Abstract
The Lorentz and Poincaré groups. The equation of motion and energy-momentum tensor for a real scalar field. Domain walls in a model with spontaneously broken \(Z_2\) symmetry. The complex scalar field with U(1) symmetry and the Mexican hat potential. The Goldstone mode of the field. Global vortex and winding number.
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Notes
- 1.
In the case of a direct product, the multiplication rule would have the form \((\hat{L}_1, a_1) ( \hat{L}_2 , a_2) = (\hat{L}_1\hat{L}_2, a_1 +a_2)\).
- 2.
The term ‘potential’ is reserved for the sum \(m^2\phi ^2/2 +V(\phi )\).
- 3.
Often another term is used, namely the classical vacuum.
- 4.
Actually, the eigenvalue problem (3.40) is explicitly solved in textbooks on quantum mechanics. It turns out that apart from the zero mode there is one bound state with \(0< \kappa < |m^2|\) and a continuum of eigenfunctions with \(\kappa \ge |m^2|\).
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Arodź, H., Hadasz, L. (2017). Scalar Fields. In: Lectures on Classical and Quantum Theory of Fields. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-55619-2_3
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DOI: https://doi.org/10.1007/978-3-319-55619-2_3
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