String theory, a leading candidate for a quantum theory of gravity, uses it to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions[clarification needed], minimal possible distance scale, and other effects that challenge intuition.
Another approach, which tries to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity.
Differential forms are used to express quantum states, using the wedge product:[2] where the position vector is the differential volume element is and x1, x2, x3 are an arbitrary set of coordinates, the upper indices indicate contravariance, lower indices indicate covariance, so explicitly the quantum state in differential form is: The overlap integral is given by: in differential form this is The probability of finding the particle in some region of space R is given by the integral over that region: provided the wave function is normalized.
Differential forms are an approach for describing the geometry of curves and surfaces in a coordinate independent way.
In quantum mechanics, idealized situations occur in rectangular Cartesian coordinates, such as the potential well, particle in a box, quantum harmonic oscillator, and more realistic approximations in spherical polar coordinates such as electrons in atoms and molecules.