nLab
coherent state

Contents

Idea

Coherent states are members of special overcomplete families of elements (usually in the projectivization) of Hilbert space of states of certain systems in quantum mechanics. This entry is mainly about Perelomov style CS, see a separate page for Hall coherent states.

Examples

Of the harmonic oscillator

The most classical coherent states of E. Schrödinger (1926, coherent refers to its meaning in optics) are the elements in L 2( n)L^2(\mathbb{R}^n) which are the eigenfunctions z=z 1,,z n|z\rangle = |z_1,\ldots,z_n\rangle of the annihilation operators a ia_i, i=1,,ni=1,\ldots,n; the eigenvalues z iz_i are complex numbers and all complex numbers appear in that way; more precisely the coherent states are parametrized by numbers z=(z 1,,z n)z=(z_1,\ldots,z_n) in n\mathbb{C}^n, and there is a measure dμd\mu on n\mathbb{C}^n providing the resolution of unity formula

nzzdμ=1 \int_{\mathbb{C}^n} |z\rangle \langle z| d\mu = 1

Coherent states however are not mutually orthogonal; they form an overcomplete family of states: not only that every vector in L 2L^2 can be expressed in terms of them but rather in more than one way and we can throw certain coherent states out and not to spoil the property (how many and which is a more subtle question). Thus expressing states in terms of coherent states is sort of Fourier analysis; in fact this approach is taken in a variant called wavelet analysis. Coherent states are minimizing the uncertainty relations, closest to classical states in their behaviour; their probability density is Gaussian. Vectors in the Hilbert space can be represented in the coherent state representation: f=zzfdμ|f\rangle = \int |z\rangle\langle z|f\rangle d\mu; if ff is in L 2L^2 then zf\langle z|f\rangle is a holomorphic function and this passage is called the Bargmann-Segal transform (referring to Irving Segal); this way certain Hilbert space of holomorphic function appears, the Bargmann-Fock space. The evolution of zf\langle z|f\rangle formally satisfies the classical Newton equations of motion.

In general, projection operators zz|z\rangle \langle z| and the scalar measure dμd\mu do not separate, in fact instead of zzdμ|z\rangle \langle z| d\mu one considers more general projection measure (descriptively a “measure” with values in self-adjoint projection operators on a separable Hilbert space) on a more general geometric space XX replacing n\mathbb{C}^n which is normalized to 1 on the whole space. Symbollically we can still pretend that the projective measure is of the form μ(A)= Azzdμ\mu(A) = \int_A |z\rangle \langle z| d\mu where dμd\mu is an ordinary (scalar-valued) measure.

Over a flag manifold

The basic example of the space XX is the homogeneous space G /B=G/KG^{\mathbb {C}}/B = G/K where GG is a compact Lie group, G G^{\mathbb{C}} its complexification, KGK\subset G maximal compact subgroup and BG B\subset G^{\mathbb{C}} the Borel subgroup. Generalized flag manifold G/KG/K is naturally a compact complex manifold, in fact Kaehler. Fix a unitary character χ:GS 1\chi:G\to S^1 which uniquely extends to a character χ:G *\chi:G^{\mathbb{C}}\to \mathbb{C}^* of the complexification; let C χ\mathbf{C}_\chi be the corresponding 1-dimensional representation. Then let L χ=G × G χL_\chi = G^{\mathbb{C}}\times_{G^{\mathbb{C}}} \mathbb{C}_\chi be the associated line bundle. It is equipped with a natural holomorphic structure and hermitean scalar product; the space of holomorphic (or antiholomorphic depending on conventions in the construction) sections of that bundle is finite-dimensional and irreducible by Borel-Weil theorem. Perelomov coherent states on that bundle are the elements of the orbit of GG of the heighest (equivalently lowest) weight vector (or equivalently of G G^{\mathbb{C}}: the real and complex orbits are equal). The covariant generalized uncertainty relations (ref. by Onofri, below) correspond to the expression for a quantity proportional to negative of the square of the moment map (see ref. by Spera, below), which is naturally extremal on symplectic orbits. These coherent states can also be realized as the duals (according to Riesz theorem, that is dual vectors in a Hilbert space) to the evaluation functionals on the space of sections: take a point qq in the space of a line bundle and then divide it by the value of the section on the projection of qq to the homogeneous space. Up to a scalar, coherent states do not depend on the choice of the point on the line, hence this yields a holomorphic embedding of G /BG^{\mathbb{C}}/B to the projectivization of the representation space, so called coherent states embedding. The rays are the coherent states and the choices with scalar multiple accounted for are the coherent vectors. These coherent vectors are normalized by the measure which is the push down of the measure form GG to G/KG/K. The classical coherent states appear in noncompact case where the group in question is the Heisenberg group, and the covariant uncertainty relations are the usual ones.

References

Related entries: coherent state (in geometric quantization), Hall coherent state, Berezin quantization?

  • A. M. Perelomov, Coherent states for arbitrary Lie groups, Comm. Math. Phys., 26 (1972), pp. 222–236; archived as arXiv:math-ph/0203002.

  • A. Perelomov, Generalized coherent states and their applications, Springer 1986

  • J. H. Rawnsley, Coherent states and Kaehler manifolds, Quart. J. Math. Oxford (2), 28 (1977), pp. 403–415

  • V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Communications on Pure and Applied Mathematics 14 (1961) 187-214 MR0157250 doi

  • F. A. Berezin, Quantization, Math. USSR Izv., 8 (1974), 1109-1163. MR 0395610 (52:16404)

  • M. Spera, On a generalized uncertainty principle, coherent states and the moment map, J. of Geometry and Physics 12 (1993) 165-182.

  • Enrico Onofri, A note on coherent state representations of Lie groups, J. Math. Phys. 16:5, 1087-1089, may 1975.

  • for physical aspects wikipedia:Coherent_state.

  • V. V. Kisil, Integral representations and coherent states, Bulletin of the Belgian Mathematical Society, v. 2 (1995), No 5, pp. 529-540.

Coherent states can be generalized to noncommutative geometry, most notably for quantum groups:

Revised on November 12, 2012 23:40:46 by Zoran Škoda (193.51.104.23)