Abstract
Scalar fields describe spinless particles, and from an algebraic point of view they are relatively simple. In the electroweak theory a doublet of scalar fields form the Higgs sector and interact with the fields of leptons, quarks and gauge bosons. Scalar fields play a pivotal role in many inflationary cosmological scenarios—they could be responsible for the anticipated exponential expansion of the early universe. This chapter is devoted to the quantization of (Euclidean) scalar field theory at zero and finite temperature. With heat kernel and zeta-function methods we calculate the thermodynamic potential of non-interacting spinless particles. In the second part we introduce and discuss the Schwinger function and effective potential to investigate the phases of interacting scalar field theory at zero and finite temperature. This includes an exhaustive discussion of the Legendre–Frenchel transformation. At the end we discretize the scalar field theory on a space-time lattice and calculate the propagator for non-interacting scalars on a hyper-cubic lattice.
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- 1.
This should not be confused with the n-point Schwinger functions at zero temperature.
- 2.
This means that its square is the identity, i.e. if a the Legendre transform takes f to g, then the Legendre transform of g will be f.
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Wipf, A. (2013). Scalar Fields at Zero and Finite Temperature. 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_5
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