physical formulas to be evaluated numerically, and the mathematical framework for all these problems, are established. Quasi-exact simulations in the full Hilbert space for relatively small system sizes are discussed. Next, it is shown that in some cases it is possible to abandon the costly treatment of the full Hilbert space while identifying subspaces that can be handled numerically and still contain all of the essential physics. Related algorithms, which are essentially effective in one spatial dimension, are discussed under the heading of “matrix product state (MPS) simulations” (or “density-matrix renormalization group (DMRG) simulations”). It is shown that this strategy works very well on short timescales, but encounters fundamental issues of quantum physics as time evolves.

Let M denote a smooth manifold. A metric on TM can be used to define a notion of the distance between any two points in M and the distance travelled along any given path in M. This chapter first explains how this is done then considers the distance minimizing paths. The discussions cover Riemannian metrics and distance; length minimizing curves; the existence of geodesics; examples of metrics with their corresponding geodesics; geodesics on SO(n); geodesics on U(n) and SU(n); and geodesics and matrix groups.

This discussion of maps and vector bundles covers the pull-back construction; pull-backs and Grassmannians; pull-back of differential forms and push-forward of vector fields; invariant forms and vector fields on Lie groups; the exponential map on a matrix group; the exponential map and right/left invariance on Gl(n; ℂ) and its subgroups; and immersion, submersion, and transversality.

This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the vector space has a multiplication, for example, if it is a Lie algebra or a matrix group, the vector-valued forms will have a corresponding product. Vector-valued forms have become indispensable in differential geometry, since connections and curvature on a principal bundle are vector-valued forms. All the vector spaces will be real vector spaces. A k-covector on a vector space T is an alternating k-linear function. If V is another vector space, a V-valued k-covector on T is an alternating k-linear function.