Classical analog of the quantum metric tensor

Area of Science:

• Classical Mechanics
• Quantum Mechanics
• Mathematical Physics

Background:

• The quantum metric tensor quantifies the distance between quantum states.
• Classical integrable systems evolve under slowly varying parameters.
• Understanding parameter space geometry is crucial in physics.

Purpose of the Study:

• To introduce and define a classical analog of the quantum metric tensor.
• To explore the properties and applications of this classical metric.
• To propose alternative formulations for quantum metric tensor and Berry's connection.

Main Methods:

• Definition of a classical metric for adiabatic evolution in classical integrable systems.
• Calculation of metric components for specific systems like the generalized harmonic oscillator.
• Perturbative calculations for the quartic anharmonic oscillator.

Main Results:

• A classical metric analogous to the quantum metric tensor is established.
• The metric's components are calculated exactly for specific oscillator models.
• The metric provides a way to measure distances in the parameter space of classical systems.

Conclusions:

• The proposed classical metric offers a new tool for analyzing adiabatic processes in classical systems.
• This work bridges classical and quantum descriptions of parameter space geometry.
• Alternative operator-based expressions for quantum metric tensor and Berry's connection are suggested.