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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|Fock space]] and [[Commutation and anti-commutation relationships, representation of|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 [[#References|[a1]]] overcomplete subsystems of this system indexed by a lattice in  $  \mathbf C $
were important. One of the earlier applications of coherent states [[#References|[a2]]], [[#References|[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 [[#References|[a4]]], [[#References|[a5]]].

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