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Quantum geometry is a Book I topic in the Quantum Collection. It describes approaches in which geometric ideas such as distance, area, curvature, topology, and spacetime structure are modified by quantum theory. The topic appears in quantum gravity, noncommutative geometry, geometric phases, coherent states, and the study of configuration spaces. It is important because many quantum systems are controlled not only by forces but also by the shape of the underlying state space and the global structure of allowed transformations.

geometry in the Quantum Collection.

Quantum gravity

Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion. String theory, a leading candidate for a quantum theory of gravity, uses the term quantum geometry 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. More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes which includes quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle.

In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase "quantum geometry" usually refers to the formalism within LQG where the observables that capture the information about the geometry are now well defined operators on a Hilbert space. In particular, certain physical observables, such as the area, have a discrete spectrum. It has also been shown that the loop quantum geometry is non-commutative.^[1]

It is possible (but considered unlikely) that this strictly quantized understanding of geometry will be consistent with the quantum picture of geometry arising from string theory.

Another, quite successful, approach, which tries to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity.

Quantum states as differential forms

Differential forms are used to express quantum states, using the wedge product:^[2]

| ψ ⟩ = \int ψ (𝐱, t) | 𝐱, t ⟩ d^{3} 𝐱

where the position vector is

𝐱 = (x^{1}, x^{2}, x^{3})

the differential volume element is

d^{3} 𝐱 = d x^{1} \land d x^{2} \land d x^{3}

and $x 1, x 2, x 3$ are an arbitrary set of coordinates, the upper indices indicate contravariance, lower indices indicate covariance, so explicitly the quantum state in differential form is:

| ψ ⟩ = \int ψ (x^{1}, x^{2}, x^{3}, t) | x^{1}, x^{2}, x^{3}, t ⟩ d x^{1} \land d x^{2} \land d x^{3}

The overlap integral is given by:

⟨ χ | ψ ⟩ = \int χ^{*} ψ d^{3} 𝐱

in differential form this is

⟨ χ | ψ ⟩ = \int χ^{*} ψ d x^{1} \land d x^{2} \land d x^{3}

The probability of finding the particle in some region of space $R$ is given by the integral over that region:

⟨ ψ | ψ ⟩ = \int_{R} ψ^{*} ψ d x^{1} \land d x^{2} \land d x^{3}

provided the wave function is normalized. When $R$ is all of 3d position space, the integral must be $1$ if the particle exists.

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. For generality, a formalism which can be used in any coordinate system is useful.

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Physics:Quantum geometry

Quantum gravity

Quantum states as differential forms

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