In condensed matter field theory, why do we use coherent states ? Why cant we do the quantization without general state (non - coherent) ? Please suggest some materials to read in this regard
Coherent states have some specific properties*) that simplify coherent-state path integrals for the partition function and various correlation functions, such as the n-particle Green function. These properties are discussed in Chapter 1 of the book Quantum Many-Particle Systems, by JW Negele, and H Orland (beware of the many typing errors in this otherwise excellent book). For orientation, you should consider a system of non-interacting bosons (say, a system of independent harmonic oscillators) and a system of non-interacting fermions (say, a system of uniform non-interacting electron gas) and seek to calculate the corresponding partition functions (the relevant calculations are presented in Chapter 2 of the above-mentioned book). In the coherent-state path-integral representation, in the former case one has to deal with ordinary variables, and in the latter with Grassmann variables whose algebraic properties are presented in the above-mentioned book; there are no (second-quantised) operators to be considered (this remains when one takes account of the interaction amongst the underlying particles).
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*) Such as that presented in Eq. (1.137) of the book by Negele and Orland.
An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short-time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many-body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first-neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function ibid
I liked the following book as an introduction to path integrals for closed systems:
Stoof, H. T., Gubbels, K. B., & Dickerscheid, D. B. (2009). Ultracold quantum fields (Vol. 1). Berlin: Springer.
The emphasis in the book is on the application of this formalism to systems in the context of ultracold atoms.
Good luck and I hope this is of some use for you.
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