Introduction to the "second quantization" formalism for nonrelativistic quantum mechanics: A possible substitution for Sections 6.7 and 6.8 of Feynman's "Statistical Mechanics"
Abstract
This is a selfcontained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for nonrelativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of manyparticle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor. (Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general $N$ body wave functions, while Feynman makes a heavy use of Slaterdeterminanttype states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.10732
 Bibcode:
 2018arXiv181210732T
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Strongly Correlated Electrons;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 18 pages. This note is not for publications