Coherent states in mathematical physics

Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also[1]). However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.[2][3][4]

A general definition

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Let   be a complex, separable Hilbert space,   a locally compact space and   a measure on  . For each   in  , denote   a vector in  . Assume that this set of vectors possesses the following properties:

  1. The mapping   is weakly continuous, i.e., for each vector   in  , the function   is continuous (in the topology of  ).
  2. The resolution of the identity   holds in the weak sense on the Hilbert space  , i.e., for any two vectors   in  , the following equality holds:  

A set of vectors   satisfying the two properties above is called a family of generalized coherent states. In order to recover the previous definition (given in the article Coherent state) of canonical or standard coherent states (CCS), it suffices to take  , the complex plane and  

Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors   simply forming a total set[clarification needed] in   and the functions  , as   runs through  , forming a reproducing kernel Hilbert space. The objective in both cases is to ensure that an arbitrary vector   be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies that   where  .

These vectors   are square integrable, continuous functions on   and satisfy the reproducing property   where   is the reproducing kernel, which satisfies the following properties

 

Some examples

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We present in this section some of the more commonly used types of coherent states, as illustrations of the general structure given above.

Nonlinear coherent states

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A large class of generalizations of the CCS is obtained by a simple modification of their analytic structure. Let   be an infinite sequence of positive numbers ( ). Define   and by convention set  . In the same Fock space in which the CCS were described, we now define the related deformed or nonlinear coherent states by the expansion

  The normalization factor   is chosen so that  . These generalized coherent states are overcomplete in the Fock space and satisfy a resolution of the identity  

  being an open disc in the complex plane of radius  , the radius of convergence of the series   (in the case of the CCS,  .) The measure   is generically of the form   (for  ), where   is related to the   through the moment condition.

Once again, we see that for an arbitrary vector   in the Fock space, the function   is of the form  , where   is an analytic function on the domain  . The reproducing kernel associated to these coherent states is  

Barut–Girardello coherent states

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By analogy with the CCS case, one can define a generalized annihilation operator   by its action on the vectors  ,   and its adjoint operator  . These act on the Fock states   as   Depending on the exact values of the quantities  , these two operators, together with the identity   and all their commutators, could generate a wide range of algebras including various types of deformed quantum algebras. The term 'nonlinear', as often applied to these generalized coherent states, comes again from quantum optics where many such families of states are used in studying the interaction between the radiation field and atoms, where the strength of the interaction itself depends on the frequency of radiation. Of course, these coherent states will not in general have either the group theoretical or the minimal uncertainty properties of the CCS (they might have more general ones).

Operators   and   of the general type defined above are also known as ladder operators . When such operators appear as generators of representations of Lie algebras, the eigenvectors of   are usually called Barut–Girardello coherent states.[5] A typical example is obtained from the representations of the Lie algebra of SU(1,1) on the Fock space.

Gazeau–Klauder coherent states

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A non-analytic extension of the above expression of the non-linear coherent states is often used to define generalized coherent states associated to physical Hamiltonians having pure point spectra. These coherent states, known as Gazeau–Klauder coherent states, are labelled by action-angle variables.[6] Suppose that we are given the physical Hamiltonian  , with  , i.e., it has the energy eigenvalues   and eigenvectors  , which we assume to form an orthonormal basis for the Hilbert space of states  . Let us write the eigenvalues as   by introducing a sequence of dimensionless quantities   ordered as:  . Then, for all   and  , the Gazeau–Klauder coherent states are defined as

  where again   is a normalization factor, which turns out to be dependent on   only. These coherent states satisfy the temporal stability condition,

  and the action identity,   While these generalized coherent states do form an overcomplete set in  , the resolution of the identity is generally not given by an integral relation as above, but instead by an integral in Bohr's sense, like it is in use in the theory of almost periodic functions.

Actually the construction of Gazeau–Klauder CS can be extended to vector CS and to Hamiltonians with degenerate spectra, as shown by Ali and Bagarello.[7]

Heat kernel coherent states

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Another type of coherent state arises when considering a particle whose configuration space is the group manifold of a compact Lie group K. Hall introduced coherent states in which the usual Gaussian on Euclidean space is replaced by the heat kernel on K.[8] The parameter space for the coherent states is the "complexification" of K; e.g., if K is SU(n) then the complexification is SL(n,C). These coherent states have a resolution of the identity that leads to a Segal-Bargmann space over the complexification. Hall's results were extended to compact symmetric spaces, including spheres, by Stenzel.[9][10] The heat kernel coherent states, in the case  , have been applied in the theory of quantum gravity by Thiemann and his collaborators.[11] Although there are two different Lie groups involved in the construction, the heat kernel coherent states are not of Perelomov type.

The group-theoretical approach

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Gilmore and Perelomov, independently, realized that the construction of coherent states may sometimes be viewed as a group theoretical problem.[12][13][14][15][16][17]

In order to see this, let us go back for a while to the case of CCS. There, indeed, the displacement operator   is nothing but the representative in Fock space of an element of the Heisenberg group (also called the Weyl–Heisenberg group), whose Lie algebra is generated by   and  . However, before going on with the CCS, take first the general case.

Let   be a locally compact group and suppose that it has a continuous, irreducible representation   on a Hilbert space   by unitary operators  . This representation is called square integrable if there exists a non-zero vector   in   for which the integral   converges. Here   is the left invariant Haar measure on  . A vector   for which   is said to be admissible, and it can be shown that the existence of one such vector guarantees the existence of an entire dense set of such vectors in  . Moreover, if the group   is unimodular, i.e., if the left and the right invariant measures coincide, then the existence of one admissible vector implies that every vector in   is admissible. Given a square integrable representation   and an admissible vector  , let us define the vectors

  These vectors are the analogues of the canonical coherent states, written there in terms of the representation of the Heisenberg group (however, see the section on Gilmore-Perelomov CS, below). Next, it can be shown that the resolution of the identity   holds on  . Thus, the vectors   constitute a family of generalized coherent states. The functions   for all vectors   in   are square integrable with respect to the measure   and the set of such functions, which in fact are continuous in the topology of  , forms a closed subspace of  . Furthermore, the mapping   is a linear isometry between   and   and under this isometry the representation   gets mapped to a subrepresentation of the left regular representation of   on  .

An example: wavelets

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A typical example of the above construction is provided by the affine group of the line,  . This is the group of all 2×2 matrices of the type,     and   being real numbers with  . We shall also write  , with the action on   given by  . This group is non-unimodular, with the left invariant measure being given by   (the right invariant measure being  ). The affine group has a unitary irreducible representation on the Hilbert space  . Vectors in   are measurable functions   of the real variable   and the (unitary) operators   of this representation act on them as   If   is a function in   such that its Fourier transform   satisfies the (admissibility) condition   then it can be shown to be an admissible vector, i.e.,   Thus, following the general construction outlined above, the vectors   define a family of generalized coherent states and one has the resolution of the identity   on  . In the signal analysis literature, a vector satisfying the admissibility condition above is called a mother wavelet and the generalized coherent states   are called wavelets. Signals are then identified with vectors   in   and the function   is called the continuous wavelet transform of the signal  .[18][19]

This concept can be extended to two dimensions, the group   being replaced by the so-called similitude group of the plane, which consists of plane translations, rotations and global dilations. The resulting 2D wavelets, and some generalizations of them, are widely used in image processing.[20]

Gilmore–Perelomov coherent states

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The construction of coherent states using group representations described above is not sufficient. Already it cannot yield the CCS, since these are not indexed by the elements of the Heisenberg group, but rather by points of the quotient of the latter by its center, that quotient being precisely  . The key observation is that the center of the Heisenberg group leaves the vacuum vector   invariant, up to a phase. Generalizing this idea, Gilmore and Perelomov[12][13][14][15] consider a locally compact group   and a unitary irreducible representation   of   on the Hilbert space  , not necessarily square integrable. Fix a vector   in  , of unit norm, and denote by   the subgroup of   consisting of all elements   that leave it invariant up to a phase, that is,   where   is a real-valued function of  . Let   be the left coset space and   an arbitrary element in  . Choosing a coset representative  , for each coset  , we define the vectors   The dependence of these vectors on the specific choice of the coset representative   is only through a phase. Indeed, if instead of  , we took a different representative   for the same coset  , then since   for some  , we would have  . Hence, quantum mechanically, both   and   represent the same physical state and in particular, the projection operator   depends only on the coset. Vectors   defined in this way are called Gilmore–Perelomov coherent states. Since   is assumed to be irreducible, the set of all these vectors as   runs through   is dense in  . In this definition of generalized coherent states, no resolution of the identity is postulated. However, if   carries an invariant measure, under the natural action of  , and if the formal operator   defined as   is bounded, then it is necessarily a multiple of the identity and a resolution of the identity is again retrieved.

Gilmore–Perelomov coherent states have been generalized to quantum groups, but for this we refer to the literature.[21][22][23][24][25][26]

Further generalization: Coherent states on coset spaces

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The Perelomov construction can be used to define coherent states for any locally compact group. On the other hand, particularly in case of failure of the Gilmore–Perelomov construction, there exist other constructions of generalized coherent states, using group representations, which generalize the notion of square integrability to homogeneous spaces of the group.[2][3]

Briefly, in this approach one starts with a unitary irreducible representation   and attempts to find a vector  , a subgroup   and a section   such that   where  ,   is a bounded, positive operator with bounded inverse and   is a quasi-invariant measure on  . It is not assumed that   be invariant up to a phase under the action of   and clearly, the best situation is when   is a multiple of the identity. Although somewhat technical, this general construction is of enormous versatility for semi-direct product groups of the type  , where   is a closed subgroup of  . Thus, it is useful for many physically important groups, such as the Poincaré group or the Euclidean group, which do not have square integrable representations in the sense of the earlier definition. In particular, the integral condition defining the operator   ensures that any vector   in   can be written in terms of the generalized coherent states   namely,   which is the primary aim of any kind of coherent states.

Coherent states: a Bayesian construction for the quantization of a measure set

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We now depart from the standard situation and present a general method of construction of coherent states, starting from a few observations on the structure of these objects as superpositions of eigenstates of some self-adjoint operator, as was the harmonic oscillator Hamiltonian for the standard CS. It is the essence of quantum mechanics that this superposition has a probabilistic flavor. As a matter of fact, we notice that the probabilistic structure of the canonical coherent states involves two probability distributions that underlie their construction. There are, in a sort of duality, a Poisson distribution ruling the probability of detecting   excitations when the quantum system is in a coherent state  , and a gamma distribution on the set   of complex parameters, more exactly on the range   of the square of the radial variable. The generalization follows that duality scheme. Let   be a set of parameters equipped with a measure   and its associated Hilbert space   of complex-valued functions, square integrable with respect to  . Let us choose in   a finite or countable orthonormal set  :   In case of infinite countability, this set must obey the (crucial) finiteness condition:   Let   be a separable complex Hilbert space with orthonormal basis   in one-to-one correspondence with the elements of  . The two conditions above imply that the family of normalized coherent states   in  , which are defined by   resolves the identity in  :   Such a relation allows us to implement a coherent state or frame quantization of the set of parameters   by associating to a function   that satisfies appropriate conditions the following operator in  :   The operator   is symmetric if   is real-valued, and it is self-adjoint (as a quadratic form) if   is real and semi-bounded. The original   is an upper symbol, usually non-unique, for the operator  . It will be called a classical observable with respect to the family   if the so-called lower symbol of  , defined as   has mild functional properties to be made precise according to further topological properties granted to the original set  . A last point of this construction of the space of quantum states concerns its statistical aspects. There is indeed an interplay between two probability distributions:

  1. For almost each  , a discrete distribution,

     

    This probability could be considered as concerning experiments performed on the system within some experimental protocol, in order to measure the spectral values of a certain self-adjoint operator  , i.e., a quantum observable, acting in   and having the discrete spectral resolution  .
  2. For each  , a continuous distribution on  ,   Here, we observe a Bayesian duality typical of coherent states. There are two interpretations: the resolution of the unity verified by the coherent states   introduces a preferred prior measure on the set  , which is the set of parameters of the discrete distribution, with this distribution itself playing the role of the likelihood function. The associated discretely indexed continuous distributions become the related conditional posterior distribution. Hence, a probabilistic approach to experimental observations concerning   should serve as a guideline in choosing the set of the  's. We note that the continuous prior distribution will be relevant for the quantization whereas the discrete posterior one characterizes the measurement of the physical spectrum from which is built the coherent superposition of quantum states  .[1]

See also

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References

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  1. ^ a b J-P. Gazeau,Coherent States in Quantum Physics, Wiley-VCH, Berlin, 2009.
  2. ^ a b S.T. Ali, J-P. Antoine, J-P. Gazeau, and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, Reviews in Mathematical Physics 7 (1995) 1013-1104.
  3. ^ a b S.T. Ali, J-P. Antoine, and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, Berlin, Heidelberg, 2000.
  4. ^ S.T. Ali, Coherent States, Encyclopedia of Mathematical Physics, pp. 537-545; Elsevier, Amsterdam, 2006.
  5. ^ Barut, A. O.; Girardello, L. (1971). "New "Coherent" States associated with non-compact groups". Communications in Mathematical Physics. 21 (1): 41–55. Bibcode:1971CMaPh..21...41B. doi:10.1007/bf01646483. ISSN 0010-3616. S2CID 122468207.
  6. ^ Gazeau, Jean Pierre; Klauder, John R (1999-01-01). "Coherent states for systems with discrete and continuous spectrum". Journal of Physics A: Mathematical and General. 32 (1): 123–132. Bibcode:1999JPhA...32..123G. doi:10.1088/0305-4470/32/1/013. ISSN 0305-4470.
  7. ^ Ali, S. Twareque; Bagarello, F. (2005). "Some physical appearances of vector coherent states and coherent states related to degenerate Hamiltonians". Journal of Mathematical Physics. 46 (5): 053518. arXiv:quant-ph/0410151. Bibcode:2005JMP....46e3518T. doi:10.1063/1.1901343. ISSN 0022-2488. S2CID 19024789.
  8. ^ Hall, B.C. (1994). "The Segal-Bargmann "Coherent State" Transform for Compact Lie Groups". Journal of Functional Analysis. 122 (1): 103–151. doi:10.1006/jfan.1994.1064. ISSN 0022-1236.
  9. ^ Stenzel, Matthew B. (1999). "The Segal–Bargmann Transform on a Symmetric Space of Compact Type" (PDF). Journal of Functional Analysis. 165 (1): 44–58. doi:10.1006/jfan.1999.3396. ISSN 0022-1236.
  10. ^ Hall, Brian C.; Mitchell, Jeffrey J. (2002). "Coherent states on spheres". Journal of Mathematical Physics. 43 (3): 1211–1236. arXiv:quant-ph/0109086. Bibcode:2002JMP....43.1211H. doi:10.1063/1.1446664. ISSN 0022-2488. S2CID 2990048.
  11. ^ Thiemann, Thomas (2001-05-16). "Gauge field theory coherent states (GCS): I. General properties". Classical and Quantum Gravity. 18 (11): 2025–2064. arXiv:hep-th/0005233. Bibcode:2001CQGra..18.2025T. doi:10.1088/0264-9381/18/11/304. ISSN 0264-9381. S2CID 16699452. and other papers in the same sequence
  12. ^ a b A. M. Perelomov, Coherent states for arbitrary Lie groups, Commun. Math. Phys. 26 (1972) 222–236; arXiv: math-ph/0203002.
  13. ^ a b A. Perelomov, Generalized coherent states and their applications, Springer, Berlin 1986.
  14. ^ a b Gilmore, Robert (1972). "Geometry of symmetrized states". Annals of Physics. 74 (2). Elsevier BV: 391–463. Bibcode:1972AnPhy..74..391G. doi:10.1016/0003-4916(72)90147-9. ISSN 0003-4916.
  15. ^ a b Gilmore, R. (1974). "On properties of coherent states" (PDF). Revista Mexicana de Física. 23: 143–187.
  16. ^ Coherent state at the nLab
  17. ^ Onofri, Enrico (1975). "A note on coherent state representations of Lie groups". Journal of Mathematical Physics. 16 (5): 1087–1089. Bibcode:1975JMP....16.1087O. doi:10.1063/1.522663. ISSN 0022-2488.
  18. ^ I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
  19. ^ S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, San Diego, 1999.
  20. ^ J-P. Antoine, R. Murenzi, P. Vandergheynst, and S.T. Ali, Two-Dimensional Wavelets and their Relatives, Cambridge University Press, Cambridge (UK), 2004.
  21. ^ Biedenharn, L C (1989-09-21). "The quantum group   and a  -analogue of the boson operators". Journal of Physics A: Mathematical and General. 22 (18): L873–L878. doi:10.1088/0305-4470/22/18/004. ISSN 0305-4470.
  22. ^ Jurčo, Branislav (1991). "On coherent states for the simplest quantum groups". Letters in Mathematical Physics. 21 (1): 51–58. Bibcode:1991LMaPh..21...51J. doi:10.1007/bf00414635. ISSN 0377-9017. S2CID 121389100.
  23. ^ Celeghini, E.; Rasetti, M.; Vitiello, G. (1991-04-22). "Squeezing and quantum groups". Physical Review Letters. 66 (16): 2056–2059. Bibcode:1991PhRvL..66.2056C. doi:10.1103/physrevlett.66.2056. ISSN 0031-9007. PMID 10043380.
  24. ^ Sazdjian, Hagop; Stanev, Yassen S.; Todorov, Ivan T. (1995). "  coherent state operators and invariant correlation functions and their quantum group counterparts". Journal of Mathematical Physics. 36 (4): 2030–2052. arXiv:hep-th/9409027. doi:10.1063/1.531100. ISSN 0022-2488. S2CID 18220520.
  25. ^ Jurĉo, B.; Ŝťovíĉek, P. (1996). "Coherent states for quantum compact groups". Communications in Mathematical Physics. 182 (1): 221–251. arXiv:hep-th/9403114. Bibcode:1996CMaPh.182..221J. doi:10.1007/bf02506391. ISSN 0010-3616. S2CID 18018973.
  26. ^ Škoda, Zoran (2007-06-22). "Coherent States for Hopf Algebras". Letters in Mathematical Physics. 81 (1): 1–17. arXiv:math/0303357. Bibcode:2007LMaPh..81....1S. doi:10.1007/s11005-007-0166-y. ISSN 0377-9017. S2CID 8470932.