# Effective Equations of Motion for Quantum Systems

@article{Bojowald2005EffectiveEO, title={Effective Equations of Motion for Quantum Systems}, author={Martin Bojowald and Aureliano Skirzewski}, journal={Reviews in Mathematical Physics}, year={2005}, volume={18}, pages={713-746} }

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 is 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…

## 60 Citations

Canonical derivation of effective potentials

- Physics
- 2014

A canonical formulation of effective equations describes quantum corrections by the back-reaction of moments on the dynamics of expectation values of a state. As a first step toward an extension to…

EFFECTIVE CONSTRAINTS FOR QUANTUM SYSTEMS

- Physics
- 2009

An effective formalism for quantum constrained systems is presented which allows manageable derivations of solutions and observables, including a treatment of physical reality conditions without…

Effective dynamics of the quantum falling particle

- PhysicsEuropean Journal of Physics
- 2020

We analyze the problem of a quantum particle falling under the influence of a one-dimensional constant gravitational field, also known as the bouncing ball, employing a semiclassical approach for the…

Effective equations for isotropic quantum cosmology including matter

- Physics
- 2007

Effective equations often provide powerful tools to develop a systematic understanding of detailed properties of a quantum system. This is especially helpful in quantum cosmology where several…

QUANTUM GRAVITY AND HIGHER CURVATURE ACTIONS

- Physics
- 2006

Effective equations are often useful to extract physical information from quantum theories without having to face all technical and conceptual difficulties. One can then describe aspects of the…

Canonical Quantum Gravity and Effective Theory

- Physics
- 2009

Effective actions provide a powerful way to understand low-energy, semiclassical or other regimes of quantum field theories. Although canonical quantum gravity theories, such as loop quantum gravity,…

Quantum phase space trajectories with application to quantum cosmology

- PhysicsPhysical Review D
- 2018

We develop an approach to quantum dynamics based on quantum phase space trajectories. The latter are built from a unitary irreducible representation of the symmetry group of the respective classical…

Effective equations of motion for constrained quantum systems: A study of the Bianchi I loop quantum cosmology

- Physics
- 2008

A new mathematical framework is formulated to derive the effective equations of motion for the constrained quantum system which possesses an internal clock. In the realm close to classical behavior,…

A no-singularity scenario in loop quantum gravity

- Physics
- 2012

Canonical methods allow the derivation of effective gravitational actions from the behavior of space-time deformations reflecting general covariance. With quantum effects, the deformations and…

## References

SHOWING 1-10 OF 33 REFERENCES

Comparison between quantum and classical dynamics in the effective action formalism

- Physics
- 1999

A major difficulty in comparing quantum and classical behavior resides in the structural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator…

Coherent states: Theory and some Applications

- Physics
- 1990

In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the…

Geometrization of quantum mechanics

- Physics
- 1979

Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional…

Quantum mechanics as a statistical theory

- Physics
- 1949

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the…

Quantum mechanics as a classical theory.

- PhysicsPhysical review. D, Particles and fields
- 1985

Basic features of quantum mechanics follow, such as the identification of observables with self-adjoint operators, and canonical quantization rules, which gives a new insight on the geometry of quantum theory.

Geometrical Formulation of Quantum Mechanics

- Physics
- 1999

States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a Uhler manifold. This leads to a geometrical formulation…

Mathematical structure of loop quantum cosmology

- Philosophy, Physics
- 2003

Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big…

Isotropic loop quantum cosmology

- Physics
- 2002

Isotropic models in loop quantum cosmology allow explicit calculations, thanks largely to a completely known volume spectrum, which is exploited in order to write down the evolution equation in a…

Relativistic field theories with symmetry-breaking solutions

- Physics
- 1964

Following some recent developments in many-body theory, we suggest a functional approach to relativistic field theories particularly suitable to treat the case of symmetry breaking solutions. The…

On the Low-Energy Ramifications and a Mathematical Extension of Loop Quantum Gravity

- Mathematics, Physics
- 2004

In this thesis we address two remaining open questions in loop quantum gravity. The first deals with the low-energy limit of the theory. We illustrate some of the conceptual difficulties and their…