Abstract
Path integrals in quantum mechanics and statistical mechanics turn into high-dimensional integrals when one discretizes time. The same happens with functional integrals for quantum fields enclosed in a finite box when one discretizes space and time. Thus we are confronted with high-dimensional integrals in quantum statistics, solid-state physics, quantum field theory, high-energy physics and numerous other branches in natural sciences or even the financial market. Beginning with simple numerical integration algorithms we introduce the Monte Carlo integration method with and without important sampling and illustrate the method with several examples. In the last section we recall some useful basic facts about probability theory. At the end we included several short C-programs for calculating integrals with the help of numerical and Monte Carlo algorithms.
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Notes
- 1.
For a discussion and proof of this law see p. 40.
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Wipf, A. (2013). High-Dimensional Integrals. In: Statistical Approach to Quantum Field Theory. Lecture Notes in Physics, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33105-3_3
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DOI: https://doi.org/10.1007/978-3-642-33105-3_3
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