Transformation Of Coherent States Research Articles

On the Connection Between the Fueter–Sce–Qian Theorem and the Generalized CK-Extension

The Fueter–Sce–Qian theorem provides a way of inducing axial monogenic functions in $$\mathbb {R}^{m+1}$$ from holomorphic intrinsic functions of one complex variable. This result was initially proved by Fueter and Sce for the cases where the dimension m is odd using pointwise differentiation, while the extension to the cases where m is even was proved by Qian using the corresponding Fourier multipliers. In this paper, we provide an alternative description of the Fueter–Sce–Qian theorem in terms of the generalized CK-extension. The latter characterizes axial null solutions of the Cauchy–Riemann operator in $$\mathbb {R}^{m+1}$$ in terms of their restrictions to the real line. This leads to a one-to-one correspondence between the space of axially monogenic functions in $$\mathbb {R}^{m+1}$$ and the space of analytic functions of one real variable. We provide explicit expressions for the Fueter–Sce–Qian map in terms of the generalized CK-extension for both cases, m even and m odd. These expressions allow for a plane wave decomposition of the Fueter–Sce–Qian map or, more in particular, a factorization of this mapping in terms of the dual Radon transform. In turn, this decomposition provides a new possibility for extending the Coherent State Transform (CST) to Clifford Analysis. In particular, we construct an axial CST defined through the Fueter–Sce–Qian mapping, and show how it is related to the axial and slice CST’s already studied in the literature.

Complex time evolution in geometric quantization and generalized coherent state transforms

For the cotangent bundle T⁎K of a compact Lie group K, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L2(K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L2(K) and a certain weighted L2-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ) within L2(K), followed by a polarization-changing geometric-quantization evolution (for complex time +τ). In this case, our construction yields the usual generalized Segal–Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemannʼs “complexifier” method (which generalizes the construction of adapted complex structures).

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

By means of the orbit method we show that, for a compact Lie group, the Blattner–Kostant–Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter–Weyl theorem for the complexification of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula, as well as behind the requisite half-form correction on which the Blatter–Kostant–Sternberg-pairing for the compact Lie group relies, and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain a number of these results [B.C. Hall, The Segal–Bargmann Coherent State Transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151; B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233–268, quant.ph/0012015].

Coherent-state analysis of the seismic head wave problem: an overcomplete representation and its relationship to rays and beams

SUMMARY The coherent-state transform (CST) is a Gaussian-windowed Fourier transform and compared with the usual plane wave expansion of seismic wavefields it provides an overcomplete basis of partial waves. These overcomplete partial waves have associated rays and the relationship of these rays to those of a physical wave front is explained here by appealing to a familiar analytic example. The exact CST of the standard Fourier plane wave summation for a point-source primary wave can be interpreted as a sum of damped plane waves forming a focussed or beam-like wavefield. This primary wave CST can also be approximated as a bundle of complex rays in a position-slowness space with higher dimension than that usually considered, a consequence of the overcompleteness. The higher-dimensional ray spreading controls the coherent-state (CS) amplitude. This complex-ray bundle in turn can be approximated as a paraxial Gaussian beam carried by a real ray associated with the physical wave front. The exact CST of the standard plane wave summation for a point-source reflection shows how the complex-ray and paraxial-beam approximations generalize to interfaces. Around a critical angle the standard plane wave summation has an asymptotic form involving the Weber function and this function also necessarily arises in the complex-ray and paraxial-beam approximations for the reflection CST. The inverse CST giving the point-source wavefield is a sum of coherent states and those contributing rays or paraxial beams that are incident around the critical angle should be given a reflection coefficient involving the Weber function. CS beams that are incident away from the critical angle collect a peripheral branch-point diffraction. In general, both types of critical-ray/branch-point signal should be included if the integrated CS response is to correctly describe the head wave. An advantage of the CST is that it combines with ray theory for gradually varying media to give a convenient solution to the caustic and pseudocaustic problems. It may be said to unify the Maslov and standard Gaussian-beam methods of seismic modelling, in part by avoiding the ray-centered coordinates conventionally employed in the latter. The general idea of such overlapping partial wave expansions, their flexibility and the smoothing they impart may have other benefits in seismic analysis and processing.

Coherent state transforms and vector bundles on elliptic curves

We extend the coherent state transform (CST) of Hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves. We show that by applying the CST to appropriate distributions, we obtain the space of level k, rank n and genus one non-abelian theta functions with the unitarity of the CST transform being preserved. Furthermore, the shift in the level k→ k+ n appears in a natural way in this finite-dimensional framework.

Coherent State Transforms and Abelian Varieties

We extend the coherent state transform (CST) of Hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group K= U(1) g . We show that this transform, applied to appropriate distributions on K, gives all classical theta functions, and that, by defining on this space of theta functions an inner product related to the K-averaged heat kernel, the unitarity of the CST transform is still preserved.

Seismic coherent states and ray geometrical spreading

SUMMARY The coherent-state transform (CST) is essentially a Gaussian-windowed Fourier transform and it yields a combined slowness–position (p, x) domain representation of seismic wavefields. Several forms of the inverse CST exist and the set of ‘coherent states’ form an overcomplete basis for wave analysis, in many ways similar to modern wavelets. The asymptotic or ‘ray’ solution to the CST of the seismic wave equation involves a phase function S(p, x) that is complex due to the Gaussian decay. Hence one must consider complex rays, as well as a higher-dimensional phase space (p, x, ∂pS, ∂xS) corresponding to the extended configuration or base space (p, x). The initial conditions and geometrical spreading of these rays involve generalizations of standard procedures, exemplified by analysis of the coherent states excited by a ray theory incident wavefield. Surfaces of constant S are ‘tangential’ to the standard ray theory wavefront T(x) and the Maslov phase fronts given by the Legendre transformation of T. Hence the real rays of T (and the Maslov phase) are particular rays of S. The geometrical spreading of an individual coherent-state (CS) wavefield requires careful consideration. Although the transport equation involves divergence in the higher-dimensional base space (p, x), Smirnov's Lemma applied in this space still gives the solution. The incorporation of initial conditions is correspondingly intricate, but the final spreading function is better behaved (‘smoother’) than either a standard ray or Maslov theory amplitude. This provides a solution to the problem of pseudo-caustics. The task of finding a ‘KMAH index’ for each ray contributing to the inverse CST is simplified to choosing a complex square root that is smoothly connected between these rays (i.e. it does not fork). It is suggested that in practice only real rays are needed and then the method can be reduced to a smoothed form of the Maslov Snell wave sum representation. This is achieved by approximating the complex CS phase S(p, x) in the vicinity of the real rays. The result also then appears similar in form to a sum of Gaussian beams, although without reference to local ray-centred coordinates. Although the derivation involves some lengthy steps, the formulae finally obtained are easy to implement. Their effectiveness is demonstrated by a numerical example of interfering caustics and pseudo-caustics caused by two nearly plane wavefront segments joined at a triplication.

Darboux transformation of coherent states for a Lobachevski plane

A Darboux transformation of coherent states is considered for a singular oscillator. A coordinate representation and a holomorphic representation are obtained for the Darboux transformation operator and the coherent states. The Hermitian metric and the Kaler potential of the transformed system are calculated. It is established that a Darboux transformation corresponds to curvature of the phase space of the classical system.

Can photon noise be reduced ?

The sensitivity of the interferometric methods for detection of gravitational waves is limited at the moment by photon noise. This noise can theoretically be reduced by entering in one input part of the interferometer a field in a particular quantum state, the so called squeezed state. A wide variety of non-linear processes, such as the degenerate four wave mixing, can realize the transformation of coherent states of the field into squeezed states. We show that the photon noise reduction is very sensitive to the spatial and temporal coherence properties of the squeezed field, and that photon noise is not appreciably reduced with incoherent sources of squeezed field.