Abstract
We present the formal procedure to quantize fields—the many-body problem—starting from the Lagrangian and Hamiltonian formulation of fields and performing the procedure of canonical quantization of systems with an infinite number of degrees of freedom. We apply this procedure to the significant examples of electrons (the Schrödinger field), plasmons, and phonons, emphasizing the analogy with systems of infinitely many quantum harmonic oscillators and also the difference between particles with integer spin and those with half-integer spin. The field associated with the former has to obey canonical commutation rules, leading to the symmetrized wavefunctions and to the Bose–Einstein statistics; on the contrary, the field associated with articles with half-integer spin must satisfy canonical anti-commutation rules, leading to antisymmetrized states and to Pauli’s exclusion principle. We also introduce the concept of Fock space. We also discuss briefly the quantization of the electromagnetic field (photons).
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Fischetti, M.V., Vandenberghe, W.G. (2016). Elementary Excitations in Solids. In: Advanced Physics of Electron Transport in Semiconductors and Nanostructures. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01101-1_9
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DOI: https://doi.org/10.1007/978-3-319-01101-1_9
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