“A proven connection exists between classical physics and quantum physics.”

The purpose of this research is to establish a theoretical formalism concerning the classical limit of quantum mechanics for damped driven oscillatory systems. I have developed a quantum formalism on the basis of a linear invariant theorem, which gives an exact quantum–classical correspondence for damped oscillatory systems perturbed by an arbitrary force. Within my formalism, the quantum trajectory and expectation values of quantum observables precisely coincide with their classical counterparts.

Our results suggest that classical and quantum mechanics are dynamically unified at a fundamental level, with the key difference residing not in the laws of motion, but in the algebraic structure of observables and the interpretation of physical quantities. In this work, we have demonstrated that the dynamical equations governing classical and quantum mechanics possess a deeper and more exact correspondence than is usually emphasized. By focusing on the Heisenberg picture, where observable quantities evolve in time, we showed that the equations of motion for quantum operators can be written in a form identical to Newton's equations of classical mechanics.

The correspondence principle states that classical physics is not a separate paradigm but an oversimplified version of something that exists on a deeper level: quantum reality. Within certain bounds, there is a guarantee between classical mechanics and quantum mechanics, where the two systems predict the same outcomes. The correspondence principle states that classical mechanics emerges from quantum mechanics in the appropriate limits.

For a quantum observable A_ℏ depending on a parameter ℏ we define the notion 'A_ℏ converges in the classical limit'. The limit is a function on phase space. For a large class of convergent Hamiltonians the ℏ-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics.

Jonathan Oppenheim at University College London has developed a new theoretical framework that aims to unify quantum mechanics and classical gravity – without the need for a theory of quantum gravity. Oppenheim's approach allows gravity to remain classical, while coupling it to the quantum world by a stochastic (random) mechanism. This allowed Oppenheim to derive an equation that describes the coupling between quantum mechanics and classical gravity, while preserving each of their unique characteristics.

The classical limit describes the emergence of classical physics from quantum theory. In the classical limit of quantum mechanics we would like to see: A compact probability distribution (a wavepacket); A 'small' value of Delta x; A 'small' value of Delta p; Time evolution such that <x(t)> and <p(t)> follow the classical x(t) and p(t).

Recently, a number of detailed examinations of the structure of classical light beams have revealed that effects widely thought to be solely quantum in origin also have a place in classical optics. These new quantum-classical connections are informing classical optics in meaningful ways specifically by expanding our understanding of optical coherence. Interference, polarization, coherence, complementarity and entanglement are a partial list of elementary notions that now appear to belong to both quantum and classical optics.

Contrary to the widespread belief, the problem of the emergence of classical mechanics from quantum mechanics is still open. In spite of many results on the ℏ → 0 asymptotics, it is not yet clear how to explain within standard quantum mechanics the classical motion of macroscopic bodies.

The Correspondence Principle implies that any new theory in physics must reduce to preceding theories that have been proven to be valid. Bohr's Correspondence Principle requires that the predictions of quantum mechanics must reproduce the predictions of classical physics in the limit of large quantum numbers. The Correspondence Principle now is used to project out the analogous classical-mechanics phenomena that underlie the observed properties of quantal systems.

In standard quantum mechanics only the wave function exists and to answer the question of how the classical world emerges from the quantum world... Due to the fact that there are no trajectories, a spread out wave function can in no way show a classical limit in standard quantum mechanics.

Classical mechanics and quantum mechanics are connected through a hamiltonian formulation of classical dynamics known as the "Poisson Bracket" formulation. Dirac first noted that quantum mechanical relations can be obtained from classical ones by simply replacing the classical poisson bracket in the classical variables by a corresponding quantum bracket multiplied by -i2pi/h. The analogous form of the equations is due to the fact that the poisson bracket and quantum mechanical commutator are both examples of Lie algebras, thus there are deep fundamental connections between classical and quantum mechanics that go beyond coincidence.

Even though nature seems to abide to the rules of quantum mechanics, classical mechanics has (almost) always given reasonable estimates. Therefore, if one gets rid of the assumption of quantization (ℏ −→0) introduced in quantum mechanics, one hopes to retrieve classical results or at least something close to it.

Classical physics and quantum physics each describe different layers of reality. Classical physics provides clear and predictable rules for the macroscopic world, while quantum physics opens the door to the mysterious and probabilistic nature of the microscopic world. Together, they form a more complete picture of how the universe works.

Ehrenfest's theorem states that the expectation values of position and momentum in quantum mechanics obey classical equations of motion, providing a direct connection between quantum and classical mechanics in the limit of narrow wave packets or macroscopic systems.

Most physicists believe that quantum physics is the right theory, even though many details are yet to be worked out. Classical physics can be derived from quantum physics in the limit that the quantum properties are hidden. That fact is called the "correspondence principle."

Most physicists and physics students understand the correspondence principle as the requirement that the results of quantum physics go over to those of classical physics in some appropriate limit, say of high quantum numbers, or of large numbers of quanta, or as Planck's constant h goes to zero.

The transition from classical to quantum is characterized by dropping the commutative law for the multiplication of physical quantities. Quantum mechanics is the noncommutative version of classical Hamiltonian mechanics.