Abstract
In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particle with integer spin and have symmetric wave functions; fermions are quantum particle with half-integer spin and are represented with anti-symmetric wave functions. The functional setting is given by symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references (Berezin in The Method of Second Quantization, 1966; Bratteli and Robinson in Operator Algebra and Quantum Statistical Mechanics II, 1981). Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions.
In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as ħ↘0 of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity.
As we have done for finite systems, we here use Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers (Ginibre and Velo in Commun. Math. Phys. 68:45–68, 1979; Ann. Phys. 128(2):243–285, 1980; Ann. Inst. Henri Poincaré, Phys. Théor. 33:363–394, 1980) Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations.
Finally, following the paper (Rodnianski and Schlein in Commun. Math. Phys. 291:31–61, 2009) one can show that, in the limit ħ↘0, the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical field equation (Hartree equation) with a uniform remainder estimates in time.
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- 1.
In physics ħ is a constant equal to 1.055×10−34 J s. As is usual in quantum mechanics we consider here ħ as an effective Planck constant obtained by scaling, for example \(\hbar\mapsto\frac{\hbar}{\sqrt{2m}}\), where m is the mass and ħ↘0 means m↗+∞.
- 2.
A c-number here is a family of time dependent complex numbers indexed by I. It be can identified with a vector in \(\mathfrak{h}\). In the language of quantum mechanics c-numbers are the opposite of \(\varGamma _{\hbar,t}^{(1)}\), operators in an Hilbert space.
- 3.
Notions of Borel summability have been defined before for complex valued series, extension to vector valued series is straightforward.
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Combescure, M., Robert, D. (2012). Bosonic Coherent States. In: Coherent States and Applications in Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0196-0_10
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DOI: https://doi.org/10.1007/978-94-007-0196-0_10
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