We analyze the stability under time evolution of complexifier coherent states (CCS) in one-dimensional mechanical systems. A system of coherent states is called stable if it evolves into another coherent state. It turns out that a system can only possess stable CCS if the classical evolution of the variable $\mathbf{z}={e}^{\ensuremath{-}i{\mathcal{L}}_{{\ensuremath{\chi}}_{C}}}\mathbf{q}$ for a given complexifier $C$ depends only on $\mathbf{z}$ itself and not on its complex conjugate. This condition is very restrictive in general so that only a few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated with certain regions in the phase space by introducing action-angle coordinates.
We analyze the stability under time evolution of complexifier coherent states (CCS) in one-dimensional mechanical systems. A system of coherent states is called stable if it evolves into another coherent state. It turns out that a system can only possess stable CCS if the classical evolution of the variable z=e\ensuremath−iL\ensuremathχCq for a given complexifier C depends only on z itself and not on its complex conjugate. This condition is very restrictive in general so that only a few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated with certain regions in the phase space by introducing action-angle coordinates.
For a system with one degree of freedom, coherent states that are parametrized by classical canonical action-angle variables are introduced. These states also possess continuity of labelling, a resolution of unity, and temporal stability. The insistence on canonical action-angle variables strongly restricts any remaining arbitrariness in the coherent state definition. Such states are introduced for semibounded Hamiltonian operators having either a discrete or a continuous spectrum. Hamiltonians that have both discrete and continuous parts in their spectrum are also discussed.
The complex geometry underlying the Schrödinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.
The exact and stable evolutions of generalized coherent states (GCS) for quantum systems are considered by making use of the time-dependent integrals of motion method and of the Klauder approach to the relationship between quantum and classical mechanics. It is shown that one can construct for any quantum system overcomplete family of states (OFS), related to the unitary representations of the Lie group G by means of integral of motion generators, and the possibility of using this group as a dynamical symmetry group is pointed out. The relation of the OFS with quantum measurement theory is also established.
Following our preceding work on spherical and linear rotors [C. Stopera and J. A. Morales, J. Chem. Phys. 152, 134112 (2020)], we reformulate an earlier rotational coherent state (CS) set to obtain a temporally stable (TS) CS set for symmetric rotors. Being TS, the new CSs remain within its own set during dynamics by evolving exclusively through their parameters. The TS CS set is now appropriate to reconstruct quantum rotational properties from classical-mechanics simulations of chemical reactions. Following literature precedents, we enforce temporal stability by incorporating action-angle-related phase factors into two parameters of the original CS set. Proofs and final expressions of the symmetric-rotor CS turn out more intricate than those of its spherical-rotor counterpart. We demonstrate and examine the key properties of the new CS set: continuity, resolution of unity, temporal stability, action identity, minimum uncertainty relationships, and quasi-classical behavior. Regarding the last property, we demonstrate that the body-fixed z-component of the CS angular momentum average evolves exactly as its classical counterpart, and that the x- and y-components display an astonishing analogy with their classical counterparts in terms of functional form, precession angular velocities, amplitudes, and phases. We elucidate some of these properties via computer simulations of a rotating benzene molecule represented with the CS set. We discuss the utilization of this CS set to reconstruct quantum rotational properties of symmetric-rotor molecules from classical-mechanics simulations. The new CS set is appropriate to establish quantum-classical connections for rotational properties in chemical dynamics, statistical mechanics, spectroscopy, nuclear physics, and quantum computing.
Recently, the coherent-state labeled by the real parameters (RCS) has been introduced in connec tion with the numerical path integration for many-fermion systems.!) This is a complement of the coherent-state labeled by complex vari ables (CCS) which gives a basis of the path inte gral in the generalized phase space leading to the quantum version of the time-dependent mean field. 2 ),3) The central formula of both types of coherent states is the relation of of unity. For the case of RCS, Dobacewski and Koonin 4 ) have very recently proved this relation based on the commutability of the projection operator with the algebra generating the coher ent-state. The proof is rather general, since one uses only the Lie algebraic concept of the highest weight vectors. However it presupposes a specific form of the integration measure from the outset. In this note, apart from such a Lie-alge braic approach, we construct the RCS for many fermion systems based on the geometric device of the Lie group and associated label space (the so called quotient space),S) thereby the partition of unity is proved through a natural construction of the invariant measure. We also point out a specific role of the RCS for path integration in comparison with the path integral in the represen tation of CCS. In the following we are concerned with the RCS of determinantal states.. We firstly show that the underlying group becomes an orthogonal group instead of the unitary group for the CCS. 2 ) Let (aI', ai) be annihilation operators of the ~'particle and hole states with It = 1 ~ m, i = 1 ~ n with respect to the initial determinantal state 1 ([}o> = IL~lai +10>. Consider the linear canonical transformation where T(g) is an irreducible unitary representa tion of a Lie group G and A, B, C and Dare m x m, m x n, n x m and n x n matrices. Now consider the real restriction of (1). Hence the canonical nature leads to g'gT=J<m+n) (2) or
Looking for a quantum-mechanical implementation of duality, we formulate a relation between coherent states and complex-differentiable structures on classical phase space C. A necessary and sufficient condition for the existence of locally-defined coherent states is the existence of an almost complex structure on C. A necessary and sufficient condition for globally-defined coherent states is a complex structure on C. The picture of quantum mechanics that emerges is conceptually close to that of a geometric manifold covered by local coordinate charts. Instead of the latter, quantum mechanics has local coherent states. A change of coordinates on C may or may not be holomorphic. Correspondingly, a transformation between quantum-mechanical states may or may not preserve coherence. Those that do not preserve coherence are duality transformations. A duality appears as the possibility of giving two or more, apparently different, descriptions of the same quantum-mechanical phenomenon. Coherence becomes a local property on classical phase space. Observers on C not connected by means of a holomorphic change of coordinates need not, and in general will not, agree on what is a semiclassical effect vs. what is a strong quantum effect