Editing Quantum geometry

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

In [[quantum gravity]], '''quantum geometry''' is the set of mathematical concepts that generalize [[geometry]] to describe physical phenomena at distance scales comparable to the [[Planck length]]. Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion.

[[String theory]] uses it to describe exotic phenomena such as [[T-duality]] and other geometric dualities, [[mirror symmetry (string theory)|mirror symmetry]], [[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 [[metric tensor]], such as the worldsheet [[instanton]]s. For example, the quantum volume of a cycle is computed from the mass of a [[Membrane (M-theory)|brane]] wrapped on this cycle.{{cn|date=May 2025}}

In an alternative approach to quantum gravity called [[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 well-defined operators on a [[Hilbert space]]. In particular, certain physical [[observable]]s, such as the area, have a [[discrete spectrum (physics)|discrete spectrum]]. LQG is [[non-commutative geometry|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 is consistent with the quantum picture of geometry arising from string theory.{{cn|date=May 2025}}

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

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

==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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{{Quantum mechanics topics|state=expanded}}
[[Category:Quantum gravity]]
[[Category:Quantum mechanics]]
[[Category:Mathematical physics]]