Coherent states

Certain overcomplete families of vectors (states) in a Hilbert space carrying an irreducible representation of a Lie group. Let $ G $ be a Lie group and let $ T ( g) $, $ g \in G $, be an irreducible unitary representation acting in a Hilbert space $ H $. Take a fixed vector $ \psi _ {0} \in H $. Consider the family of all vectors $ \psi _ {g} = T ( g) \psi _ {0} $. This is a set of coherent vectors. Recall that two vectors (of norm 1) $ \psi _ {1} $ and $ \psi _ {2} $ of $ H $ define the same (quantum mechanical) state if they differ by a phase factor: $ \psi _ {2} = \mathop{\rm exp} ( ia) \psi _ {1} $. Let $ H $ be the subgroup of $ G $ consisting of all $ h $ such that $ T ( h) \psi _ {0} = \mathop{\rm exp} ( ia ( h)) \psi _ {0} $, the isotropy subgroup of the state $ \psi _ {0} $. Then the coherent states $ \psi _ {g} $ are parametrized by the homogeneous space $ G/H $.

The classical case, which goes back to E. Schrödinger and J. von Neumann, concerns the Heisenberg–Weyl group acting by the usual Schrödinger (or Fock) representation (cf. Fock space and Commutation and anti-commutation relationships, representation of) with for $ \psi _ {0} $ the Fock vacuum. In the case of one degree of freedom (the quantum harmonic oscillator) the Heisenberg–Weyl group can be written $ W _ {1} = \{ {( t, \alpha ) } : {t \in \mathbf R, \alpha \in \mathbf C } \} $, $ ( t, \alpha ) ( s, \beta ) = ( t + s + \mathop{\rm Im} ( \alpha \overline \beta \; ), \alpha + \beta ) $. In this case $ H $ is the normal subgroup $ \{ ( t, 0) \} $, so that the coherent states are parametrized by the plane $ \mathbf C $. In von Neumann's use of coherent states to investigate quantum measurements [a1] overcomplete subsystems of this system indexed by a lattice in $ \mathbf C $ were important. One of the earlier applications of coherent states [a2], [a3] was to the description of a coherent laser beam and that is where the name comes from. Since then coherent states have become an important tool in both mathematics (Lie group representations, special functions, theta-functions, reproducing-kernel Hilbert spaces) and in various branches of theoretical physics [a4], [a5].

References

[a1]	J. von Neumann, "Mathematische Grundlagen der Quantenmechanik" , Springer (1932) Zbl 58.0929.06
[a2]	R.J. Glauber, "The quantum theory of optical coherence" Phys. Rev. , 130 (1963) pp. 2529–2539
[a3]	R.J. Glauber, "Coherent and incoherent states of the radiation field" Phys. Rev. , 131 (1963) pp. 2766–2788
[a4]	A. Perelomov, "Generalized coherent states and their applications" , Springer (1986)
[a5]	J.R. Klander, B.-S. Skagerstam, "Coherent states. Applications in physics and mathematical physics" , World Sci. (1985)