Physics:Quantum topology - Revision history

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	== Overview ==		== Overview ==
	In ordinary geometry, two shapes may differ by distances, angles, or curvature. In topology, the emphasis is instead on properties that remain unchanged under continuous deformation. Quantum systems can possess similar robust features, where a phase, winding number, knot structure, band invariant, or boundary state cannot be removed without changing the underlying quantum state in a discontinuous way.<ref name="nobel2016" />		In ordinary geometry, two shapes may differ by distances, angles, or curvature. In topology, the emphasis is instead on properties that remain unchanged under continuous deformation. Quantum systems can possess similar robust features, where a phase, winding number, knot structure, band invariant, or boundary state cannot be removed without changing the underlying quantum state in a discontinuous way.<ref name="nobel2016">{{cite web \|title=The Nobel Prize in Physics 2016 \|url=https://www.nobelprize.org/prizes/physics/2016/summary/ \|website=NobelPrize.org \|publisher=Nobel Prize Outreach \|access-date=7 May 2026}}</ref>

	This makes topology especially important in [[Physics:Quantum mechanics\|quantum mechanics]], [[Physics:Quantum field theory\|quantum field theory]], and [[Physics:Quantum materials\|quantum materials]]. In such systems, physical effects may depend less on local details and more on the global structure of the quantum state.<ref name="schirber2016">{{cite journal \|last=Schirber \|first=Michael \|title=Nobel Prize—Topological Phases of Matter \|journal=Physics \|volume=9 \|pages=116 \|date=7 October 2016 \|doi=10.1103/Physics.9.116}}</ref>		This makes topology especially important in [[Physics:Quantum mechanics\|quantum mechanics]], [[Physics:Quantum field theory\|quantum field theory]], and [[Physics:Quantum materials\|quantum materials]]. In such systems, physical effects may depend less on local details and more on the global structure of the quantum state.<ref name="schirber2016">{{cite journal \|last=Schirber \|first=Michael \|title=Nobel Prize—Topological Phases of Matter \|journal=Physics \|volume=9 \|pages=116 \|date=7 October 2016 \|doi=10.1103/Physics.9.116}}</ref>
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	=References=		= References =
	{{reflist\|3}}		{{reflist\|3}}

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	'''topology''' ~~is a Book I topic in the Quantum Collection. Quantum~~ topology refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states. Quantum topology refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states. In ordinary geometry, two shapes may differ by distances, angles, or curvature.		'''topology''' quantum topology refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states. Quantum topology refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states. In ordinary geometry, two shapes may differ by distances, angles, or curvature.

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	'''~~Quantum~~ topology''' refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states.~~<ref name="nobel2016">{{cite web \|title=The Nobel Prize~~ in ~~Physics 2016 \|url=https://www~~.~~nobelprize~~.~~org/prizes/physics/2016/summary/ \|website=NobelPrize~~.~~org \|publisher=Nobel Prize Outreach \|access-date=7 May 2026}}</ref>~~		'''topology''' is a Book I topic in the Quantum Collection. Quantum topology refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states. Quantum topology refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states. In ordinary geometry, two shapes may differ by distances, angles, or curvature.

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	A '''topological phase''' is a quantum phase of matter distinguished by global invariants rather than by ordinary local order parameters. Such phases may have protected boundary states, unusual quasiparticles, or quantized transport properties.<ref name="nobel2016" />		A '''topological phase''' is a quantum phase of matter distinguished by global invariants rather than by ordinary local order parameters. Such phases may have protected boundary states, unusual quasiparticles, or quantized transport properties.<ref name="nobel2016" />

	Topological phases occur in systems such as quantum Hall states, topological insulators, topological superconductors, and certain strongly correlated many-body systems. Their robustness makes them important in studies of ~~[[Physics:Quantum information and computing\|~~quantum information]] and fault-tolerant quantum computation.<ref name="nayak2008">{{cite journal \|last1=Nayak \|first1=Chetan \|last2=Simon \|first2=Steven H. \|last3=Stern \|first3=Ady \|last4=Freedman \|first4=Michael \|last5=Das Sarma \|first5=Sankar \|title=Non-Abelian anyons and topological quantum computation \|journal=Reviews of Modern Physics \|volume=80 \|issue=3 \|pages=1083–1159 \|year=2008 \|doi=10.1103/RevModPhys.80.1083}}</ref>		Topological phases occur in systems such as quantum Hall states, topological insulators, topological superconductors, and certain strongly correlated many-body systems. Their robustness makes them important in studies of quantum information and fault-tolerant quantum computation.<ref name="nayak2008">{{cite journal \|last1=Nayak \|first1=Chetan \|last2=Simon \|first2=Steven H. \|last3=Stern \|first3=Ady \|last4=Freedman \|first4=Michael \|last5=Das Sarma \|first5=Sankar \|title=Non-Abelian anyons and topological quantum computation \|journal=Reviews of Modern Physics \|volume=80 \|issue=3 \|pages=1083–1159 \|year=2008 \|doi=10.1103/RevModPhys.80.1083}}</ref>

	== Wave functions and phase ==		== Wave functions and phase ==

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

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