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</ul> </li> <li id="toc-q-Deformed_spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#q-Deformed_spacetime"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span><i>q</i>-Deformed spacetime</span> </div> </a> <ul id="toc-q-Deformed_spacetime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fuzzy_or_spin_model_spacetime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fuzzy_or_spin_model_spacetime"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Fuzzy or spin model spacetime</span> </div> </a> <ul id="toc-Fuzzy_or_spin_model_spacetime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heisenberg_model_spacetimes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heisenberg_model_spacetimes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Heisenberg model spacetimes</span> </div> 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data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">October 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>, the concept of <b>quantum spacetime</b> is a generalization of the usual concept of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> in which some variables that ordinarily <a href="/wiki/Commutative_property" title="Commutative property">commute</a> are assumed not to commute and form a different <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>. The choice of that algebra varies from one theory to another. As a result of this change, some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies. </p><p>The idea of quantum spacetime was proposed in the early days of quantum theory by <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a> and <a href="/wiki/Dmitri_Ivanenko" title="Dmitri Ivanenko">Ivanenko</a> as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to <a href="/wiki/Rudolf_Peierls" title="Rudolf Peierls">Rudolf Peierls</a>, who noted that electrons in a magnetic field can be regarded as moving in a quantum spacetime, and to <a href="/wiki/Robert_Oppenheimer" class="mw-redirect" title="Robert Oppenheimer">Robert Oppenheimer</a>, who carried it to <a href="/wiki/Hartland_Snyder" title="Hartland Snyder">Hartland Snyder</a>, who published the first concrete example.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Snyder's <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> was made simple by <a href="/wiki/C._N._Yang" class="mw-redirect" title="C. N. Yang">C. N. Yang</a> in the same year. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Physical spacetime is a quantum spacetime when in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> position and momentum variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95a7ed3d2b1822eb6a28970329fa7b428219be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.533ex; height:2.009ex;" alt="{\displaystyle x,p}"></span> are already <a href="/wiki/Noncommutative" class="mw-redirect" title="Noncommutative">noncommutative</a>, obey the <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a>, and are continuous. Because of the Heisenberg uncertainty relations, greater energy is needed to probe smaller distances. Ultimately, according to gravity theory, the probing particles form <a href="/wiki/Black_holes" class="mw-redirect" title="Black holes">black holes</a> that destroy what was to be measured. The process cannot be repeated, so it cannot be considered to be a measurement. This limited measurability led many to expect that the usual picture of continuous commutative spacetime breaks down at <a href="/wiki/Planck_scale" class="mw-redirect" title="Planck scale">Planck scale</a> distances, if not sooner. </p><p>Physical spacetime is expected to be quantum because physical coordinates are slightly noncommutative. The astronomical coordinates of a star are modified by gravitational fields between the observer and the star, as in the deflection of light by the sun, one of the classic tests of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. Therefore, the coordinates actually depend on gravitational field variables. According to quantum theories of gravity, these field variables do not commute; therefore coordinates that depend on them likely do not commute. </p><p>Both arguments are based on pure gravity and quantum theory, and they limit the measurement of time by the only time constant in pure <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>, the <a href="/wiki/Planck_time" class="mw-redirect" title="Planck time">Planck time</a>. Instruments, however, are not purely gravitational but are made of particles. They may set a more severe, larger, limit than the Planck time. </p> <div class="mw-heading mw-heading2"><h2 id="Criteria">Criteria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=2" title="Edit section: Criteria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum spacetimes are often described mathematically using the <a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a> of Connes, <a href="/wiki/Quantum_geometry" title="Quantum geometry">quantum geometry</a>, or <a href="/wiki/Quantum_groups" class="mw-redirect" title="Quantum groups">quantum groups</a>. </p><p>Any noncommutative algebra with at least four generators could be interpreted as a quantum spacetime, but the following desiderata have been suggested: </p> <ul><li>Local <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> and <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> symmetries should be retained, possibly in a generalised form. Their generalisation often takes the form of a <a href="/wiki/Quantum_group" title="Quantum group">quantum group</a> acting on the quantum spacetime algebra.</li> <li>The algebra might plausibly arise in an effective description of quantum gravity effects in some regime of that theory. For example, a physical parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, perhaps the <a href="/wiki/Planck_length" class="mw-redirect" title="Planck length">Planck length</a>, might control the deviation from commutative classical spacetime, so that ordinary Lorentzian spacetime arises as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c36a80bb2503b9d3f5abf6c3b227fc214e23daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.132ex; height:2.176ex;" alt="{\displaystyle \lambda \to 0}"></span>.</li> <li>There might be a notion of <a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">quantum differential calculus</a> on the quantum spacetime algebra, compatible with the (quantum) symmetry and preferably reducing to the usual differential calculus as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c36a80bb2503b9d3f5abf6c3b227fc214e23daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.132ex; height:2.176ex;" alt="{\displaystyle \lambda \to 0}"></span>.</li></ul> <p>This would permit wave equations for particles and fields and facilitate predictions for experimental deviations from classical spacetime physics that can then be tested experimentally. </p> <ul><li>The Lie algebra should be <a href="/wiki/Semisimple" class="mw-redirect" title="Semisimple">semisimple</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This makes it easier to formulate a finite theory.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Models">Models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=3" title="Edit section: Models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several models were found in the 1990s more or less meeting most of the above criteria. </p> <div class="mw-heading mw-heading3"><h3 id="Bicrossproduct_model_spacetime">Bicrossproduct model spacetime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=4" title="Edit section: Bicrossproduct model spacetime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bicrossproduct model spacetime was introduced by <a href="/wiki/Shahn_Majid" title="Shahn Majid">Shahn Majid</a> and Henri Ruegg<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and has Lie algebra relations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{i},x_{j}]=0,\quad [x_{i},t]=i\lambda x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{i},x_{j}]=0,\quad [x_{i},t]=i\lambda x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588bc2270f33d7269df9442e97a158030bcb51dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.996ex; height:3.009ex;" alt="{\displaystyle [x_{i},x_{j}]=0,\quad [x_{i},t]=i\lambda x_{i}}"></span></dd></dl> <p>for the spatial variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> and the time variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> has dimensions of time and is therefore expected to be something like the Planck time. The Poincaré group here is correspondingly deformed, now to a certain <a href="/wiki/Bicrossproduct" class="mw-redirect" title="Bicrossproduct">bicrossproduct</a> quantum group with the following characteristic features. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2dhyp.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2dhyp.png/260px-2dhyp.png" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/2dhyp.png/390px-2dhyp.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f7/2dhyp.png 2x" data-file-width="462" data-file-height="462" /></a><figcaption>Orbits for the action of the Lorentz group on momentum space in the construction of the bicrossproduct model in units of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e339fe3b3bd9b9883154cd2d5363f3ecf4ef7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:2.676ex;" alt="{\displaystyle \lambda ^{-1}}"></span>. Mass-shell hyperboloids are 'squashed' into a cylinder.</figcaption></figure> <p>The momentum generators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> commute among themselves but addition of momenta, reflected in the quantum group structure, is deformed (momentum space becomes a <a href="/wiki/Non-abelian_group" title="Non-abelian group">non-abelian group</a>). Meanwhile, the Lorentz group generators enjoy their usual relations among themselves but act non-linearly on the momentum space. The orbits for this action are depicted in the figure as a cross-section of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span> against one of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span>. The on-shell region describing particles in the upper center of the image would normally be hyperboloids but these are now 'squashed' into the cylinder </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}}<\lambda ^{-1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo><</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}}<\lambda ^{-1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b92173cb3a3a9cb4262d86c9561bb54006d8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.849ex; height:4.843ex;" alt="{\displaystyle {\sqrt {p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}}<\lambda ^{-1}\,}"></span></dd></dl> <p>in simplified units. The upshot is that Lorentz-boosting a momentum will never increase it above the Planck momentum. The existence of a highest momentum scale or lowest distance scale fits the physical picture. This squashing comes from the non-linearity of the Lorentz boost and is an endemic feature of bicrossproduct quantum groups known since their introduction in 1988.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Some physicists dub the bicrossproduct model <a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">doubly special relativity</a>, since it sets an upper limit to both speed and momentum. </p><p>Another consequence of the squashing is that the propagation of particles is deformed, even of light, leading to a <a href="/wiki/Variable_speed_of_light" title="Variable speed of light">variable speed of light</a>. This prediction requires the particular <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0},p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0},p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71802c7c7ee8e3b5eccedb5c2fa1fea4308f0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.316ex; height:2.009ex;" alt="{\displaystyle p_{0},p_{i}}"></span> to be the physical energy and spatial momentum (as opposed to some other function of them). Arguments for this identification were provided in 1999 by <a href="/wiki/Giovanni_Amelino-Camelia" title="Giovanni Amelino-Camelia">Giovanni Amelino-Camelia</a> and Majid<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> through a study of plane waves for a quantum differential calculus in the model. They take the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\sum _{i}p_{i}x_{i}}e^{ip_{0}t}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\sum _{i}p_{i}x_{i}}e^{ip_{0}t}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5545c835a0adc9ff909af2d631c52eacd6222542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.557ex; height:2.843ex;" alt="{\displaystyle e^{i\sum _{i}p_{i}x_{i}}e^{ip_{0}t}\,}"></span></dd></dl> <p>In other words, a form which is sufficiently close to classical that one might plausibly believe the interpretation. At the moment, such wave analysis represents the best hope to obtain physically testable predictions from the model. </p><p>Prior to this work there were a number of unsupported claims to make predictions from the model based solely on the form of the Poincaré quantum group. There were also claims based on an earlier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span>-Poincaré quantum group introduced by Jurek Lukierski and co-workers<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> which were important precursors to the bicrossproduct, albeit without the actual quantum spacetime and with different proposed generators for which the above picture does not apply. The bicrossproduct model spacetime has also been called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span>-deformed spacetime with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa =\lambda ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =\lambda ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac48475164922d0306b55fc2623d3a0cd11b400a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.125ex; height:2.676ex;" alt="{\displaystyle \kappa =\lambda ^{-1}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="q-Deformed_spacetime"><i>q</i>-Deformed spacetime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=5" title="Edit section: q-Deformed spacetime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This model was introduced independently by a team<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> working under <a href="/wiki/Julius_Wess" title="Julius Wess">Julius Wess</a> in 1990 and by <a href="/wiki/Shahn_Majid" title="Shahn Majid">Shahn Majid</a> and coworkers in a series of papers on braided matrices starting a year later.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The point of view in the second approach is that usual Minkowski spacetime has a description via <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> as the space of 2 x 2 hermitian matrices. In quantum group theory and using <a href="/wiki/Braided_monoidal_category" title="Braided monoidal category">braided monoidal category</a> methods, a natural q-version of this is defined here for real values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> as a 'braided hermitian matrix' of generators and relations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}^{\dagger },\quad \beta \alpha =q^{2}\alpha \beta ,\ [\alpha ,\delta ]=0,\ [\beta ,\gamma ]=(1-q^{-2})\alpha (\delta -\alpha ),\ [\delta ,\beta ]=(1-q^{-2})\alpha \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>β</mi> <mi>α</mi> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>α</mi> <mi>β</mi> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>α</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>δ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>α</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}^{\dagger },\quad \beta \alpha =q^{2}\alpha \beta ,\ [\alpha ,\delta ]=0,\ [\beta ,\gamma ]=(1-q^{-2})\alpha (\delta -\alpha ),\ [\delta ,\beta ]=(1-q^{-2})\alpha \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b558c64a5c3018e924147ca5eb04d55761b6a9fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:96.656ex; height:6.676ex;" alt="{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}^{\dagger },\quad \beta \alpha =q^{2}\alpha \beta ,\ [\alpha ,\delta ]=0,\ [\beta ,\gamma ]=(1-q^{-2})\alpha (\delta -\alpha ),\ [\delta ,\beta ]=(1-q^{-2})\alpha \beta }"></span></dd></dl> <p>These relations say that the generators commute as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\to 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec80cb4e51140ce942dc15b9b857f6a5fb203c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.846ex; height:2.509ex;" alt="{\displaystyle q\to 1}"></span> thereby recovering usual Minkowski space. Working with more familiar variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z,t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z,t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df423d847ba8083b4882c6cffdaf9ace0bc0d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.515ex; height:2.343ex;" alt="{\displaystyle x,y,z,t}"></span> as linear combinations of these, in particular, time </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\text{Trace}}_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=q\delta +q^{-1}\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>Trace</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>q</mi> <mi>δ</mi> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\text{Trace}}_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=q\delta +q^{-1}\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2ad47d091caa1976a1824d3a0fb319a8a042bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.015ex; height:6.176ex;" alt="{\displaystyle t={\text{Trace}}_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=q\delta +q^{-1}\alpha }"></span></dd></dl> <p>is given by a natural braided trace of the matrix and commutes with the other generators (so this model is different from the bicrossproduct one). The braided-matrix picture also leads naturally to a quantity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\det }_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=\alpha \delta -q^{2}\gamma \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">det</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> </mtd> <mtd> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>α</mi> <mi>δ</mi> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>γ</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\det }_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=\alpha \delta -q^{2}\gamma \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c993c5dbc393fd975475a74b47eb08797b8a6920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.736ex; height:6.176ex;" alt="{\displaystyle {\det }_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=\alpha \delta -q^{2}\gamma \beta }"></span></dd></dl> <p>which as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\to 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec80cb4e51140ce942dc15b9b857f6a5fb203c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.846ex; height:2.509ex;" alt="{\displaystyle q\to 1}"></span> returns the usual <a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a> (this translates to a metric in the quantum differential geometry). The parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=e^{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23e9dcdee6365fae2f6ea4cbe624e3048f77bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.442ex; height:3.009ex;" alt="{\displaystyle q=e^{\lambda }}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=e^{i\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>λ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{i\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10209c83cfa678fa95fd03604cf226ed8add7871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.009ex; height:3.009ex;" alt="{\displaystyle q=e^{i\lambda }}"></span> is dimensionless and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is thought to be a ratio of the Planck scale and the cosmological length. That is, there are indications that this model relates to quantum gravity <b>with</b> a non-zero <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a>, the choice of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> depending on whether this is positive or negative. This describes the mathematically better understood but perhaps less physically justified positive case. </p><p>A full understanding of this model requires (and was concurrent with the development of) a full theory of 'braided linear algebra' for such spaces. The momentum space for the theory is another copy of the same algebra and there is a certain 'braided addition' of momentum on it expressed as the structure of a <a href="/wiki/Braided_Hopf_algebra" title="Braided Hopf algebra">braided Hopf algebra</a> or quantum group <i>in</i> a certain braided monoidal category). This theory, by 1993, had provided the corresponding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>-deformed Poincaré group as generated by such translations and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>-Lorentz transformations, completing the interpretation as a quantum spacetime.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In the process it was discovered that the Poincaré group not only had to be deformed but had to be extended to include dilations of the quantum spacetime. For such a theory to be exact, all particles in the theory need to be massless, which is consistent with experiment, as masses of elementary particles are vanishingly small compared to the <a href="/wiki/Planck_mass" class="mw-redirect" title="Planck mass">Planck mass</a>. If current thinking in cosmology is correct, then this model is more appropriate, but it is significantly more complicated and for this reason its physical predictions have yet to be worked out.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Please clarify the preceding statement or statements with a good explanation from a reliable source. (September 2017)">how?</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fuzzy_or_spin_model_spacetime">Fuzzy or spin model spacetime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=6" title="Edit section: Fuzzy or spin model spacetime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This refers in modern usage to the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> algebra </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{1},x_{2}]=2i\lambda x_{3},\ [x_{2},x_{3}]=2i\lambda x_{1},\ [x_{3},x_{1}]=2i\lambda x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> <mi>i</mi> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> <mi>i</mi> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> <mi>i</mi> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{1},x_{2}]=2i\lambda x_{3},\ [x_{2},x_{3}]=2i\lambda x_{1},\ [x_{3},x_{1}]=2i\lambda x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5214d9db230a71ac8a2ad8db8647823237ce515e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.923ex; height:2.843ex;" alt="{\displaystyle [x_{1},x_{2}]=2i\lambda x_{3},\ [x_{2},x_{3}]=2i\lambda x_{1},\ [x_{3},x_{1}]=2i\lambda x_{2}}"></span></dd></dl> <p>familiar from <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> but interpreted in this context as coordinates of a quantum space or spacetime. These relations were proposed by <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> in his earliest <a href="/wiki/Spin_network" title="Spin network">spin network</a> theory of space. It is a toy model of quantum gravity in 3 spacetime dimensions (not the physical 4) with a Euclidean (not the physical Minkowskian) signature. It was again proposed<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in this context by <a href="/wiki/Gerardus_%27t_Hooft" class="mw-redirect" title="Gerardus 't Hooft">Gerardus 't Hooft</a>. A further development including a quantum differential calculus and an action of a certain 'quantum double' quantum group as deformed Euclidean group of motions was given by Majid and E. Batista.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>A striking feature of the noncommutative geometry, is that the smallest covariant quantum differential calculus has one dimension higher than expected, namely 4, suggesting that the above can also be viewed as the spatial part of a 4-dimensional quantum spacetime. The model should not be confused with <a href="/wiki/Fuzzy_sphere" title="Fuzzy sphere">fuzzy spheres</a> which are finite-dimensional matrix algebras which can be thought of as spheres in the spin model spacetime of fixed radius. </p> <div class="mw-heading mw-heading3"><h3 id="Heisenberg_model_spacetimes">Heisenberg model spacetimes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=7" title="Edit section: Heisenberg model spacetimes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quantum spacetime of <a href="/wiki/Hartland_Snyder" title="Hartland Snyder">Hartland Snyder</a> proposes that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{\mu },x_{\nu }]=iM_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{\mu },x_{\nu }]=iM_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d817b0376a0cbd4997de4ac7368807044e528a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.564ex; height:3.009ex;" alt="{\displaystyle [x_{\mu },x_{\nu }]=iM_{\mu \nu }}"></span></dd></dl> <p>where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a692c0659d85500284eb84671066217b7a40e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.349ex; height:2.843ex;" alt="{\displaystyle M_{\mu \nu }}"></span> generate the Lorentz group. This quantum spacetime and that of <a href="/wiki/C._N._Yang" class="mw-redirect" title="C. N. Yang">C. N. Yang</a> entail a radical unification of spacetime, energy-momentum, and angular momentum. </p><p>The idea was revived in a modern context by <a href="/wiki/Sergio_Doplicher" title="Sergio Doplicher">Sergio Doplicher</a>, <a href="/wiki/Klaus_Fredenhagen" title="Klaus Fredenhagen">Klaus Fredenhagen</a> and John Roberts in 1995,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> by letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a692c0659d85500284eb84671066217b7a40e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.349ex; height:2.843ex;" alt="{\displaystyle M_{\mu \nu }}"></span> simply be viewed as some function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec9df53adf1f4825f65856a0bcdb42ed8c599bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.553ex; height:2.343ex;" alt="{\displaystyle x_{\mu }}"></span> as defined by the above relation, and any relations involving it viewed as higher order relations among the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec9df53adf1f4825f65856a0bcdb42ed8c599bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.553ex; height:2.343ex;" alt="{\displaystyle x_{\mu }}"></span>. The Lorentz symmetry is arranged so as to transform the indices as usual and without being deformed. </p><p>An even simpler variant of this model is to let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> be a numerical antisymmetric tensor, in which context it is usually denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>, so the relations are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{\mu },x_{\nu }]=i\theta _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{\mu },x_{\nu }]=i\theta _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a978efb883961e76659c3576edc99e8b7bc37d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.4ex; height:3.009ex;" alt="{\displaystyle [x_{\mu },x_{\nu }]=i\theta _{\mu \nu }}"></span>. In even dimensions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, any nondegenerate such theta can be transformed to a normal form in which this really is just the <a href="/wiki/Heisenberg_algebra" class="mw-redirect" title="Heisenberg algebra">Heisenberg algebra</a> but the difference that the variables are being proposed as those of spacetime. This proposal was once popular because of its familiar form of relations and because it has been argued<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> that it emerges from the theory of open strings landing on D-branes, see <a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">noncommutative quantum field theory</a> and <a href="/w/index.php?title=Moyal_plane&action=edit&redlink=1" class="new" title="Moyal plane (page does not exist)">Moyal plane</a>. However, this D-brane lives in some of the higher spacetime dimensions in the theory and hence it is not physical spacetime that string theory suggests to be effectively quantum in this way. It also requires subscribing to D-branes as an approach to quantum gravity in the first place. When posited as quantum spacetime, it is hard to obtain physical predictions and one reason for this is that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is a tensor, then by dimensional analysis, it should have dimensions of length<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d19a0bee8c4bfc83c25449369b10341886e6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.054ex; height:2.509ex;" alt="{\displaystyle {}^{2}}"></span>, and if this length is speculated to be the Planck length, then the effects would be harder to ever detect than for other models. </p> <div class="mw-heading mw-heading2"><h2 id="Noncommutative_extensions_to_spacetime">Noncommutative extensions to spacetime</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=8" title="Edit section: Noncommutative extensions to spacetime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although not quantum spacetime in the sense above, another use of noncommutative geometry is to tack on 'noncommutative extra dimensions' at each point of ordinary spacetime. Instead of invisible curled up extra dimensions as in string theory, <a href="/wiki/Alain_Connes" title="Alain Connes">Alain Connes</a> and coworkers have argued that the coordinate algebra of this extra part should be replaced by a finite-dimensional noncommutative algebra. For a certain reasonable choice of this algebra, its representation and extended Dirac operator, the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of elementary particles can be recovered. In this point of view, the different kinds of matter particles are manifestations of geometry in these extra noncommutative directions. Connes's first works here date from 1989<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> but has been developed considerably since then. Such an approach can theoretically be combined with quantum spacetime as above. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Anabelian_geometry" title="Anabelian geometry">Anabelian topology</a></li> <li><a href="/wiki/Quantum_reference_frame" title="Quantum reference frame">Quantum reference frame</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSnyder1947" class="citation cs2">Snyder, H. (1947), "Quantized space-time", <i>Physical Review</i>, <b>67</b> (1): <span class="nowrap">38–</span>41, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1947PhRv...71...38S">1947PhRv...71...38S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.71.38">10.1103/PhysRev.71.38</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Quantized+space-time&rft.volume=67&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E38-%3C%2Fspan%3E41&rft.date=1947&rft_id=info%3Adoi%2F10.1103%2FPhysRev.71.38&rft_id=info%3Abibcode%2F1947PhRv...71...38S&rft.aulast=Snyder&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Yang, I. E. Segal 1947.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajidRuegg1994" class="citation cs2">Majid, S.; Ruegg, H. (1994), "Bicrossproduct structure of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span>-Poincaré group and noncommutative geometry", <i>Physics Letters B</i>, <b>334</b> (<span class="nowrap">3–</span>4): <span class="nowrap">348–</span>354, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9405107">hep-th/9405107</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994PhLB..334..348M">1994PhLB..334..348M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2894%2990699-8">10.1016/0370-2693(94)90699-8</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10927240">10927240</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Bicrossproduct+structure+of+the+MATH+RENDER+ERROR-Poincar%C3%A9+group+and+noncommutative+geometry&rft.volume=334&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E348-%3C%2Fspan%3E354&rft.date=1994&rft_id=info%3Aarxiv%2Fhep-th%2F9405107&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10927240%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2894%2990699-8&rft_id=info%3Abibcode%2F1994PhLB..334..348M&rft.aulast=Majid&rft.aufirst=S.&rft.au=Ruegg%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajid1988" class="citation cs2">Majid, Shahn (1988), "Hopf algebras for physics at the Planck scale", <i>Classical and Quantum Gravity</i>, <b>5</b> (12): <span class="nowrap">1587–</span>1607, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988CQGra...5.1587M">1988CQGra...5.1587M</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.125.6178">10.1.1.125.6178</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F5%2F12%2F010">10.1088/0264-9381/5/12/010</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250857702">250857702</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=Hopf+algebras+for+physics+at+the+Planck+scale&rft.volume=5&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1587-%3C%2Fspan%3E1607&rft.date=1988&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.125.6178%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250857702%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F5%2F12%2F010&rft_id=info%3Abibcode%2F1988CQGra...5.1587M&rft.aulast=Majid&rft.aufirst=Shahn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmelino-CameliaMajid2000" class="citation cs2">Amelino-Camelia, G.; Majid, S. (2000), "Waves on noncommutative spacetime and gamma-ray bursts", <i>International Journal of Modern Physics A</i>, <b>15</b> (27): <span class="nowrap">4301–</span>4323, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9907110">hep-th/9907110</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000IJMPA..15.4301A">2000IJMPA..15.4301A</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fs0217751x00002779">10.1142/s0217751x00002779</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5730928">5730928</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+A&rft.atitle=Waves+on+noncommutative+spacetime+and+gamma-ray+bursts&rft.volume=15&rft.issue=27&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4301-%3C%2Fspan%3E4323&rft.date=2000&rft_id=info%3Aarxiv%2Fhep-th%2F9907110&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5730928%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2Fs0217751x00002779&rft_id=info%3Abibcode%2F2000IJMPA..15.4301A&rft.aulast=Amelino-Camelia&rft.aufirst=G.&rft.au=Majid%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLukierskiNowickiRueggTolstoy1991" class="citation cs2">Lukierski, J.; Nowicki, A.; Ruegg, H.; Tolstoy, V. N. (1991), "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>-Deformation of Poincaré algebras", <i>Physics Letters B</i>, <b>264</b> (<span class="nowrap">3–</span>4): <span class="nowrap">331–</span>338, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhLB..264..331L">1991PhLB..264..331L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2891%2990358-w">10.1016/0370-2693(91)90358-w</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=MATH+RENDER+ERROR-Deformation+of+Poincar%C3%A9+algebras&rft.volume=264&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E331-%3C%2Fspan%3E338&rft.date=1991&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2891%2990358-w&rft_id=info%3Abibcode%2F1991PhLB..264..331L&rft.aulast=Lukierski&rft.aufirst=J.&rft.au=Nowicki%2C+A.&rft.au=Ruegg%2C+H.&rft.au=Tolstoy%2C+V.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarow-WatamuraSchliekerSchollWatamura1990" class="citation cs2">Carow-Watamura, U.; Schlieker, M.; Scholl, M.; Watamura, S. (1990), "Tensor representation of the quantum group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL_{q}(2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SL_{q}(2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408b51150e420ddea34243f2a91fb6316aa81506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.754ex; height:3.009ex;" alt="{\displaystyle SL_{q}(2,\mathbb {C} )}"></span> and quantum Minkowski space", <i>Zeitschrift für Physik C</i>, <b>48</b> (1): 159, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01565619">10.1007/BF01565619</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120260589">120260589</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik+C&rft.atitle=Tensor+representation+of+the+quantum+group+MATH+RENDER+ERROR+and+quantum+Minkowski+space&rft.volume=48&rft.issue=1&rft.pages=159&rft.date=1990&rft_id=info%3Adoi%2F10.1007%2FBF01565619&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120260589%23id-name%3DS2CID&rft.aulast=Carow-Watamura&rft.aufirst=U.&rft.au=Schlieker%2C+M.&rft.au=Scholl%2C+M.&rft.au=Watamura%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajid1991" class="citation cs2">Majid, S. (1991), "Examples of braided groups and braided matrices", <i>Journal of Mathematical Physics</i>, <b>32</b> (12): <span class="nowrap">3246–</span>3253, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991JMP....32.3246M">1991JMP....32.3246M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.529485">10.1063/1.529485</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Examples+of+braided+groups+and+braided+matrices&rft.volume=32&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3246-%3C%2Fspan%3E3253&rft.date=1991&rft_id=info%3Adoi%2F10.1063%2F1.529485&rft_id=info%3Abibcode%2F1991JMP....32.3246M&rft.aulast=Majid&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajid1993" class="citation cs2">Majid, S. (1993), "Braided momentum in the q-Poincaré group", <i>Journal of Mathematical Physics</i>, <b>34</b> (5): <span class="nowrap">2045–</span>2058, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9210141">hep-th/9210141</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993JMP....34.2045M">1993JMP....34.2045M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.530154">10.1063/1.530154</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3138714">3138714</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Braided+momentum+in+the+q-Poincar%C3%A9+group&rft.volume=34&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2045-%3C%2Fspan%3E2058&rft.date=1993&rft_id=info%3Aarxiv%2Fhep-th%2F9210141&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3138714%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.530154&rft_id=info%3Abibcode%2F1993JMP....34.2045M&rft.aulast=Majid&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREF't_Hooft1996" class="citation cs2">'t Hooft, G. (1996), "Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness", <i>Classical and Quantum Gravity</i>, <b>13</b> (5): <span class="nowrap">1023–</span>1039, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9601014">gr-qc/9601014</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996CQGra..13.1023T">1996CQGra..13.1023T</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F13%2F5%2F018">10.1088/0264-9381/13/5/018</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250858614">250858614</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=Quantization+of+point+particles+in+%282+%2B+1%29-dimensional+gravity+and+spacetime+discreteness&rft.volume=13&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1023-%3C%2Fspan%3E1039&rft.date=1996&rft_id=info%3Aarxiv%2Fgr-qc%2F9601014&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250858614%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F13%2F5%2F018&rft_id=info%3Abibcode%2F1996CQGra..13.1023T&rft.aulast=%27t+Hooft&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatistaMajid2003" class="citation cs2">Batista, E.; Majid, S. (2003), "Noncommutative geometry of angular momentum space U(su_2)", <i>Journal of Mathematical Physics</i>, <b>44</b> (1): <span class="nowrap">107–</span>137, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0205128">hep-th/0205128</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003JMP....44..107B">2003JMP....44..107B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1517395">10.1063/1.1517395</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16315442">16315442</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Noncommutative+geometry+of+angular+momentum+space+U%28su_2%29&rft.volume=44&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E107-%3C%2Fspan%3E137&rft.date=2003&rft_id=info%3Aarxiv%2Fhep-th%2F0205128&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16315442%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.1517395&rft_id=info%3Abibcode%2F2003JMP....44..107B&rft.aulast=Batista&rft.aufirst=E.&rft.au=Majid%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoplicherFredenhagenRoberts1995" class="citation cs2">Doplicher, S.; Fredenhagen, K.; Roberts, J. E. (1995), "The quantum structure of spacetime at the Planck scale and quantum fields", <i>Communications in Mathematical Physics</i>, <b>172</b> (1): <span class="nowrap">187–</span>220, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0303037">hep-th/0303037</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995CMaPh.172..187D">1995CMaPh.172..187D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02104515">10.1007/BF02104515</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16206928">16206928</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Mathematical+Physics&rft.atitle=The+quantum+structure+of+spacetime+at+the+Planck+scale+and+quantum+fields&rft.volume=172&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E187-%3C%2Fspan%3E220&rft.date=1995&rft_id=info%3Aarxiv%2Fhep-th%2F0303037&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16206928%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02104515&rft_id=info%3Abibcode%2F1995CMaPh.172..187D&rft.aulast=Doplicher&rft.aufirst=S.&rft.au=Fredenhagen%2C+K.&rft.au=Roberts%2C+J.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeibergWitten1999" class="citation cs2">Seiberg, N.; Witten, E. (1999), "String theory and noncommutative geometry", <i>Journal of High Energy Physics</i>, <b>1999</b> (9): 9909, 032, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9908142">hep-th/9908142</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999JHEP...09..032S">1999JHEP...09..032S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1126-6708%2F1999%2F09%2F032">10.1088/1126-6708/1999/09/032</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:668885">668885</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=String+theory+and+noncommutative+geometry&rft.volume=1999&rft.issue=9&rft.pages=9909%2C+032&rft.date=1999&rft_id=info%3Aarxiv%2Fhep-th%2F9908142&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A668885%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1126-6708%2F1999%2F09%2F032&rft_id=info%3Abibcode%2F1999JHEP...09..032S&rft.aulast=Seiberg&rft.aufirst=N.&rft.au=Witten%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConnesLott1989" class="citation cs2">Connes, A.; Lott, J. (1989), <a rel="nofollow" class="external text" href="https://deepblue.lib.umich.edu/bitstream/2027.42/29524/1/0000611.pdf">"Particle models and noncommutative geometry"</a> <span class="cs1-format">(PDF)</span>, <i>Nuclear Physics B: Proceedings Supplements</i>, <b>18</b> (2): 29, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991NuPhS..18...29C">1991NuPhS..18...29C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0920-5632%2891%2990120-4">10.1016/0920-5632(91)90120-4</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F29524">2027.42/29524</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Physics+B%3A+Proceedings+Supplements&rft.atitle=Particle+models+and+noncommutative+geometry&rft.volume=18&rft.issue=2&rft.pages=29&rft.date=1989&rft_id=info%3Ahdl%2F2027.42%2F29524&rft_id=info%3Adoi%2F10.1016%2F0920-5632%2891%2990120-4&rft_id=info%3Abibcode%2F1991NuPhS..18...29C&rft.aulast=Connes&rft.aufirst=A.&rft.au=Lott%2C+J.&rft_id=https%3A%2F%2Fdeepblue.lib.umich.edu%2Fbitstream%2F2027.42%2F29524%2F1%2F0000611.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajid1995" class="citation cs2">Majid, S. (1995), <i>Foundations of Quantum Group Theory</i>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Quantum+Group+Theory&rft.pub=Cambridge+University+Press&rft.date=1995&rft.aulast=Majid&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._Oriti2009" class="citation cs2">D. Oriti, ed. (2009), <i>Approaches to Quantum Gravity</i>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Approaches+to+Quantum+Gravity&rft.pub=Cambridge+University+Press&rft.date=2009&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConnesMarcolli2007" class="citation cs2"><a href="/wiki/Alain_Connes" title="Alain Connes">Connes, A.</a>; <a href="/wiki/Matilde_Marcolli" title="Matilde Marcolli">Marcolli, M.</a> (2007), <i>Noncommutative Geometry, Quantum Fields and Motives</i>, Colloquium Publications</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Noncommutative+Geometry%2C+Quantum+Fields+and+Motives&rft.pub=Colloquium+Publications&rft.date=2007&rft.aulast=Connes&rft.aufirst=A.&rft.au=Marcolli%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMajidSchroers2009" class="citation cs2">Majid, S.; Schroers, B.J. (2009), "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>-Deformation and semidualization in 3D quantum gravity", <i>Journal of Physics A: Mathematical and Theoretical</i>, <b>42</b> (42): 425402 (40pp), <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009JPhA...42P5402M">2009JPhA...42P5402M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1751-8113%2F42%2F42%2F425402">10.1088/1751-8113/42/42/425402</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17160062">17160062</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+A%3A+Mathematical+and+Theoretical&rft.atitle=MATH+RENDER+ERROR-Deformation+and+semidualization+in+3D+quantum+gravity&rft.volume=42&rft.issue=42&rft.pages=425402+%2840pp%29&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17160062%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1751-8113%2F42%2F42%2F425402&rft_id=info%3Abibcode%2F2009JPhA...42P5402M&rft.aulast=Majid&rft.aufirst=S.&rft.au=Schroers%2C+B.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span>.</li> <li>R. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, 4th Ed. Addison-Wesley 1999.</li> <li>J. Matousek, J. Nesetril, Invitation to Discrete Mathematics. Oxford University Press 1998.</li> <li>Taylor E. F., John A. Wheeler, Spacetime Physics, publisher W. H. Freeman, 1963.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhoshbin-e-Khoshnazar2013" class="citation journal cs1">Khoshbin-e-Khoshnazar, M.R. (2013). "Binding Energy of the Very Early Universe: Abandoning Einstein for a Discretized Three–Torus Poset.A Proposal on the Origin of Dark Energy". <i>Gravitation and Cosmology</i>. <b>19</b> (2): <span class="nowrap">106–</span>113. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013GrCo...19..106K">2013GrCo...19..106K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2Fs0202289313020059">10.1134/s0202289313020059</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121288092">121288092</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Gravitation+and+Cosmology&rft.atitle=Binding+Energy+of+the+Very+Early+Universe%3A+Abandoning+Einstein+for+a+Discretized+Three%E2%80%93Torus+Poset.A+Proposal+on+the+Origin+of+Dark+Energy&rft.volume=19&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E106-%3C%2Fspan%3E113&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121288092%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1134%2Fs0202289313020059&rft_id=info%3Abibcode%2F2013GrCo...19..106K&rft.aulast=Khoshbin-e-Khoshnazar&rft.aufirst=M.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_spacetime&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://plus.maths.org/issue43/features/noncom/index-gifd.html">Plus Magazine article on quantum geometry</a> by Marianne Freiberger</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._Majid2008" class="citation cs2">S. Majid, ed. (2008), <i>On Space and Time</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-64168-6" title="Special:BookSources/978-1-107-64168-6"><bdi>978-1-107-64168-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+Space+and+Time&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-1-107-64168-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+spacetime" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics332" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a class="mw-selflink selflink">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" 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