I am starting to studying condensed matter theory and I do not understand if Many-Body Quantum Mechanics and Quantum Field Theory are just synonyms or are two different methods.

It seems to me that the first is used when the starting point is a many-body wavefunction and then the occupation numbers formalism is adopted. In the second case, the starting point is a classical field which is quantized to describe some features of a material. However, in the end, it seems to me that the types of techniques are the same (Green's functions, Path integrals, Faymann diagrams and so on).

Are just two synonyms and QFT is just the cooler, newer name of the same topic or are they different methods?

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    $\begingroup$ Many-body theory and quantum field theory are just synonyms. They're both quantum mechanical theories of many particles. Any difference in connotation is just due to history. $\endgroup$ – knzhou Dec 21 '18 at 14:35
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    $\begingroup$ Usually the term "many body theory" is used for non-relativistic theories of many particles, while the term "QFT" is mostly used for relativistic quantum theories. The main difference is that in many body theories the total number of particles is conserved (at least in closed systems), and thus the tools of second quantozation, fock spaces, etc. are a mathematical convenience that allows to do more efficient computations. On the other hand, in relativistic theories the number of particles is variable due to the possibility of particle creation and annihilation. $\endgroup$ – yuggib Dec 22 '18 at 6:15
  • $\begingroup$ In this context the aforementioned tools are necessary to describe the theory in a proper way. $\endgroup$ – yuggib Dec 22 '18 at 6:16

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