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International Journal of Geometric Methods in Modern Physics

ORDINARY DIFFERENTIAL EQUATIONS, FIRST INTEGRALS, CONFORMAL INVARIANTS, BOSE OPERATORS, COHERENT STATES AND ENTANGLEMENT

Willi-Hans Steeb

Published Nov 1, 2008• 0 citations
Publisher

Abstract

Autonomous systems of first order ordinary differential equations can be embedded into a Hilbert space description by using Bose operators and Glauber coherent states. The first integrals and conformal invariants of the autonomous system can be represented by states in a Hilbert space. Thus the embedding in a Hilbert space allows us to study entanglement of the states. We apply it here to first integrals and conformal invariants which are expressed as states.

DOI
10.1142/s0219887808003193

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Autonomous systems of first order ordinary differential equations can be embedded into a Hilbert space description by using Bose operators and Glauber coherent states. The first integrals and conformal invariants of the autonomous system can be represented by states in a Hilbert space. Thus the embedding in a Hilbert space allows us to study entanglement of the states. We apply it here to first integrals and conformal invariants which are expressed as states.
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture are developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.
The mapping of operator equations into CCC-number differential form by means of the atomic coherent-state representation leads to mathematical difficulties that have no counterpart in the case of the Glauber-Sudarshan representation for Bose systems. We give sufficient conditions for the solutions of CCC-number equations in the atomic coherent-state representation to be physically meaningful.
We outline the basic questions that are being studied in any theory of entanglement. Following a brief review of some of the main achievements of entanglement theory for finite-dimensional systems such as qubits, we will then consider entanglement in infinite-dimensional systems. Asking for a theory of entanglement in such systems under experimentally feasible operations leads to the development of the theory of entanglement of Gaussian states. Results of this theory are presented and the tools that have been developed for this theory are then applied to a number of problems.
A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.