Generalized uncertainty principle

{{Short description|Physics generalization}}

The generalized uncertainty principle (GUP) is a proposed extension of the Heisenberg uncertainty principle that incorporates potential effects of gravitational interactions into quantum mechanical systems. It emerges from several approaches to quantum gravity, including string theory, loop quantum gravity, and quantum geometry, and suggests the existence of a minimum measurable length, typically associated with the Planck scale.

A commonly used formulation of the GUP is:

:$\backslash Delta\; x\; \backslash Delta\; p\; \backslash geq\; \backslash frac\{\backslash hbar\}\{2\}\; +\; \backslash beta\; \backslash Delta\; p^2$,

where $\backslash Delta\; x$ and $\backslash Delta\; p$ represent the uncertainties in position and momentum, $\backslash hbar$ is the reduced Planck constant, and $\backslash beta$ is a parameter related to the minimal length scale. This modification implies that position measurements cannot be made with arbitrary precision, as there exists a fundamental lower bound to spatial resolution. The concept is motivated by the expectation that classical notions of spacetime may break down at extremely small scales, such as the Planck length.{{cite journal | arxiv=1203.6191 | doi=10.12942/lrr-2013-2 | title=Minimal Length Scale Scenarios for Quantum Gravity | year=2013 | last1=Hossenfelder | first1=Sabine | journal=Living Reviews in Relativity | volume=16 | issue=1 | page=2 | doi-access=free | pmid=28179841 | pmc=5255898 | bibcode=2013LRR....16....2H }}{{cite journal | doi=10.1142/S0217732399001462 | title=On Gravity and the Uncertainty Principle | year=1999 | last1=Adler | first1=Ronald J. | last2=Santiago | first2=David I. | journal=Modern Physics Letters A | volume=14 | issue=20 | pages=1371–1381 | arxiv=gr-qc/9904026 | bibcode=1999MPLA...14.1371A | s2cid=23960215 }}

Several forms of the GUP have been proposed in the literature, varying in mathematical structure and underlying theoretical assumptions, depending on the specific model of quantum gravity being considered.{{cite journal | last1=Snyder | first1=Hartland S. | doi=10.1103/PhysRev.71.38 | title=Quantized Space-Time | journal=Phys. Rev. | volume=71 | pages=38–41 | date=1 January 1947 | issue=1 | bibcode=1947PhRv...71...38S }}{{cite journal | last1=Amati | first1=D. | last2=Ciafaloni | first2=M. | last3=Veneziano | first3=G. | doi=10.1016/0370-2693(89)91366-X | title=Can spacetime be probed below the string size? | year=1989 | journal=Physics Letters B | volume=216 | issue=1–2 | pages=41–47 | bibcode=1989PhLB..216...41A | url=http://cds.cern.ch/record/191788/files/198811126.pdf }}{{cite journal | last1=Garay | first1=L. J. | doi=10.1142/S0217751X95000085 | title=Quantum gravity and minimum length | year=1995 | journal=International Journal of Modern Physics A | volume=10 | issue=2 | pages=145–166 | arxiv=gr-qc/9403008 | bibcode=1995IJMPA..10..145G }}{{cite journal | last1=Scardigli | first1=F. | doi=10.1016/S0370-2693(99)00167-7 | title=Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment | year=1999 | journal=Physics Letters B | volume=452 | issue=1–2 | pages=39–44 | arxiv=hep-th/9904025 | bibcode=1999PhLB..452...39S }}{{cite journal | last1=Brau | first1=F. | doi=10.1088/0305-4470/32/44/308 | title=Minimal Length Uncertainty Relation and Hydrogen Atom | year=1999 | journal=Journal of Physics A: Mathematical and General | volume=32 | issue=44 | pages=7691–7696 | arxiv=quant-ph/9905033 | bibcode=1999JPhA...32.7691B }}{{cite journal | last1=Kempf | first1=A. | last2=Mangano | first2=G. | last3=Mann | first3=R. B. | doi=10.1103/PhysRevD.52.1108| title=Hilbert space representation of the minimal length uncertainty relation | year=1995 | journal=Physical Review D | volume=52 | issue=2 | pages=1108–1118 | pmid=10019328 | arxiv=hep-th/9412167 | bibcode=1995PhRvD..52.1108K }}{{cite journal | last1=Maggiore | first1=M.| doi=10.1016/0370-2693(93)91401-8 | title=A Generalized Uncertainty Principle in Quantum Gravity | year=1993 | journal=Physics Letters B | volume=304 | issue=1–2 | pages=65–69 | arxiv=hep-th/9301067 | bibcode=1993PhLB..304...65M}}

{{cite journal | last1=Capozziello | first1=S. | last2=Lambiase | first2=G. | last3=Scarpetta | first3=G. | doi=10.1023/A:1003634814685 | title=The Generalized Uncertainty Principle from Quantum Geometry | year=1999 | journal=International Journal of Theoretical Physics | volume=39 | issue=1 | pages=15–22 }}{{cite journal | last1=Todorinov | first1=V. | last2=Bosso | first2=P. | last3=Das | first3=S. | doi=10.1016/j.aop.2019.03.014 | title=Relativistic Generalized Uncertainty Principle | year=2019 | journal=Annals of Physics | volume=405 | pages=92–100 | arxiv=1810.11761 | bibcode=2019AnPhy.405...92T }}{{cite journal | last1=Ali | first1=A. F. | last2=Das | first2=S. | last3=Vagenas | first3=E. C. | doi=10.1016/j.physletb.2009.06.061 | title=Discreteness of Space from the Generalized Uncertainty Principle | year=2009 | journal=Physics Letters B | volume=678 | issue=5 | pages=497–499 | arxiv=0906.5396 | bibcode=2009PhLB..678..497A }}