View source for Physics:Quantum geometry

{{short description|Set of mathematical concepts propagating geometric concepts}}
{{Quantum mechanics}}

In [[Physics:Theoretical physics|theoretical physics]], '''quantum geometry''' is the set of mathematical concepts generalizing the concepts of [[Geometry|geometry]] whose understanding is necessary to describe the physical phenomena at distance scales comparable to the [[Planck length]]. At these distances, [[Physics:Quantum mechanics|quantum mechanics]] has a profound effect on physical phenomena.

==Quantum gravity==

{{Main|Physics:Quantum gravity}}

Each theory of [[Physics:Quantum gravity|quantum gravity]] uses the term "quantum geometry" in a slightly different fashion. [[Physics:String theory|String theory]], a leading candidate for a quantum theory of gravity, uses the term quantum geometry to describe exotic phenomena such as [[Physics:T-duality|T-duality]] and other geometric dualities, [[Mirror symmetry (string theory)|mirror symmetry]], [[Topology|topology]]-changing transitions{{clarify|date=May 2016}}, 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 [[Physics:Metric tensor|metric tensor]], such as the worldsheet [[Physics:Instanton|instanton]]s. 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 [[Physics:Loop quantum gravity|loop quantum gravity]] (LQG), the phrase "quantum geometry" usually refers to the [[Scientific formalism|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 [[Physics:Observable|observable]]s, such as the area, have a discrete spectrum. It has also been shown that the loop quantum geometry is non-commutative.<ref>{{citation
 | last1 = Ashtekar | first1 = Abhay
 | last2 = Corichi | first2 = Alejandro
 | last3 = Zapata | first3 = José A.
 | doi = 10.1088/0264-9381/15/10/006
 | issue = 10
 | journal = Classical and Quantum Gravity
 | mr = 1662415
 | pages = 2955–2972
 | title = Quantum theory of geometry. III. Non-commutativity of Riemannian structures
 | volume = 15
 | year = 1998|arxiv = gr-qc/9806041 |bibcode = 1998CQGra..15.2955A | s2cid = 250895945
 }}.</ref>

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:<ref>''The Road to Reality'', Roger Penrose, Vintage books, 2007, {{ISBN|0-679-77631-1}}</ref>

:<math>|\psi\rangle = \int \psi(\mathbf{x},t) \, |\mathbf{x},t\rangle \, \mathrm{d}^3\mathbf{x} </math>

where the position vector is

:<math>\mathbf{x} = (x^1,x^2,x^3) </math>

the differential [[Volume element|volume element]] is

:<math>\mathrm{d}^3\mathbf{x} = \mathrm{d}x^1 \!\wedge \mathrm{d}x^2 \!\wedge \mathrm{d}x^3</math>

and {{math|''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup>}} are an arbitrary set of coordinates, the upper [[Index notation|indices]] indicate [[Covariance and contravariance of vectors|contravariance]], lower indices indicate [[Covariance and contravariance of vectors|covariance]], so explicitly the quantum state in differential form is:

:<math>|\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</math>

The overlap integral is given by:

:<math>\langle\chi|\psi\rangle = \int \chi^* \psi ~ \mathrm{d}^3\mathbf{x}</math>

in differential form this is

:<math>\langle\chi|\psi\rangle = \int \chi^* \psi ~ \mathrm{d}x^1 \!\wedge \mathrm{d}x^2 \!\wedge \mathrm{d}x^3</math>

The probability of finding the particle in some region of space {{math|''R''}} is given by the integral over that region:

:<math>\langle\psi|\psi\rangle = \int_R \psi^* \psi ~ \mathrm{d}x^1 \!\wedge \mathrm{d}x^2 \!\wedge \mathrm{d}x^3</math>

provided the wave function is [[Wave function|normalized]]. When {{math|''R''}} is all of 3d position space, the integral must be {{math|1}} if the particle exists.

Differential forms are an approach for describing the geometry of curves and [[Surface (mathematics)|surface]]s in a coordinate independent way. In [[Physics:Quantum mechanics|quantum mechanics]], idealized situations occur in rectangular Cartesian coordinates, such as the [[Physics:Potential well|potential well]], [[Physics:Particle in a box|particle in a box]], [[Physics:Quantum harmonic oscillator|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.

==See also==
* [[Noncommutative geometry]]

==References==
{{reflist}}

==Further reading==

* ''Supersymmetry'', Demystified, P. Labelle, McGraw-Hill (USA), 2010, {{ISBN|978-0-07-163641-4}}
* ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{ISBN|9780131461000}}
* ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, {{ISBN|0-07-145546 9}}
* ''Quantum Field Theory'', D. McMahon, Mc Graw Hill (USA), 2008, {{ISBN|978-0-07-154382-8}}

==External links==
*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/spaceandtime.pdf Space and Time: From Antiquity to Einstein and Beyond]
*[http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf Quantum Geometry and its Applications]

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