Coherent states in mathematical physics

A large class of generalizations of the CCS is obtained by a simple modification of their analytic structure. Let ${\displaystyle \varepsilon _{1}\leq \varepsilon _{2}\leq \ldots \leq \varepsilon _{n}\leq \ldots }$ be an infinite sequence of positive numbers (${\displaystyle \varepsilon _{1}\neq 0}$). Define ${\displaystyle \varepsilon _{n}!=\varepsilon _{1}\varepsilon _{2}\ldots \varepsilon _{n}}$ and by convention set ${\displaystyle \varepsilon _{0}!=1}$. In the same Fock space in which the CCS were described, we now define the related deformed or nonlinear coherent states by the expansion

${\displaystyle \vert \alpha \rangle ={\mathcal {N}}(\vert \alpha \vert ^{2})^{-{\frac {1}{2}}}\;\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{\sqrt {\varepsilon _{n}!}}}|n\rangle \,.}$

The normalization factor ${\displaystyle {\mathcal {N}}(\vert \alpha \vert ^{2})}$ is chosen so that ${\displaystyle \langle \alpha \vert \alpha \rangle =1}$. These generalized coherent states are overcomplete in the Fock space and satisfy a resolution of the identity

${\displaystyle \int _{\mathcal {D}}\vert \alpha \rangle \langle \alpha \vert \;{\mathcal {N}}(\vert \alpha \vert ^{2})\;d\nu (\alpha ,{\overline {\alpha }})=I\;,}$

${\displaystyle {\mathcal {D}}}$ being an open disc in the complex plane of radius ${\displaystyle L}$, the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{\sqrt {\varepsilon _{n}!}}}}$ (in the case of the CCS, ${\displaystyle L=\infty }$.) The measure ${\displaystyle d\nu }$ is generically of the form ${\displaystyle d\theta \;d\lambda (r)}$ (for ${\displaystyle \alpha =re^{i\theta }}$), where ${\displaystyle d\lambda }$ is related to the ${\displaystyle \varepsilon _{n}!}$ through the moment condition.

Once again, we see that for an arbitrary vector ${\displaystyle |\phi \rangle }$ in the Fock space, the function ${\displaystyle \Phi (\alpha )=\langle \phi |\alpha \rangle }$ is of the form ${\displaystyle \Phi (\alpha )={\mathcal {N}}(\vert \alpha \vert ^{2})^{-{\frac {1}{2}}}f(\alpha )}$, where ${\displaystyle f\,}$ is an analytic function on the domain ${\displaystyle {\mathcal {D}}}$. The reproducing kernel associated to these coherent states is

${\displaystyle K({\overline {\alpha }},\alpha ')=\langle \alpha |\alpha '\rangle =\left[{\mathcal {N}}(\vert \alpha \vert ^{2}){\mathcal {N}}(\vert \alpha '\vert ^{2})\right]^{-{\frac {1}{2}}}\sum _{n=0}^{\infty }{\frac {({\overline {\alpha }}\alpha ')^{n}}{\varepsilon _{n}!}}\;.}$

By analogy with the CCS case, one can define a generalized annihilation operator ${\displaystyle A}$ by its action on the vectors ${\displaystyle |\alpha \rangle }$,

${\displaystyle A|\alpha \rangle =\alpha |\alpha \rangle \;,}$

and its adjoint operator ${\displaystyle A^{\dagger }}$. These act on the Fock states ${\displaystyle |n\rangle }$ as

${\displaystyle A|n\rangle ={\sqrt {\varepsilon _{n}}}|n-1\rangle \;,\qquad A^{\dagger }|n\rangle ={\sqrt {\varepsilon _{n+1}}}|n+1\rangle \;.}$

Depending on the exact values of the quantities ${\displaystyle \varepsilon _{n}}$, these two operators, together with the identity ${\displaystyle I}$ 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 (there might have more general ones).

Operators ${\displaystyle A}$ and ${\displaystyle A^{\dagger }}$ 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 ${\displaystyle A}$ 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.

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 ${\displaystyle H=\sum _{n=0}^{\infty }E_{n}|n\rangle \langle n|}$, with ${\displaystyle E_{0}=0}$, i.e., it has the energy eigenvalues ${\displaystyle E_{n}}$ and eigenvectors ${\displaystyle |n\rangle }$, which we assume to form an orthonormal basis for the Hilbert space of states ${\displaystyle {\mathfrak {H}}}$. Let us write the eigenvalues as ${\displaystyle E_{n}=\omega \varepsilon _{n}}$ by introducing a sequence of dimensionless quantities ${\displaystyle \{\varepsilon _{n}\}}$ ordered as: ${\displaystyle 0=\varepsilon _{0}<\varepsilon _{1}<\varepsilon _{2}<\ldots \;}$. Then, for all ${\displaystyle J\geq 0}$ and ${\displaystyle \gamma \in \mathbb {R} }$, the Gazeau–Klauder coherent states are defined as

${\displaystyle |J,\gamma \rangle ={\mathcal {N}}(J)^{-{\frac {1}{2}}}\sum _{n=0}^{\infty }\,{\frac {J^{n/2}e^{-i\varepsilon _{n}\gamma }}{\sqrt {\varepsilon _{n}!}}}|n\rangle \;,}$

where again ${\displaystyle {\mathcal {N}}}$ is a normalization factor, which turns out to be dependent on ${\displaystyle J}$ only. These coherent states satisfy the temporal stability condition,

${\displaystyle e^{-iHt}\vert J,\gamma \rangle =\vert J,\gamma +\omega t\rangle \;,}$

and the action identity,

${\displaystyle \langle J,\gamma |H|J,\gamma \rangle _{\mathfrak {H}}=\omega J\;.}$

While these generalized coherent states do form an overcomplete set in ${\displaystyle {\mathfrak {H}}}$, 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]

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 ${\displaystyle K=\mathrm {SU} (2)}$, 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.

A typical example of the above construction is provided by the affine group of the line, ${\displaystyle G_{\text{Aff}}}$. This is the group of all 2${\displaystyle \times }$2 matrices of the type,

${\displaystyle g={\begin{pmatrix}a&b\\0&1\end{pmatrix}}\;,}$

${\displaystyle a}$ and ${\displaystyle b}$ being real numbers with ${\displaystyle a\neq 0}$. We shall also write ${\displaystyle g=(b,a)}$, with the action on ${\displaystyle \mathbb {R} }$ given by ${\displaystyle (b,a)\cdot x=b+ax}$. This group is non-unimodular, with the left invariant measure being given by ${\displaystyle d\mu (b,a)=a^{-2}\;db\;da}$ (the right invariant measure being ${\displaystyle a^{-1}\;db\;da}$). The affine group has a unitary irreducible representation on the Hilbert space ${\displaystyle L^{2}(\mathbb {R} ,dx)}$. Vectors in ${\displaystyle L^{2}(\mathbb {R} ,dx)}$ are measurable functions ${\displaystyle \varphi (x)}$ of the real variable ${\displaystyle x}$ and the (unitary) operators ${\displaystyle U(b,a)}$ of this representation act on them as

${\displaystyle (U(b,a)\varphi )(x)={\frac {1}{\sqrt {\vert a\vert }}}\;\varphi \left({\frac {x-b}{a}}\right)={\frac {1}{\sqrt {\vert a\vert }}}\;\varphi \left((b,a)^{-1}\cdot x\right)\;.}$

If ${\displaystyle \psi }$ is a function in ${\displaystyle L^{2}(\mathbb {R} ,dx)}$ such that its Fourier transform ${\displaystyle {\widehat {\psi }}}$ satisfies the (admissibility) condition

${\displaystyle \int _{\mathbb {R} }{\frac {\vert {\widehat {\psi }}(k)\vert ^{2}}{\vert k\vert }}\;dk<\infty \;,}$

then it can be shown to be an admissible vector, i.e.,

${\displaystyle c(\psi )=\int _{G_{\text{Aff}}}\vert \langle \psi |U(b,a)\psi \rangle \vert ^{2}\;{\frac {db\;da}{a^{2}}}<\infty \;.}$

Thus, following the general construction outlined above, the vectors

${\displaystyle |b,a\rangle ={\frac {1}{\sqrt {c(\psi )}}}\;U(b,a)\psi \;,\qquad (b,a)\in G_{\text{Aff}}}$

define a family of generalized coherent states and one has the resolution of the identity

${\displaystyle \int _{G_{\text{Aff}}}|b,a\rangle \langle b,a|\;{\frac {db\;da}{a^{2}}}=I}$

on ${\displaystyle L^{2}(\mathbb {R} ,dx)}$. In the signal analysis literature, a vector satisfying the admissibility condition above is called a mother wavelet and the generalized coherent states ${\displaystyle |b,a\rangle }$ are called wavelets. Signals are then identified with vectors ${\displaystyle |\varphi \rangle }$ in ${\displaystyle L^{2}(\mathbb {R} ,dx)}$ and the function

${\displaystyle F(b,a)=\langle b,a|\varphi \rangle \;,}$

is called the continuous wavelet transform of the signal ${\displaystyle \varphi }$. [18] [19]

This concept can be extended to two dimensions, the group ${\displaystyle G_{\text{Aff}}\;}$ 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]

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 ${\displaystyle \mathbb {R} ^{2}}$. The key observation is that the center of the Heisenberg group leaves the vacuum vector ${\displaystyle |0\rangle }$ invariant, up to a phase. Generalizing this idea, Gilmore and Perelomov [12] [13] [14] [15] consider a locally compact group ${\displaystyle G}$ and a unitary irreducible representation ${\displaystyle U}$ of ${\displaystyle G}$ on the Hilbert space ${\displaystyle {\mathfrak {H}}}$, not necessarily square integrable. Fix a vector ${\displaystyle |\psi \rangle }$ in ${\displaystyle {\mathfrak {H}}}$, of unit norm, and denote by ${\displaystyle H}$ the subgroup of ${\displaystyle G}$ consisting of all elements ${\displaystyle h}$ that leave it invariant up to a phase, that is,

${\displaystyle U(h)\mid \psi \rangle =e^{i\omega (h)}\mid \psi \rangle \,,}$

where ${\displaystyle \omega }$ is a real-valued function of ${\displaystyle h}$. Let ${\displaystyle X=G/H}$ be the left coset space and ${\displaystyle x}$ an arbitrary element in ${\displaystyle X}$. Choosing a coset representative ${\displaystyle g(x)\in G}$, for each coset ${\displaystyle x}$, we define the vectors

${\displaystyle |x\rangle =U(g(x))|\psi \rangle \in {\mathfrak {H}}.}$

The dependence of these vectors on the specific choice of the coset representative ${\displaystyle g(x)}$ is only through a phase. Indeed, if instead of ${\displaystyle g(x)}$, we took a different representative ${\displaystyle g(x)'\in G}$ for the same coset ${\displaystyle x}$, then since ${\displaystyle g(x)'=g(x)h}$ for some ${\displaystyle h\in H}$, we would have ${\displaystyle U(g(x)')|\psi \rangle =e^{i\omega (h)}|x\rangle }$. Hence, quantum mechanically, both ${\displaystyle |x\rangle }$ and ${\displaystyle U(g(x)')|\psi \rangle }$ represent the same physical state and in particular, the projection operator ${\displaystyle |x\rangle \langle x|}$ depends only on the coset. Vectors ${\displaystyle |x\rangle }$ defined in this way are called Gilmore–Perelomov coherent states. Since ${\displaystyle U}$ is assumed to be irreducible, the set of all these vectors as ${\displaystyle x}$ runs through ${\displaystyle G/H}$ is dense in ${\displaystyle {\mathfrak {H}}}$. In this definition of generalized coherent states, no resolution of the identity is postulated. However, if ${\displaystyle X}$ carries an invariant measure, under the natural action of ${\displaystyle G}$, and if the formal operator ${\displaystyle B}$ defined as

${\displaystyle B=\int _{X}|x\rangle \langle x|\;d\mu (x)\;,}$

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]