Quantum geometry

Intheoretical physics,quantum geometryis the set of mathematical concepts generalizing the concepts ofgeometrywhose understanding is necessary to describe the physical phenomena at distance scales comparable to thePlanck length.At these distances,quantum mechanicshas a profound effect on physical phenomena.

Quantum gravity

Main article:quantum gravity

Each theory ofquantum gravityuses 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 asT-dualityand othergeometric dualities,mirror symmetry,topology-changing transitions,minimal possible distance scale, and other effects that challenge intuition. More technically, quantum geometry refers to the shape of aspacetime manifoldas experienced byD-braneswhich includes quantum corrections to themetric tensor,such as the worldsheetinstantons.For example, the quantum volume of a cycle is computed from the mass of abranewrapped on this cycle.

In an alternative approach to quantum gravity calledloop quantum gravity(LQG), the phrase "quantum geometry" usually refers to theformalismwithin LQG where the observables that capture the information about the geometry are now well defined operators on aHilbert space.In particular, certain physicalobservables,such as the area, have adiscrete spectrum.It has also been shown that the loop quantum geometry isnon-commutative.

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" isDiscrete Lorentzian quantum gravity.

Quantum states as differential forms

Main article:Wavefunction

See also:Differential forms

Differential formsare used to expressquantum states,using thewedge product:

$|\psi \rangle =\int \psi (\mathbf {x},t)\,|\mathbf {x},t\rangle \,\mathrm {d} ^{3}\mathbf {x}$

where theposition vectoris

$\mathbf {x} =(x^{1},x^{2},x^{3})$

the differentialvolume elementis

$\mathrm {d} ^{3}\mathbf {x} =\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}$

andx¹,x²,x³are an arbitrary set of coordinates, the upperindicesindicatecontravariance,lower indices indicatecovariance,so explicitly the quantum state in differential form is:

$|\psi \rangle =\int \psi (x^{1},x^{2},x^{3},t)\,|x^{1},x^{2},x^{3},t\rangle \,\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}$

The overlap integral is given by:

$\langle \chi |\psi \rangle =\int \chi ^{*}\psi ~\mathrm {d} ^{3}\mathbf {x}$

in differential form this is

$\langle \chi |\psi \rangle =\int \chi ^{*}\psi ~\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}$

The probability of finding the particle in some region of spaceRis given by the integral over that region:

$\langle \psi |\psi \rangle =\int _{R}\psi ^{*}\psi ~\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}$

provided the wave function isnormalized.WhenRis all of 3d position space, the integral must be1if the particle exists.

Differential forms are an approach for describing the geometry ofcurvesandsurfacesin a coordinate independent way. Inquantum mechanics,idealized situations occur in rectangularCartesian coordinates,such as thepotential well,particle in a box,quantum harmonic oscillator,and more realistic approximations inspherical polar coordinatessuch aselectronsinatomsandmolecules.For generality, a formalism which can be used in any coordinate system is useful.

See also

• Noncommutative geometry
• Quantum spacetime

Further reading

• Supersymmetry,Demystified, P. Labelle, McGraw-Hill (USA), 2010,ISBN978-0-07-163641-4
• Quantum Mechanics,E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,ISBN9780131461000
• Quantum Mechanics Demystified,D. McMahon, Mc Graw Hill (USA), 2006,ISBN0-07-145546 9
• Quantum Field Theory,D. McMahon, Mc Graw Hill (USA), 2008,ISBN978-0-07-154382-8

External links

• Space and Time: From Antiquity to Einstein and Beyond
• Quantum Geometry and its Applications