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<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="3.8.5">Jekyll</generator><link href="http://localhost:4000/feed.xml" rel="self" type="application/atom+xml" /><link href="http://localhost:4000/" rel="alternate" type="text/html" /><updated>2018-11-17T15:57:04+00:00</updated><id>http://localhost:4000/feed.xml</id><title type="html">Paul J Druce’s site</title><subtitle>A place to collect my notes on the maths of physics and anything related.</subtitle><entry><title type="html">A footing in Geometric Quantisation</title><link href="http://localhost:4000/academic/2018/11/16/GeometricQuantisation.html" rel="alternate" type="text/html" title="A footing in Geometric Quantisation" /><published>2018-11-16T00:00:00+00:00</published><updated>2018-11-16T00:00:00+00:00</updated><id>http://localhost:4000/academic/2018/11/16/GeometricQuantisation</id><content type="html" xml:base="http://localhost:4000/academic/2018/11/16/GeometricQuantisation.html">&lt;h1 id=&quot;weekly-meetingsgeometric-quantisation&quot;&gt;Weekly Meetings:Geometric Quantisation&lt;/h1&gt;
&lt;h2 id=&quot;week-1---axel-polaczek&quot;&gt;Week 1 - Axel Polaczek&lt;/h2&gt;
&lt;p&gt;What is the purpose of geometric quantisation? To construct a methodology of taking a classical theory on a manifold and creating a quantum theory. This is done by making use of symplectic geometry (the stage for classical mechanics) and associating to a symplectic manifold a U(1)-principal bundle with connection. Then construction a Hilbert space of states is the space of sections of the associated line bundle.&lt;/p&gt;
&lt;p&gt;There are some subtleties and obstructions that will present themselves which will prevent us from constructing such structures for any symplectic manifold.&lt;/p&gt;
&lt;h4 id=&quot;symplectic-geometry&quot;&gt;Symplectic Geometry&lt;/h4&gt;
&lt;p&gt;We need to have some basic understand of symplectic geometry, so there will be a lot of definitions following:&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt; A pair &lt;span class=&quot;math inline&quot;&gt;\((M,\omega)\)&lt;/span&gt; consisting of a manifold &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt; and a &lt;a href=&quot;https://en.wikipedia.org/wiki/Differential_form#Intrinsic_definitions+&quot; target=&quot;_blank&quot;&gt;two-form&lt;/a&gt; &lt;span class=&quot;math inline&quot;&gt;\(\omega\)&lt;/span&gt; is said to be symplectic manifold if we have that the two form satisfies - &lt;span class=&quot;math inline&quot;&gt;\(d\omega = 0\)&lt;/span&gt; (closedness) - if &lt;span class=&quot;math inline&quot;&gt;\(\omega(X,Y) = 0\)&lt;/span&gt; for all &lt;span class=&quot;math inline&quot;&gt;\(X\in \Gamma(TM)\)&lt;/span&gt; then we must have that &lt;span class=&quot;math inline&quot;&gt;\(Y = 0\)&lt;/span&gt;. (non degenerate )&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;We can also have &lt;strong&gt;symplectic vector fields&lt;/strong&gt;, &lt;span class=&quot;math display&quot;&gt;\[X\in \Gamma(TM)\]&lt;/span&gt; if we have that &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{L}_X \omega = 0\)&lt;/span&gt;. Using the &lt;a href=&quot;https://en.wikipedia.org/wiki/Interior_product&quot; target=&quot;_blank&quot;&gt;interior product&lt;/a&gt;, &lt;span class=&quot;math inline&quot;&gt;\((i_X \omega) (Y) = \omega(X,Y)\)&lt;/span&gt; we can express this condition as: &lt;span class=&quot;math inline&quot;&gt;\((i_X \circ d + d \circ i_X)\omega = 0\)&lt;/span&gt;. Which when combined with the closeness property of the symplectic form, we have that a vector field is symplectic if &lt;span class=&quot;math inline&quot;&gt;\((d \circ i_X) \omega =0\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;An important concept is that of a Hamiltonian vector field.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt; A function, &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt; defines a vector field called the Hamiltonian vector field (of &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt;) &lt;span class=&quot;math inline&quot;&gt;\(X_f \in \Gamma(TM)\)&lt;/span&gt; which obeys the relation: &lt;span class=&quot;math inline&quot;&gt;\(d f = - i_{X_f} \omega\)&lt;/span&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Expressing this relationship a bit more fully, we see that &lt;span class=&quot;math inline&quot;&gt;\(df (Y) = Y(f)\)&lt;/span&gt; by definition of the differential and by the definition of the Hamiltonian vector field we can write this as: &lt;span class=&quot;math inline&quot;&gt;\(df(Y) = (- i_{X_f} \omega)(Y) = -\omega(X_f, Y)\)&lt;/span&gt;. The minus sign seems to be a quirk of Axel’s (the person giving this talk) as most online resources I’ve come across don’t include it. Something to keep an eye on as we proceed, as we will change speakers.&lt;/p&gt;
&lt;p&gt;The next structure we need is that of a Poisson Bracket.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt; A map &lt;span class=&quot;math inline&quot;&gt;\(\{\cdot,\cdot\}: C^\infty(M) \to C^\infty(M) \to C^\infty(M)\)&lt;/span&gt; is called the Poisson bracket of the symplectic form &lt;span class=&quot;math inline&quot;&gt;\(\omega\)&lt;/span&gt; if we have that &lt;span class=&quot;math inline&quot;&gt;\(\{f,g\} = i_{X_f}i_{X_g} \omega= - \omega(X_f,X_g)\)&lt;/span&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;This is called the &lt;em&gt;natural Poisson structure of a symplectic manifold&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition&lt;/strong&gt;: A function &lt;span class=&quot;math inline&quot;&gt;\(g\)&lt;/span&gt; is constant along the integral curves of the hamiltonian vector field &lt;span class=&quot;math inline&quot;&gt;\(X_f\)&lt;/span&gt; &lt;em&gt;if and only if&lt;/em&gt; &lt;span class=&quot;math inline&quot;&gt;\(\{f,g\} = 0\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;&lt;em&gt;&lt;u&gt;Proof&lt;/u&gt;&lt;/em&gt;: Take &lt;span class=&quot;math inline&quot;&gt;\(\gamma:[0,1] \to M\)&lt;/span&gt; to be the integral curve of &lt;span class=&quot;math inline&quot;&gt;\(X_f\)&lt;/span&gt;. So we can then look at &lt;span class=&quot;math inline&quot;&gt;\(\frac{d}{dt} g(\gamma(t)) = X_f(g(\gamma(t)))\)&lt;/span&gt; (which is just definition of an integral curve), then we can examine the right hand side as the following: &lt;span class=&quot;math display&quot;&gt;\[
X_f(g(\gamma(t))) = d(g\circ\gamma)(t) (X_f) = i_{X_f} d(g\circ \gamma)(t)
\]&lt;/span&gt; We then take the hamiltonian vector field associated to &lt;span class=&quot;math inline&quot;&gt;\(g\)&lt;/span&gt;, and we can express &lt;span class=&quot;math inline&quot;&gt;\(dg = -i_{X_g}\omega\)&lt;/span&gt; this as: &lt;span class=&quot;math display&quot;&gt;\[
X_f(g(\gamma(t))) = i_{X_f} (dg \circ d\gamma) (t) = - i_{X_f} (i_{X_g} \omega) (\gamma(t)) = -\{f,g\} (\gamma(t))
\]&lt;/span&gt; As this holds for arbitrary values of &lt;span class=&quot;math inline&quot;&gt;\(t\)&lt;/span&gt; this must hold along the entire curve &lt;span class=&quot;math inline&quot;&gt;\(\gamma\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;Another handy result is that &lt;span class=&quot;math inline&quot;&gt;\(X_{\{f,g\}} = - [X_f, X_g]\)&lt;/span&gt;, which can be shown if we invert the statement above for the Lie derivative interns of the interior and exterior product. To get that &lt;span class=&quot;math inline&quot;&gt;\(i_{[X,Y]} =[\mathcal{L}_X,Y]\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;Also, as we have that &lt;span class=&quot;math inline&quot;&gt;\(d\omega = 0\)&lt;/span&gt; then by Poincare lemma we have that &lt;span class=&quot;math inline&quot;&gt;\(\omega = d\theta\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;Time for the classic example:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Examples&lt;/strong&gt;: Take the manifold to be the total space of the cotangent bundle &lt;span class=&quot;math inline&quot;&gt;\(M =T^*(Q)\)&lt;/span&gt; over some manifold &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt;? (I think we just treat &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; as a vector space, but I’m not sure where this becomes important if at all). So then points in this space are of the form &lt;span class=&quot;math inline&quot;&gt;\(m = (q,p) \in M\)&lt;/span&gt; and we can form a bundle structure over &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; by taking projection onto the first coordinate &lt;span class=&quot;math inline&quot;&gt;\(\pi:M \to Q\)&lt;/span&gt; (so we are just taking the cotangent bundle over &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt;). We need a map &lt;span class=&quot;math inline&quot;&gt;\(\theta: TM \to \mathbb{R}\)&lt;/span&gt; (i.e. &lt;span class=&quot;math inline&quot;&gt;\(\theta \in \Omega^1(M)\)&lt;/span&gt;) such that the following diagram commutes&lt;/p&gt;
&lt;figure&gt;
&lt;img src=&quot;/Users/pauldruce/Google%20Drive/Notes/Seminars/GeometricQuantisation.assets/image-20181113102905387.png&quot; alt=&quot;image-20181113102905387&quot; /&gt;&lt;figcaption&gt;image-20181113102905387&lt;/figcaption&gt;
&lt;/figure&gt;
&lt;p&gt;Where &lt;span class=&quot;math inline&quot;&gt;\(p:TQ \to \mathbb{R}\)&lt;/span&gt; is what exactly? My notes from the time don’t say, but lets figure this out. Given a point &lt;span class=&quot;math inline&quot;&gt;\(m\in M = T^*Q\)&lt;/span&gt; we know that it gets mapped to &lt;span class=&quot;math inline&quot;&gt;\(q\)&lt;/span&gt; under the map &lt;span class=&quot;math inline&quot;&gt;\(\pi\)&lt;/span&gt;. Alternatively we can view &lt;span class=&quot;math inline&quot;&gt;\(m\)&lt;/span&gt; as a map from &lt;span class=&quot;math inline&quot;&gt;\(T_qQ \to \mathbb{R}\)&lt;/span&gt; as its a one-form after all. Which in Axels notes must be the map &lt;span class=&quot;math inline&quot;&gt;\(p\)&lt;/span&gt; above, alternatively I’ll write it as &lt;span class=&quot;math inline&quot;&gt;\(m\)&lt;/span&gt; below.&lt;/p&gt;
&lt;figure&gt;
&lt;img src=&quot;/Users/pauldruce/Google%20Drive/Notes/Seminars/GeometricQuantisation.assets/image-20181113103859073.png&quot; alt=&quot;image-20181113103859073&quot; /&gt;&lt;figcaption&gt;image-20181113103859073&lt;/figcaption&gt;
&lt;/figure&gt;
&lt;p&gt;So what is the combination &lt;span class=&quot;math inline&quot;&gt;\(m\circ d\pi\)&lt;/span&gt; , it acts on an element of &lt;span class=&quot;math inline&quot;&gt;\(TM\)&lt;/span&gt; corresponding to the point &lt;span class=&quot;math inline&quot;&gt;\(m\in M\)&lt;/span&gt; and sends it to &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}\)&lt;/span&gt;. So it’s a map from &lt;span class=&quot;math inline&quot;&gt;\(T_mM \to \mathbb{R}\)&lt;/span&gt; which is means in the diagrams above, we need to put a specific on &lt;span class=&quot;math inline&quot;&gt;\(\theta = \theta_m\)&lt;/span&gt;. Then we vary &lt;span class=&quot;math inline&quot;&gt;\(m\)&lt;/span&gt; over the entire manifold and see that the formula holds.&lt;/p&gt;
&lt;p&gt;In terms of coordinates we have that &lt;span class=&quot;math inline&quot;&gt;\(\theta = p_i dq_i\)&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(p = p_idx^i\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(q_i:M \to \mathbb{R}\)&lt;/span&gt; are the coordinates in a local chart of &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt; treated as functions on &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;: Take &lt;span class=&quot;math inline&quot;&gt;\(M = T^*\mathbb{R}\)&lt;/span&gt;, with points &lt;span class=&quot;math inline&quot;&gt;\((q,p) = m\in M\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\omega = dp \wedge dq\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(dH = -i_{X_H}\omega\)&lt;/span&gt; . We can write any vector field in terms of it’s components &lt;span class=&quot;math inline&quot;&gt;\(X_H = X^p_H \frac{\partial}{\partial p} + X^q_H \frac{\partial}{\partial q}\)&lt;/span&gt; . By looking at the definition of a Hamiltonian vector field we have &lt;span class=&quot;math display&quot;&gt;\[
dH(Y)= -i_{X_H} \omega(Y) = - \omega(X_H,Y) = -(X^p_H dq(Y) - X^q_H dp(Y))
\]&lt;/span&gt; so that &lt;span class=&quot;math inline&quot;&gt;\(dH = - (X^p_H dq - X^q_H dp)\)&lt;/span&gt;, but we also have that &lt;span class=&quot;math inline&quot;&gt;\(dH = \frac{\partial H}{\partial p} dp + \frac{\partial H}{\partial q} dq\)&lt;/span&gt;. So we have that $X^p_H = -  $ and &lt;span class=&quot;math inline&quot;&gt;\(X^q_H = \frac{\partial H}{\partial p}\)&lt;/span&gt; and so combining we get: &lt;span class=&quot;math display&quot;&gt;\[
X_H = - \frac{\partial H}{\partial q}\frac{\partial}{\partial p} + \frac{\partial H}{\partial p}\frac{\partial}{\partial q}
\]&lt;/span&gt;&lt;/p&gt;
&lt;h4 id=&quot;quantisation&quot;&gt;Quantisation&lt;/h4&gt;
&lt;ul&gt;
&lt;li&gt;[ ] This was fairly rushed and I may embellish the notes bit later on.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;What we want is a map &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{Q}: Obs \to End(\mathcal{H})\)&lt;/span&gt; which maps us from the space of classical observables on our symplectic manifold to some operators on a Hilbert space. The space of observables is just the algebra of functions &lt;span class=&quot;math inline&quot;&gt;\(C^\infty(M)\)&lt;/span&gt;. We want this map to follow some general rules&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;there was a fair bit of discussion about whether or not we want this map to satisfy all of the requirements or not. I’m sure it will come up again later&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Q1:&lt;/strong&gt; &lt;span class=&quot;math inline&quot;&gt;\(Q(1) = Q_1 = id\)&lt;/span&gt; , so the identity is mapped to the identity.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Q2:&lt;/strong&gt; &lt;span class=&quot;math inline&quot;&gt;\(Q_{\alpha f + g} = \alpha Q_f + Q_g\)&lt;/span&gt; , so the map preserves the linear structure.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Q3:&lt;/strong&gt; &lt;span class=&quot;math inline&quot;&gt;\([Q_f, Q_g] = i\hbar Q_{\{f,g\}}\)&lt;/span&gt; , this is the condition that required all the background on symplectic manifolds, but you can see that it maps the symplectic structure to a commutator. We will see why this is important later I guess.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;There was one other condition and it caused some disagreement between the audience of the meeting.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Q4:&lt;/strong&gt; For &lt;span class=&quot;math inline&quot;&gt;\(\phi: M^{(1)} \to M^{(2)}\)&lt;/span&gt; such the &lt;span class=&quot;math inline&quot;&gt;\(\phi^*\omega^{(2)} = \omega^{(1)}\)&lt;/span&gt; and for &lt;span class=&quot;math inline&quot;&gt;\(f\in C^\infty(M^{(2)}) = Obs^{(2)}\)&lt;/span&gt; we want
&lt;ul&gt;
&lt;li&gt;&lt;span class=&quot;math inline&quot;&gt;\(f\circ \phi \in Obs^{(1)}\)&lt;/span&gt;&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;math inline&quot;&gt;\(\exists U_\phi :\mathcal{H^{(1)}} \to \mathcal{H^{(2)}}\)&lt;/span&gt; such that &lt;span class=&quot;math inline&quot;&gt;\(Q^{(1)}_{f\circ \phi} = U^*_\phi Q^{(2)}_f U_\phi\)&lt;/span&gt;&lt;/li&gt;
&lt;/ul&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;These are statements about &lt;em&gt;irreducibility&lt;/em&gt;. Which I understand to be that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem&quot; target=&quot;_blank&quot;&gt;Stone vonNeumann theorem&lt;/a&gt; tells us that there is unique (up to unitary equivalence) representation that is irreducible (I.e. that map &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; is an irreducible representation of &lt;span class=&quot;math inline&quot;&gt;\(Obs\)&lt;/span&gt;) that satisfies the standard canonical commutation relations. However, the way it’s formalised above is supposed to exists for a general manifold, even when the split of &lt;span class=&quot;math inline&quot;&gt;\(M = T^*Q\)&lt;/span&gt; (where &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; is some general symplectic manifold now instead of a vector space as above) into global coordinates &lt;span class=&quot;math inline&quot;&gt;\((q,p)\)&lt;/span&gt; doesn’t exist. However, it’s not clear that this is true to me.&lt;/p&gt;
&lt;h2 id=&quot;week-2---axel-polaczek&quot;&gt;Week 2 - Axel Polaczek&lt;/h2&gt;
&lt;p&gt;We examine the above example, where &lt;span class=&quot;math inline&quot;&gt;\(M = T^*Q\)&lt;/span&gt; for &lt;span class=&quot;math inline&quot;&gt;\(Q = \mathbb{R^d}\)&lt;/span&gt;. Then we can take &lt;span class=&quot;math inline&quot;&gt;\(\omega = d\theta\)&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\theta\)&lt;/span&gt; is known globally so we can write it as above &lt;span class=&quot;math inline&quot;&gt;\(\theta = p_i dq^i\)&lt;/span&gt;. In this case we have an expression for the map &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt;. &lt;span class=&quot;math display&quot;&gt;\[
Q_f = - i \hbar X_f - \theta(X_f) + f
\]&lt;/span&gt; Where the Hilbert space in question is &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H} = L^2(M)\)&lt;/span&gt;, where the norm is given by the inner product &lt;span class=&quot;math inline&quot;&gt;\(\langle \phi,\psi \rangle = \int_M \omega^n \phi^* \psi\)&lt;/span&gt; or something similar. (I think d = 2n but not 100% sure).&lt;/p&gt;
&lt;p&gt;We then can check that this map satisfied &lt;strong&gt;Q1 - Q4&lt;/strong&gt;.&lt;/p&gt;
&lt;p&gt;However, in the general case of a symplectic manifold &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt;. We can’t choose global coordinates for &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; or &lt;span class=&quot;math inline&quot;&gt;\(M = T^*Q\)&lt;/span&gt;. However by &lt;a href=&quot;https://en.wikipedia.org/wiki/Darboux%27s_theorem&quot; target=&quot;_blank&quot;&gt;Darboux theorem&lt;/a&gt; we can always choose a covering of &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; such that in each coordinate patch &lt;span class=&quot;math inline&quot;&gt;\(\omega = dq_i \wedge dp_i\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;So if we take two charts &lt;span class=&quot;math inline&quot;&gt;\(U_a, U_b\)&lt;/span&gt; , take &lt;span class=&quot;math inline&quot;&gt;\(\theta\)&lt;/span&gt; to be defined on &lt;span class=&quot;math inline&quot;&gt;\(U_a\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\theta^\prime\)&lt;/span&gt; to be defined on &lt;span class=&quot;math inline&quot;&gt;\(U_b\)&lt;/span&gt; and look at &lt;span class=&quot;math inline&quot;&gt;\(\omega\)&lt;/span&gt; on the overlap of &lt;span class=&quot;math inline&quot;&gt;\(U_a \cap U_b\)&lt;/span&gt;. Then we have that &lt;span class=&quot;math inline&quot;&gt;\(\omega = d\theta = d\theta^\prime\)&lt;/span&gt;. So we know then that &lt;span class=&quot;math inline&quot;&gt;\(\theta = \theta^\prime + du_{ab}\)&lt;/span&gt; and the functions &lt;span class=&quot;math inline&quot;&gt;\(u_{ab}\)&lt;/span&gt; depends on the coordinate patches in question.&lt;/p&gt;
&lt;p&gt;As the definition of &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; above depends on the potentials &lt;span class=&quot;math inline&quot;&gt;\(\theta\)&lt;/span&gt; we should examine how the terms in the expression above differ in the different coordinate patches. &lt;span class=&quot;math display&quot;&gt;\[
\theta(X_f) - \theta^\prime(X_f) = du(X_f) = X_f(u)
\]&lt;/span&gt; We can then write &lt;span class=&quot;math inline&quot;&gt;\(X_f(u)\)&lt;/span&gt; as the following expression involving operators &lt;span class=&quot;math inline&quot;&gt;\(e^{-u}\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(e^u\)&lt;/span&gt;. We do this as we want to relate the two expression of &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; to each other via unitary operators and this step will help us do that. &lt;span class=&quot;math display&quot;&gt;\[
X_f(u) = -e^u X_f e^{-u}
\]&lt;/span&gt; Then we can render the expression &lt;span class=&quot;math inline&quot;&gt;\(Q_f\)&lt;/span&gt; in the chart &lt;span class=&quot;math inline&quot;&gt;\(U_a\)&lt;/span&gt; and use the relation for the potentials. &lt;span class=&quot;math display&quot;&gt;\[
\begin{align}
e^{\frac{u}{i\hbar}} Q_f^a e^{-\frac{u}{i\hbar}} \phi &amp;amp;= e^{\frac{u}{i\hbar}} \left( -i\hbar X_f - \theta(X_f)+f\right) e^{\frac{u}{i\hbar} } \phi\\
&amp;amp;= e^{- \frac{u}{i\hbar}}\left( -i\hbar X_f + f - \theta^\prime(X_f) - X_f(u) \right)e^{\frac{u}{i\hbar}} \phi\\
&amp;amp;=e^{- \frac{u}{i\hbar}} \left(  -i\hbar X_f + f -\theta^\prime(X_f) - X_f(u)  \right)e^{\frac{u}{i\hbar}} \phi\\
&amp;amp;=\left( i\hbar e^{- \frac{u}{i\hbar}}X_f e^{-\frac{u}{i\hbar}} + f - \theta^\prime(X_f) - X_f(u) \right)\phi
\end{align}
\]&lt;/span&gt; As &lt;span class=&quot;math inline&quot;&gt;\(X_f(u)\)&lt;/span&gt; is just a function, the exponentials can commute with it. However, for the &lt;span class=&quot;math inline&quot;&gt;\(-ihX_f\)&lt;/span&gt; we need to act the vector field. &lt;span class=&quot;math display&quot;&gt;\[
\begin{align}
-i\hbar e^{\frac{u}{i\hbar}} X_f (e^{-\frac{u}{i\hbar}}\phi) &amp;amp;= -i\hbar e^{\frac{u}{i\hbar}} X_f (e^{-\frac{u}{i\hbar}}) \phi + -i\hbar e^{\frac{u}{i\hbar}}e^{-\frac{u}{i\hbar}} X_f (\phi) \\
&amp;amp;= (+X_f(u) -i\hbar X_f) \phi
\end{align}
\]&lt;/span&gt; So substituting this into the expression above gives us that &lt;span class=&quot;math display&quot;&gt;\[
e^{\frac{u}{i\hbar}} Q_f^a e^{-\frac{u}{i\hbar}} \phi = \left(-i\hbar X_f + f -\theta^\prime(X_f)\right)\phi = Q^b_f \phi
\]&lt;/span&gt; ##### Complex Line Bundles&lt;/p&gt;
&lt;p&gt;We know switch gears, and view all of the above construction as being data in a complex line bundle with connection/covariant derivative.&lt;/p&gt;
&lt;p&gt;Given a 1 dimensional complex vector bundle &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{L} \overset{\pi}{\longrightarrow} M\)&lt;/span&gt;, let &lt;span class=&quot;math inline&quot;&gt;\(\varphi_u: \pi^{-1}(U) \to U\times \mathbb{C}\)&lt;/span&gt; be the local trivialisations and on the intersections &lt;span class=&quot;math inline&quot;&gt;\(U_{\alpha \beta} = U_\alpha \cap U_\beta \neq \emptyset\)&lt;/span&gt; let &lt;span class=&quot;math inline&quot;&gt;\(g_{\alpha\beta}: \mathbb{C} \to \mathbb{C}\)&lt;/span&gt; be the transitions functions such that: &lt;span class=&quot;math display&quot;&gt;\[
\begin{align}
\varphi_\alpha^{-1} \circ \varphi_\beta : &amp;amp;U_{\alpha \beta} \times \mathbb{C} \to  &amp;amp;U_{\alpha \beta} \times \mathbb{C}\\
&amp;amp; (m,\phi_\beta)\mapsto  &amp;amp; (m,g_{\alpha\beta} \psi_\beta )
\end{align}
\]&lt;/span&gt; And such that they satisfy the cocycle conditions for triple intersections &lt;span class=&quot;math inline&quot;&gt;\(U_{\alpha\beta\gamma}\)&lt;/span&gt;: &lt;span class=&quot;math inline&quot;&gt;\(g_{ab}g_{bc}g_{ca} = 1\)&lt;/span&gt;, &lt;span class=&quot;math inline&quot;&gt;\(g_{\alpha \beta}g_{\beta \gamma} = g_{\alpha \gamma}\)&lt;/span&gt;. &lt;span class=&quot;math inline&quot;&gt;\(g_{\alpha \alpha } = 1\)&lt;/span&gt; and also &lt;span class=&quot;math inline&quot;&gt;\(g_{\alpha\beta} = \frac{1}{g_{\beta \alpha}}\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;If we view &lt;span class=&quot;math inline&quot;&gt;\(exp(\frac{u_{ab}}{i\hbar}) = g_{ab}\)&lt;/span&gt; as transition functions. Then the fact that transition functions have to satisfy the &lt;em&gt;cocycle condition&lt;/em&gt; that &lt;span class=&quot;math inline&quot;&gt;\(g_{ab}g_{bc}g_{ca} = 1\)&lt;/span&gt;, then we need that &lt;span class=&quot;math inline&quot;&gt;\(u_{ab} + u_{bc} + u_{ca} = n_{abc}\in \mathbb{Z}\)&lt;/span&gt;. Which is called the integrability condition or &lt;em&gt;prequantisation condition&lt;/em&gt;.&lt;/p&gt;</content><author><name></name></author><category term="[&quot;academic&quot;]" /><category term="academic" /><summary type="html">Weekly Meetings:Geometric Quantisation</summary></entry><entry><title type="html">First Year Report</title><link href="http://localhost:4000/catagory%201/catagory%202/2018/11/12/First-Year-Report-HTML.html" rel="alternate" type="text/html" title="First Year Report" /><published>2018-11-12T00:00:00+00:00</published><updated>2018-11-12T00:00:00+00:00</updated><id>http://localhost:4000/catagory%201/catagory%202/2018/11/12/First-Year-Report-HTML</id><content type="html" xml:base="http://localhost:4000/catagory%201/catagory%202/2018/11/12/First-Year-Report-HTML.html">&lt;!DOCTYPE html&gt;
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&lt;h1 class=&quot;title&quot;&gt;First Year Report - Noncommutative Geometry and Quantum Gravity&lt;/h1&gt;
&lt;p class=&quot;author&quot;&gt;Paul Druce&lt;/p&gt;
&lt;/header&gt;

&lt;h1 id=&quot;motivation&quot;&gt;Motivation&lt;/h1&gt;
&lt;p&gt;Noncommutative geometry&lt;a href=&quot;#fn1&quot; class=&quot;footnote-ref&quot; id=&quot;fnref1&quot;&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;/a&gt; is an attempt to generalise the notion of differential geometry. The generalisation is taken by considering the fact that the differential nature of a manifold, &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;, is encoded in a commutative algebra of smooth functions &lt;span class=&quot;math inline&quot;&gt;\(C^{\infty}(M)\)&lt;/span&gt;. We then pose the question of what sort of structure has similar properties but where the corresponding algebra is noncommutative. This question has been extended to consider Riemannian spin manifolds and their appropriate generalisations with the aim of developing a good model for reality. The standard model of particle physics has a very natural origin within noncommutative geometry and current research is looking into incorporating gravity into the overall picture. To make sense of the basic objects in noncommutative geometry it is useful to cast the corresponding “normal&amp;quot; commutative picture into the correct framework. As the object we wish to generalise is that of a Riemannian spin manifold, below is a brief walk through the basics of spin geometry in the appropriate view point.&lt;/p&gt;
&lt;h1 id=&quot;introduction-to-spin-geometry&quot;&gt;Introduction to Spin Geometry&lt;/h1&gt;
&lt;p&gt;Spin geometry has far reaching applications in mathematics and physics and is the topic on many books and papers. A very thorough and most famous is that by Lawson and Michelsohn &lt;span class=&quot;citation&quot; data-cites=&quot;LawsonJr:1989ub&quot;&gt;&lt;/span&gt; however for out applications it is very mathematically dense and there are more digestible books. For the more geometrically incline the book by Friedrich &lt;span class=&quot;citation&quot; data-cites=&quot;Friedrich:2000wd&quot;&gt;&lt;/span&gt; is very good and for those who prefer to stay closely related to physics the books by Lawrie &lt;span class=&quot;citation&quot; data-cites=&quot;Lawrie:2012uz&quot;&gt;&lt;/span&gt;, Benn and Tucker &lt;span class=&quot;citation&quot; data-cites=&quot;Benn:1987td&quot;&gt;&lt;/span&gt; and specifically for those interested in noncommutative geometry and physics the book by Suijlekom &lt;span class=&quot;citation&quot; data-cites=&quot;vanSuijlekom:2014kl&quot;&gt;&lt;/span&gt; is excellent.&lt;/p&gt;
&lt;p&gt;The basic object of interest are called Riemannian spin manifolds. These are manifolds with a riemannian metric and spin structure on them. T he formalism for spinors and therefore particle physics requires such an object, however, there is considerable prerequisite material needed in order to define a spin manifold. To begin with, we start with the definition of a Clifford algebra bundle&lt;a href=&quot;#fn2&quot; class=&quot;footnote-ref&quot; id=&quot;fnref2&quot;&gt;&lt;sup&gt;2&lt;/sup&gt;&lt;/a&gt;. Given a Riemannian manifold, &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;, we can define a Clifford algebra, &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}(T_xM,Q_g)\)&lt;/span&gt;, at each point of the manifold, where &lt;span class=&quot;math inline&quot;&gt;\(T_xM\)&lt;/span&gt; is the tangent space at the point &lt;span class=&quot;math inline&quot;&gt;\(x\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(Q_g\)&lt;/span&gt; is a quadratic form induced by the metric &lt;span class=&quot;math inline&quot;&gt;\(g\)&lt;/span&gt; defined as follows: &lt;span class=&quot;math display&quot;&gt;\[Q_g(X_x) = g_x(X_x,X_x) \quad \forall X_x \in T_xM\]&lt;/span&gt; by letting &lt;span class=&quot;math inline&quot;&gt;\(x\)&lt;/span&gt; vary we can construct the Clifford algebra bundle:&lt;/p&gt;
&lt;p&gt;The Clifford Algebra bundle &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}^+(TM)\)&lt;/span&gt; is the fibre bundle&lt;a href=&quot;#fn3&quot; class=&quot;footnote-ref&quot; id=&quot;fnref3&quot;&gt;&lt;sup&gt;3&lt;/sup&gt;&lt;/a&gt; over &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;, where the fibres are the clifford algebras &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}(T_xM,Q_g)\)&lt;/span&gt; and the transition functions are inherited from the tangent bundle &lt;span class=&quot;math inline&quot;&gt;\(TM\)&lt;/span&gt;, &lt;span class=&quot;math inline&quot;&gt;\(t_{ij}\colon U_i\cap U_j \to SO(n)\)&lt;/span&gt;, where &lt;span class=&quot;math inline&quot;&gt;\(n = dim(M)\)&lt;/span&gt; and their action on &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}(T_xM, Q_g)\)&lt;/span&gt; is given by: &lt;span class=&quot;math display&quot;&gt;\[t_{ij}( v_1 v_2 \dots v_k) = t_{ij}(v_1) \dots t_{ij}(v_k)\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Given a Clifford algebra bundle we can define the algebra of continuous real-valued sections denoted by &lt;span class=&quot;math inline&quot;&gt;\(\text{Cliff}^+(M) := \Gamma (\text{Cl}^+(TM))\)&lt;/span&gt;. By replacing &lt;span class=&quot;math inline&quot;&gt;\(Q_g\)&lt;/span&gt; with &lt;span class=&quot;math inline&quot;&gt;\(-Q_g\)&lt;/span&gt; we can analogously define &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}^-(TM)\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\text{Cliff}^-(M)\)&lt;/span&gt;. We can also define: &lt;span class=&quot;math display&quot;&gt;\[\mathbb{C}\text{liff}(M) \vcentcolon=\text{Cliff}^+(M) \otimes_{\mathbb{R}} \mathbb{C}\]&lt;/span&gt; which is the space of continuous sections of the bundle of complexified algebras &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{C}\text{l}(TM)\)&lt;/span&gt;. Now we are ready to start describing what spin structures are and therefore what spinors are. As spinors are one of the most fundamental objects in quantum physics, a precise notion of what we mean is desirable.&lt;/p&gt;
&lt;p&gt;A Riemannian manifold, &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;, is said to be &lt;span class=&quot;math inline&quot;&gt;\(spin^{c}\)&lt;/span&gt; if there exists a vector bundle&lt;br /&gt;
&lt;span class=&quot;math inline&quot;&gt;\((S, M, End(S), f)\)&lt;/span&gt; such that there is an algebra bundle isomorphism: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
\mathbb{C}\text{l}(TM) &amp;amp;\simeq End(S)  \qquad \text{$M$ is even dimensional}\\
\mathbb{C}\text{l}^0(TM) &amp;amp;\simeq End(S) \qquad \text{$M$ is odd dimensional}\end{aligned}\]&lt;/span&gt; In such a case, the pair &lt;span class=&quot;math inline&quot;&gt;\((M,S)\)&lt;/span&gt; is said to be a &lt;em&gt;&lt;span class=&quot;math inline&quot;&gt;\(spin^c\)&lt;/span&gt; structure&lt;/em&gt;&lt;a href=&quot;#fn4&quot; class=&quot;footnote-ref&quot; id=&quot;fnref4&quot;&gt;&lt;sup&gt;4&lt;/sup&gt;&lt;/a&gt; for &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;If a spin&lt;span class=&quot;math inline&quot;&gt;\(^c\)&lt;/span&gt; structure &lt;span class=&quot;math inline&quot;&gt;\((M,S)\)&lt;/span&gt; exists, the we refer to the bundle as the &lt;em&gt;spinor bundle&lt;/em&gt; and the sections as &lt;em&gt;spinors&lt;/em&gt;. However, not all of these spinors will be physical spinors. As we require physical spinors to be &lt;em&gt;square integrable&lt;/em&gt; and so we now define the space of spinors we want.&lt;/p&gt;
&lt;p&gt;The &lt;em&gt;Hilbert space of square-integrable spinors&lt;/em&gt;, denoted &lt;span class=&quot;math inline&quot;&gt;\(L^2(S)\)&lt;/span&gt;, is define by the completion of &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(S)\)&lt;/span&gt; under the norm induced by the following inner product for &lt;span class=&quot;math inline&quot;&gt;\(\phi_i \in \Gamma(S)\)&lt;/span&gt;. : &lt;span class=&quot;math display&quot;&gt;\[(\phi_1, \phi_2)_M = \int\limits_M \langle \phi_1, \phi_2 \rangle_S \sqrt{g} \mathop{}\!\mathrm{d}x\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The inner product of spinors is just the clifford inner product at each point of the manifold, where the inner product on on the Clifford algebras can be defined on basis elements from its quadratic form as follows:&lt;/p&gt;
&lt;p&gt;Let &lt;span class=&quot;math inline&quot;&gt;\(\{ e_i\}\)&lt;/span&gt; be an orthonormal basis&lt;a href=&quot;#fn5&quot; class=&quot;footnote-ref&quot; id=&quot;fnref5&quot;&gt;&lt;sup&gt;5&lt;/sup&gt;&lt;/a&gt; of V. Setting &lt;span class=&quot;math inline&quot;&gt;\(\alpha = e_1 \dots e_p\)&lt;/span&gt; let &lt;span class=&quot;math inline&quot;&gt;\(\hat{\alpha} =  e_p \dots e_1\)&lt;/span&gt;, then the inner product is &lt;span class=&quot;math display&quot;&gt;\[\langle e_{i_1} e_{i_2} \dots e_{i_p}, e_{j_1} e_{j_2} \dots e_{j_q} \rangle \vcentcolon=\langle 1 ,  e_{i_p}  \dots e_{i_2} e_{i_1} e_{j_1} e_{j_2} \dots e_{j_q}\rangle =
\begin{cases}
0 \quad \text{if $p \neq q$} \\
0 \quad \text{if $e_{i_k} \neq e_{j_k} $ for any $k$}\\
Q(e_{i_1}) \dots Q(e_{i_p}) \quad \text{otherwise}
\end{cases}\]&lt;/span&gt; and can be extended linearly to arbitrary elements of &lt;span class=&quot;math inline&quot;&gt;\(Cl(V,Q)\)&lt;/span&gt; by &lt;span class=&quot;math inline&quot;&gt;\(\langle \alpha , \beta  \rangle =\langle 1, \hat{\alpha} \beta  \rangle\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;However, the model we have for reality requires us to be able to pair up each spinor with its anti-particle spinor. To be able to do this mathematically requires us to have not just a spin&lt;span class=&quot;math inline&quot;&gt;\(^{\mathbb{C}}\)&lt;/span&gt; manifold but a &lt;em&gt;spin manifold&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;A Riemannian spin&lt;span class=&quot;math inline&quot;&gt;\(^c\)&lt;/span&gt; manifold is called a &lt;em&gt;Riemannian spin manifold&lt;/em&gt; if there exists an anti-unitary operator &lt;span class=&quot;math inline&quot;&gt;\(J_M\colon \Gamma(S) \to \Gamma(S)\)&lt;/span&gt; such that:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(J_M\)&lt;/span&gt; commutes with the action of real valued continuous functions on &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(S)\)&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(J_M\)&lt;/span&gt; commutes with &lt;span class=&quot;math inline&quot;&gt;\(\text{Cliff}^-(M)\)&lt;/span&gt; in the even cases and with &lt;span class=&quot;math inline&quot;&gt;\(\text{Cliff}^-(M)^0\)&lt;/span&gt; in the odd cases.&lt;/p&gt;&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The pair &lt;span class=&quot;math inline&quot;&gt;\((S,J_M)\)&lt;/span&gt; is referred to as a &lt;em&gt;spin structure&lt;/em&gt; on &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(J_M\)&lt;/span&gt; is referred to as the &lt;em&gt;charge conjugation&lt;/em&gt; on &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;. Another important structure which is necessary for understanding the basic objects in noncommutative geometry is the notion of a &lt;em&gt;chirality operator&lt;/em&gt;. Let &lt;span class=&quot;math inline&quot;&gt;\(\{\gamma_a\}_{a=1}^n\)&lt;/span&gt; be the generators of &lt;span class=&quot;math inline&quot;&gt;\(\text{Cliff}^+(M)\big|_U\)&lt;/span&gt; for some some open neighbourhood, U, of M, where &lt;span class=&quot;math inline&quot;&gt;\(\{x_a\}_{a=1}^n\)&lt;/span&gt; are the local coordinates. The &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; satisfy: &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\gamma_b + \gamma_b\gamma_a = 2 g(\partial_a, \partial_b)\)&lt;/span&gt; and if we choose an orthonormal basis for &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(TM)\big|_U\)&lt;/span&gt; then the &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; satisfy the relation: &lt;span class=&quot;math display&quot;&gt;\[\gamma_a \gamma_b + \gamma_b \gamma_a = 2 \delta_{ab}\]&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;The &lt;em&gt;chiratity operator&lt;/em&gt; can be constructed from the &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; as follows: &lt;span class=&quot;math display&quot;&gt;\[\gamma_M = (-i)^m \gamma_1 \dots \gamma_n\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(m = n/2\)&lt;/span&gt; (when &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt; is even) or &lt;span class=&quot;math inline&quot;&gt;\(m = (n-1)/2\)&lt;/span&gt; (when n is odd).&lt;/p&gt;
&lt;h2 id=&quot;the-dirac-operator&quot;&gt;The Dirac Operator&lt;/h2&gt;
&lt;p&gt;The Dirac operator can be thought of a “square root” of the Laplacian of a space and contains a lot of information. The Dirac operator describes the dynamics of spinors but also encodes the metric information. Thus if we are given the Dirac operator we can extract the full metric and thus know everything we need to about the space we are making calculations in. Do define it however, the notion of a spin connection is required. Given a Riemannian spin manifold &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt; with spin structure &lt;span class=&quot;math inline&quot;&gt;\((S,J_M)\)&lt;/span&gt;, let &lt;span class=&quot;math inline&quot;&gt;\(\{E_a\}\)&lt;/span&gt; be a local orthonormal basis for &lt;span class=&quot;math inline&quot;&gt;\(TM\big|_U\)&lt;/span&gt; then &lt;span class=&quot;math inline&quot;&gt;\(g(E_a, E_b) = \delta_{ab}\)&lt;/span&gt;. Let &lt;span class=&quot;math inline&quot;&gt;\(\theta^a\)&lt;/span&gt; be the duals to &lt;span class=&quot;math inline&quot;&gt;\(E_a\)&lt;/span&gt;, then the Levi-Civita connection in this basis acts on vectors and one forms in the following way: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
\nabla E_a &amp;amp;=  \tensor{\tilde{\Gamma}}{_a^b_{c}} \mathop{}\!\mathrm{d}{x^c}\otimes E_b \\
\nabla \theta^a &amp;amp;= - \tensor{\tilde{\Gamma}}{^a_{bc}} \mathop{}\!\mathrm{d}{x^b} \otimes \theta^c\end{aligned}\]&lt;/span&gt; And recall that if we have an orthonormal basis for &lt;span class=&quot;math inline&quot;&gt;\(TM|_U\)&lt;/span&gt; then we have that &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a \gamma_b + \gamma_b \gamma_a = 2 \delta_{ab}\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;The spin connection &lt;span class=&quot;math inline&quot;&gt;\(\nabla^S\)&lt;/span&gt; on the Spinor bundle &lt;span class=&quot;math inline&quot;&gt;\((S, M, End(S), f)\)&lt;/span&gt; is the lift of the Levi-Civita connection and in locally is given by: &lt;span class=&quot;math display&quot;&gt;\[\nabla^S_{a} \psi(x) =  \left( \partial_{a} - \frac{1}{4} \tensor{\tilde{\Gamma}}{^b_{ac}} \gamma^d \gamma_d \right) \psi(x)\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The notion of Clifford multiplication is necessary to define the Dirac operator so we include its definition in this setting below:&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Clifford multiplication&lt;/em&gt; is defined as the linear map: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
 c\colon &amp;amp;\Omega_{\text{dR}}^1(M) \times \Gamma(S) \to \Gamma(S) \\
&amp;amp;(\omega, \psi) \to \omega^{\#} \cdot \psi\end{aligned}\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\Omega_{\text{dR}}^1(M)\)&lt;/span&gt; is the space of 1-forms on &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\omega^{\#}\)&lt;/span&gt; vector field in &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(TM)\)&lt;/span&gt; associated to &lt;span class=&quot;math inline&quot;&gt;\(\omega\)&lt;/span&gt;. The vector field acts an endomorphism on the fibres of &lt;span class=&quot;math inline&quot;&gt;\(S\)&lt;/span&gt; via the embedding &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(TM) \to \text{Cliff}^+(M)\subset\Gamma(\text{End}(S))\)&lt;/span&gt;. Choosing local coordinates for &lt;span class=&quot;math inline&quot;&gt;\(U\subset M\)&lt;/span&gt; we can write &lt;span class=&quot;math inline&quot;&gt;\(\omega|_U = \omega_a \mathop{}\!\mathrm{d}{x}^a\)&lt;/span&gt; and we can write the Clifford multiplication as follows &lt;span class=&quot;math display&quot;&gt;\[c(\omega,\psi)|_U = \omega_a (\gamma^a \psi)|_U\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;where the &lt;span class=&quot;math inline&quot;&gt;\(\mathop{}\!\mathrm{d}{x}^a\)&lt;/span&gt; have been identified with &lt;span class=&quot;math inline&quot;&gt;\(\partial_a\)&lt;/span&gt; via the metric and then embedded in &lt;span class=&quot;math inline&quot;&gt;\(\text{Cliff}^+(M)\)&lt;/span&gt; and become the &lt;span class=&quot;math inline&quot;&gt;\(\gamma^a\)&lt;/span&gt;. The pair &lt;span class=&quot;math inline&quot;&gt;\(\left(\Gamma(S),c\right)\)&lt;/span&gt; is a &lt;em&gt;Clifford module&lt;/em&gt; and will play an important roll in the noncommutative setting. For completeness the definition for a Clifford module has been included below:&lt;/p&gt;
&lt;p&gt;A &lt;em&gt;Clifford module&lt;/em&gt; over a compact Riemannian manifold &lt;span class=&quot;math inline&quot;&gt;\((M,g)\)&lt;/span&gt; is a pair &lt;span class=&quot;math inline&quot;&gt;\((\Gamma(E),c)\)&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(E)\)&lt;/span&gt; is the sections of a smooth vector bundle, &lt;span class=&quot;math inline&quot;&gt;\(E\)&lt;/span&gt;, and &lt;span class=&quot;math inline&quot;&gt;\(c\)&lt;/span&gt; is a &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}(TM)\)&lt;/span&gt;-module homomorphism from &lt;span class=&quot;math inline&quot;&gt;\(\Gamma\left(\text{Cl}(TM)\right) \to  \Gamma\left( \text{End}\left(\Gamma(E)\right)\right)\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;We are now ready to define the Dirac operator for a Riemannian manifold, I have included its expression in local coordinates to indicate why so much preparatory work was necessary.&lt;/p&gt;
&lt;p&gt;The &lt;em&gt;Dirac operator&lt;/em&gt; for a spin manifold &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt; with spin structure &lt;span class=&quot;math inline&quot;&gt;\((S,J_M)\)&lt;/span&gt; is the composition of spin connection with Clifford multiplication and can be expressed in local coordinates as: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
\slashed{D}_M &amp;amp;= c \circ \nabla^S\colon \Gamma(S) \to \\
\slashed{D}_M \psi(x) &amp;amp;= -i \gamma^a \left(\partial_a - \frac{1}{4}\tensor{\tilde{\Gamma}}{^b_{ac}} \gamma^c\gamma_b \right) \psi(x)\end{aligned}\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;To see how we can unwrap the Dirac operator and retrieve the metric it is useful to look at the definition of a Dirac operator in terms of vielbeins.&lt;/p&gt;
&lt;p&gt;Vielbeins are defined as follows: Take an orthonormal basis &lt;span class=&quot;math inline&quot;&gt;\(\{e_a \}\)&lt;/span&gt; for &lt;span class=&quot;math inline&quot;&gt;\(\Gamma(TM)\)&lt;/span&gt; such that&lt;a href=&quot;#fn6&quot; class=&quot;footnote-ref&quot; id=&quot;fnref6&quot;&gt;&lt;sup&gt;6&lt;/sup&gt;&lt;/a&gt; &lt;span class=&quot;math inline&quot;&gt;\(g(e_a,e_b)(x) = \delta_{ab}\)&lt;/span&gt;, we can express this basis in terms of the coordinate basis &lt;span class=&quot;math inline&quot;&gt;\(\{ \partial_{\mu} \}\)&lt;/span&gt; as follows &lt;span class=&quot;math inline&quot;&gt;\(e_a(x) = \tensor{e}{^{\mu}_a}(x) \partial_{\mu}(x)\)&lt;/span&gt;. A vielbein is defined to be &lt;span class=&quot;math inline&quot;&gt;\(\tensor{e}{^{\mu}_a}\)&lt;/span&gt;; the invertible transformation matrix, however the name often extends to the orthonormal basis also. Also note that the metric equation about can now be rewritten as &lt;span class=&quot;math inline&quot;&gt;\(g_{\mu \nu}(x) \tensor{e}{^{\mu}_a}(x) \tensor{e}{^{\nu}_b}(x) = \delta_{ab}\)&lt;/span&gt; or equivalently &lt;span class=&quot;math display&quot;&gt;\[g_{\mu \nu}(x) = \tensor{e}{_\mu^{a}}(x) \tensor{e}{_{\nu}^b}(x) \delta_{ab}\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\tensor{e}{^\mu_a}(x) \tensor{e}{^{a}_{\nu}}(x) = \tensor{\delta}{^\mu_\nu}\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\tensor{e}{^\mu_a}(x) \tensor{e}{^{b}_{\mu}}(x) = \tensor{\delta}{_a^b}\)&lt;/span&gt;. The Dirac operator in terms of these vielbeins is defined as below: &lt;span class=&quot;math display&quot;&gt;\[\slashed{D}_M = -i \gamma^a \tensor{e}{_a^b} \nabla^S_b
\label{tetdir}\]&lt;/span&gt; So given a Dirac operator we can extract the vielbeins and therefore we can reconstruct the full metric.&lt;/p&gt;
&lt;h1 id=&quot;noncommutative-geometry&quot;&gt;Noncommutative Geometry&lt;/h1&gt;
&lt;p&gt;We are now ready to delve into noncommutative geometry and we start with a definition which took many years to refine and is still only appropriate for Riemannian spin geometries , although current research is investigating formalisms for “noncommutative psudeo-Riemannian spin geometry”. A noncommutative geometry is described by a collection of objects called a spectral triple:&lt;/p&gt;
&lt;p&gt;A &lt;em&gt;spectral triple&lt;/em&gt; is a triple &lt;span class=&quot;math inline&quot;&gt;\(\left( \mathcal{H},\mathcal{A}, D\right)\)&lt;/span&gt;, where:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; is a Hilbert space&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A}\)&lt;/span&gt; is a &lt;span class=&quot;math inline&quot;&gt;\(\ast\)&lt;/span&gt;-unital algebra represented as bounded operators on &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(D\)&lt;/span&gt; is a self-adjoint operator on &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; such that the resolvent &lt;span class=&quot;math inline&quot;&gt;\((i+D)^{-1}\)&lt;/span&gt; is a compact operator and &lt;span class=&quot;math inline&quot;&gt;\(\left[D, a \right]\)&lt;/span&gt; is bounded for each &lt;span class=&quot;math inline&quot;&gt;\(a\in \mathcal{A}\)&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;We often require additional structure to our noncommutative geometries if we require them to be appropriate generalisations of Riemannian spin geometries. The first is a &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{Z}_2\)&lt;/span&gt;-grading &lt;span class=&quot;math inline&quot;&gt;\(\gamma\)&lt;/span&gt; on the Hilbert space &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; such that: &lt;span class=&quot;math display&quot;&gt;\[\gamma a = a \gamma \quad (\forall a\in \mathcal{A}), \qquad \gamma D = -D \gamma\]&lt;/span&gt; If such a &lt;span class=&quot;math inline&quot;&gt;\(\gamma\)&lt;/span&gt; exists then we the spectral triple is said to be &lt;em&gt;even&lt;/em&gt;. The second is an anti-linear map &lt;span class=&quot;math inline&quot;&gt;\(J\colon \mathcal{H} \to \mathcal{H}\)&lt;/span&gt; such that: &lt;span class=&quot;math display&quot;&gt;\[J^2 = \epsilon, \qquad JD = \epsilon^\prime DJ, \qquad J\gamma = \epsilon^{\prime \prime} \gamma J \quad (\text{when $\gamma$ exists})\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\epsilon, \epsilon^{\prime}, \epsilon^{\prime \prime}\)&lt;/span&gt; take values from Table &lt;a href=&quot;#epsilonval&quot; data-reference-type=&quot;ref&quot; data-reference=&quot;epsilonval&quot;&gt;[epsilonval]&lt;/a&gt;. We also require If such a &lt;span class=&quot;math inline&quot;&gt;\(J\)&lt;/span&gt; exists then the spectral triple is said to be &lt;em&gt;real&lt;/em&gt; and &lt;span class=&quot;math inline&quot;&gt;\(J\)&lt;/span&gt; is referred to the as the &lt;em&gt;real structure&lt;/em&gt;.&lt;/p&gt;

&lt;p&gt;&lt;span&gt;Y Y Y Y Y Y Y Y Y &lt;/span&gt;&lt;br /&gt;
&lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt; &amp;amp; 0 &amp;amp; 1 &amp;amp; 2 &amp;amp; 3 &amp;amp; 4 &amp;amp; 5 &amp;amp; 6 &amp;amp; 7&lt;br /&gt;
&lt;span class=&quot;math inline&quot;&gt;\(\epsilon\)&lt;/span&gt; &amp;amp; 1&amp;amp; 1 &amp;amp;-1 &amp;amp;-1 &amp;amp;-1 &amp;amp;-1 &amp;amp;1 &amp;amp;1&lt;br /&gt;
&lt;span class=&quot;math inline&quot;&gt;\(\epsilon^{\prime}\)&lt;/span&gt; &amp;amp; 1 &amp;amp;-1 &amp;amp;1 &amp;amp;1 &amp;amp;1 &amp;amp;-1 &amp;amp;1 &amp;amp;1&lt;br /&gt;
&lt;span class=&quot;math inline&quot;&gt;\(\epsilon^{\prime\prime}\)&lt;/span&gt; &amp;amp;1 &amp;amp; 1 &amp;amp;-1 &amp;amp; 1 &amp;amp;1 &amp;amp; 1 &amp;amp; -1 &amp;amp; 1&lt;br /&gt;
&lt;/p&gt;
&lt;p&gt;&lt;span id=&quot;epsilonval&quot; label=&quot;epsilonval&quot;&gt;[epsilonval]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The values for &lt;span class=&quot;math inline&quot;&gt;\(\epsilon, \epsilon^{\prime}, \epsilon^{\prime\prime}\)&lt;/span&gt; are determined by the &lt;em&gt;KO dimension&lt;/em&gt; of the real spectral triple, which for the spectral triple of a Riemannian spin manifold will be equal to the dimension of the manifold modulo 8.&lt;/p&gt;
&lt;p&gt;We also impose to two extra conditions which are chosen to keep up from straying too far away from commutative geometry. Firstly, using the &lt;span class=&quot;math inline&quot;&gt;\(J\)&lt;/span&gt; operator we can define the &lt;em&gt;opposite algebra&lt;/em&gt; which entails the right action of the algebra. If &lt;span class=&quot;math inline&quot;&gt;\(a\in\mathcal{A}\)&lt;/span&gt; then we can define &lt;span class=&quot;math inline&quot;&gt;\(a^{\circ} = J a^\ast J^{-1}\)&lt;/span&gt;. Thus we can define the right action of the algebra on the Hilbert space to be &lt;span class=&quot;math inline&quot;&gt;\(\psi b = b^{\circ} \psi\)&lt;/span&gt;. The first requirement is that: &lt;span class=&quot;math display&quot;&gt;\[\left[a,b^\circ \right] = 0 \hspace{1cm} \text{\emph{Order zero condition}}\]&lt;/span&gt; which is used to model the fact that if we take a function &lt;span class=&quot;math inline&quot;&gt;\(f\in C^{\infty}(M)\)&lt;/span&gt; for some Riemannian spin manifold and &lt;span class=&quot;math inline&quot;&gt;\(\psi\)&lt;/span&gt; some spinor, then &lt;span class=&quot;math inline&quot;&gt;\((f\psi)(x)  = f(x)\psi(x) = (\psi f )(x)\)&lt;/span&gt;. In other words, in commutative geometry we cannot distinguish between left and right actions of the algebra of functions and we want to keep some property of this commutative geometry feature even when the ‘functions’ no longer commute. The other condition is to do with the fact that commutation with the Dirac operator should also commute with the right action: &lt;span class=&quot;math display&quot;&gt;\[\left[ \left[D,a\right], b^\circ \right] = 0 \hspace{1cm} \text{\emph{First order condition}}.\]&lt;/span&gt; If we again look at commutative geometry, the Dirac operator can be expressed as in Eq &lt;a href=&quot;#tetdir&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;tetdir&quot;&gt;[tetdir]&lt;/a&gt;, and thus: &lt;span class=&quot;math inline&quot;&gt;\(\left[ D, f \right]\psi = -i\gamma^a e_a^b (\nabla^S_b f) \psi\)&lt;/span&gt; and so acts as just multiplying pointwise by a function and so will commute with a right multiplication by a function. It is this property we wish to persist in our noncommutative geometry. We also note that there are noncommutative geometry models where higher order conditions are available, notably the work by Boyle and Farnsworth &lt;span class=&quot;citation&quot; data-cites=&quot;Boyle:2016ut&quot;&gt;&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;We are now ready to write down the spectral triple for a commutative geometry (a Riemannian spin manifold):&lt;/p&gt;
&lt;p&gt;The real spectral triple for a Riemannian spin manifold, &lt;span class=&quot;math inline&quot;&gt;\(M\)&lt;/span&gt;, with spin structure &lt;span class=&quot;math inline&quot;&gt;\((S,J_M)\)&lt;/span&gt; is the structure &lt;span class=&quot;math inline&quot;&gt;\(\left(L^2(S), C^{\infty}(M), \slashed{D}_M; \gamma_M, J_M \right)\)&lt;/span&gt;, where &lt;span class=&quot;math inline&quot;&gt;\(L^2(S)\)&lt;/span&gt; is the Hilbert space as the space of square integrable spinors and &lt;span class=&quot;math inline&quot;&gt;\(C^{\infty}(M)\)&lt;/span&gt; is the algebra of smooth functions. And where &lt;span class=&quot;math inline&quot;&gt;\(\slashed{D}_M\)&lt;/span&gt;, &lt;span class=&quot;math inline&quot;&gt;\(\gamma_M\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(J_M\)&lt;/span&gt; are the Dirac operator, chirality operator and charge conjugation as defined in the previous section.&lt;/p&gt;
&lt;h1 id=&quot;fuzzy-space&quot;&gt;Fuzzy Space&lt;/h1&gt;
&lt;p&gt;We are going to look at a specific type of noncommutative geometry, referred to as a &lt;em&gt;fuzzy space&lt;/em&gt;. Fuzzy spaces belong to a class of noncommutative geometries called &lt;em&gt;matrix geometries&lt;/em&gt;, which briefly are a spectral triple that is a product of a type &lt;span class=&quot;math inline&quot;&gt;\((0,0)\)&lt;/span&gt; matrix geometry with a Clifford module. I will go onto define this is more detail below. Fuzzy spaces can be thought of as approximating Riemannian spin manifold with &lt;span class=&quot;math inline&quot;&gt;\(n \times n\)&lt;/span&gt; matrices where &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt; is an integer free to choose. Firstly, some important facts about Clifford modules have been described:&lt;/p&gt;
&lt;p&gt;To successfully describe a fuzzy space we have to deal with Clifford algebras over &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^n\)&lt;/span&gt; such that &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}(\mathbb{R}^n)\)&lt;/span&gt; has generators &lt;span class=&quot;math inline&quot;&gt;\(\{\gamma_a\}_{a=1}^n\)&lt;/span&gt;which satisfy: &lt;span class=&quot;math display&quot;&gt;\[\gamma_a \gamma_b + \gamma_b \gamma_a = 2 \eta_{ab}\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\eta\)&lt;/span&gt; is a diagonal matrix with &lt;span class=&quot;math inline&quot;&gt;\(\pm 1\)&lt;/span&gt; in its entries. However, as we still require a Hilbert space in our spectral triple, which in turn requires us to have a positive definite inner product. We require that all the &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt;’s are unitary and we have the standard hermitian inner product &lt;span class=&quot;math inline&quot;&gt;\((u,v) = \sum_i \bar{u}_i v_i\)&lt;/span&gt; on the Hilbert space. This requirement doesn’t impose any restrictions on the &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; as they form a finite group and finite groups always have a unitary representation &lt;span class=&quot;citation&quot; data-cites=&quot;Hamermesh:1962wm&quot;&gt;&lt;/span&gt;. Which just leaves us to choose a basis of the vector space these elements act on, &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt;, which will be the Hilbert space of our spectral triple, such that the hermitian form is the standard one. Thus, if &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a^2 = 1\)&lt;/span&gt; then &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; is Hermitian and if &lt;span class=&quot;math inline&quot;&gt;\(\gamma_b^2 = -1\)&lt;/span&gt; then &lt;span class=&quot;math inline&quot;&gt;\(\gamma_b\)&lt;/span&gt; is anti-Hermitian. Note that if &lt;span class=&quot;math inline&quot;&gt;\(\eta\)&lt;/span&gt; has &lt;span class=&quot;math inline&quot;&gt;\(p\)&lt;/span&gt; entries of &lt;span class=&quot;math inline&quot;&gt;\(+1\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(q\)&lt;/span&gt; entries of &lt;span class=&quot;math inline&quot;&gt;\(-1\)&lt;/span&gt;, then we say the Clifford module is of type &lt;span class=&quot;math inline&quot;&gt;\((p,q)\)&lt;/span&gt;, and the number &lt;span class=&quot;math inline&quot;&gt;\(s = p-q\)&lt;/span&gt; (mod 8), called the signature of the clifford module, determines much of the characteristics of the Clifford module. The signature, &lt;span class=&quot;math inline&quot;&gt;\(s\)&lt;/span&gt;, also coincides with the KO-dimension of the associated spectral triple. Representing this Clifford algebra as complex matrices&lt;a href=&quot;#fn7&quot; class=&quot;footnote-ref&quot; id=&quot;fnref7&quot;&gt;&lt;sup&gt;7&lt;/sup&gt;&lt;/a&gt;, which act on a vector space &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt;, provides us with the Clifford module we require to define a fuzzy space.&lt;/p&gt;
&lt;p&gt;Now let us define a type &lt;span class=&quot;math inline&quot;&gt;\((0,0)\)&lt;/span&gt; matrix geometry:&lt;/p&gt;
&lt;p&gt;A type &lt;span class=&quot;math inline&quot;&gt;\((0,0)\)&lt;/span&gt; matrix geometry&lt;a href=&quot;#fn8&quot; class=&quot;footnote-ref&quot; id=&quot;fnref8&quot;&gt;&lt;sup&gt;8&lt;/sup&gt;&lt;/a&gt; is a real spectral triple of KO dimension &lt;span class=&quot;math inline&quot;&gt;\(s_0 = 0\)&lt;/span&gt; and the following objects: &lt;span class=&quot;math inline&quot;&gt;\((\mathcal{H}_0, \mathcal{A}_0, D_0=0; \gamma_0 = 1, J_0)\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;It can be shown (see &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;) that all type &lt;span class=&quot;math inline&quot;&gt;\((0,0)\)&lt;/span&gt; matrix geometries are isomorphic to the case when &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}_0\)&lt;/span&gt; is a &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{C}\)&lt;/span&gt;-linear vector subspace of &lt;span class=&quot;math inline&quot;&gt;\(M_n(\mathbb{C})\)&lt;/span&gt; such that if &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{C}^n\)&lt;/span&gt; with its standard Hermitian inner product is a faithful left module of the algebra &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A}_0\)&lt;/span&gt;, we have that &lt;span class=&quot;math inline&quot;&gt;\(am \in \mathcal{H}_0\)&lt;/span&gt; and also &lt;span class=&quot;math inline&quot;&gt;\(m^*\in \mathcal{H}_0\)&lt;/span&gt;. If this is the case then the representation of &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A}_0\)&lt;/span&gt; on to &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}_0\)&lt;/span&gt; is just matrix multiplication, the real structure is just Hermitian conjugation &lt;span class=&quot;math inline&quot;&gt;\(J_0(\cdot) = (\cdot)^\ast\)&lt;/span&gt; and the inner product is just &lt;span class=&quot;math inline&quot;&gt;\((m_1,m_2) = \text{Tr}( m_1^\ast m_2)\)&lt;/span&gt;. It can be shown that all the axioms for a type &lt;span class=&quot;math inline&quot;&gt;\((0,0)\)&lt;/span&gt; matrix geometry are satisfied by this collection of objects (see &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;).&lt;/p&gt;
&lt;p&gt;We are now ready to define a fuzzy geometry. First we take all of the algebras in question to be &lt;em&gt;simple algebras&lt;/em&gt;. Thats means &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A} = M_n(\mathbb{C}), M_n(\mathbb{R})\)&lt;/span&gt; or &lt;span class=&quot;math inline&quot;&gt;\(M_{n/2}(\mathbb{H})\)&lt;/span&gt;. For the sake of brevity, the assumption that &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A} = M_N(\mathbb{C})\)&lt;/span&gt; will be taken. Furthermore we impose that &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}_0 = M_n(\mathbb{C})\)&lt;/span&gt;, where in the case of &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A} = M_{n/2}(\mathbb{H})\)&lt;/span&gt; we express the quaternions as &lt;span class=&quot;math inline&quot;&gt;\(2\times 2\)&lt;/span&gt; complex matrices, such that &lt;span class=&quot;math inline&quot;&gt;\(M_{n/2}(\mathbb{H}) \subset M_n(\mathbb{C})\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Let &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt; be a Clifford module of type &lt;span class=&quot;math inline&quot;&gt;\((p,q)\)&lt;/span&gt; with chirality operator &lt;span class=&quot;math inline&quot;&gt;\(\gamma\)&lt;/span&gt; and real structure &lt;span class=&quot;math inline&quot;&gt;\(C\)&lt;/span&gt;. For &lt;span class=&quot;math inline&quot;&gt;\(p+q\)&lt;/span&gt; even let &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt; be irreducible and for &lt;span class=&quot;math inline&quot;&gt;\(p+q\)&lt;/span&gt; odd let the chiral subspaces &lt;span class=&quot;math inline&quot;&gt;\(V_\pm\)&lt;/span&gt; be irreducible. A &lt;em&gt;fuzzy space&lt;/em&gt; is a real spectral triple with the following objects&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H} = V \otimes M_n(\mathbb{C})\)&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A} = M_n(\mathbb{C})\)&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt; &lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\Gamma = \gamma \otimes 1\)&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(J = C\otimes J_0\)&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;where the inner product on &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; is defined by: &lt;span class=&quot;math display&quot;&gt;\[\langle v_1 \otimes m_1, v_2 \otimes m_2  \rangle = (v_1,v_2) \text{Tr}(m_1^\ast m_2)\]&lt;/span&gt; and the action of the algebra is just my multiplication on &lt;span class=&quot;math inline&quot;&gt;\(M_n(\mathbb{C})\)&lt;/span&gt;, i.e. &lt;span class=&quot;math inline&quot;&gt;\(\rho(a)(v\otimes m) = v \otimes (am)\)&lt;/span&gt;. The actions of &lt;span class=&quot;math inline&quot;&gt;\(\Gamma\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(J\)&lt;/span&gt; on an element of &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; are as follows: &lt;span class=&quot;math display&quot;&gt;\[\Gamma(v\otimes m) = \gamma v \otimes m, \qquad J(v \otimes m) = Cv \otimes m^{\ast}\]&lt;/span&gt; And the final object is the Dirac operator, &lt;span class=&quot;math inline&quot;&gt;\(D\)&lt;/span&gt;, where it can be shown (see &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;) that depending on the sign of &lt;span class=&quot;math inline&quot;&gt;\(\epsilon^{\prime}\)&lt;/span&gt;, takes the following forms:&lt;/p&gt;

&lt;p&gt;For this case we have that &lt;span class=&quot;math inline&quot;&gt;\(J\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(D\)&lt;/span&gt; commute and that &lt;span class=&quot;math inline&quot;&gt;\(\langle D(v_1 \otimes m_1), v_2 \otimes m_2  \rangle= \langle v_1 \otimes m_1, D(v_2 \otimes m_2 ) \rangle\)&lt;/span&gt;, we have that the Dirac operator has to be of the form &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;: &lt;span class=&quot;math display&quot;&gt;\[D (v\otimes m ) = \sum_i \alpha^i v \otimes \left[ L_i, m \right] + \sum_j \tau^{j} v \otimes \{H_j, m \}
\label{D1}\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(\alpha^i\)&lt;/span&gt; are products of gamma matrices and both &lt;span class=&quot;math inline&quot;&gt;\(\alpha^i\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(L_i\)&lt;/span&gt; are anti-Hermitian matrices, and where &lt;span class=&quot;math inline&quot;&gt;\(\tau^j\)&lt;/span&gt; are products of gamma matrices and both &lt;span class=&quot;math inline&quot;&gt;\(\tau^j\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(H_j\)&lt;/span&gt; are Hermitian matrices.&lt;/p&gt;

&lt;p&gt;For this case, we have that &lt;span class=&quot;math inline&quot;&gt;\(J\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(D\)&lt;/span&gt; now anti-commute and also note that &lt;span class=&quot;math inline&quot;&gt;\(C\)&lt;/span&gt; anti-commutes with the &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; in such cases, so we need to split the sums into sums where we have a product of even or odd number of &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt;. However we still require &lt;span class=&quot;math inline&quot;&gt;\(D\)&lt;/span&gt; to be self-adjoint, so we still require &lt;span class=&quot;math inline&quot;&gt;\(D\)&lt;/span&gt; to have either entirely Hermitian or entirely anti-Hermitian entries. This leaves us with the following form &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
D (v\otimes m ) &amp;amp;= \sum_i \alpha_-^i v \otimes \left[ L_i, m \right] + \sum_j \tau_-^{j} v \otimes \left\{H_j, m \right\} \\
              &amp;amp;+ \sum_k \alpha_+^k v \otimes \left\{L_k, m \right\} + \sum_l \tau_+^{l} v \otimes \left[H_l, m\right]
            \label{D-1}\end{aligned}\]&lt;/span&gt; where the &lt;span class=&quot;math inline&quot;&gt;\(+\)&lt;/span&gt; or &lt;span class=&quot;math inline&quot;&gt;\(-\)&lt;/span&gt; subscript on &lt;span class=&quot;math inline&quot;&gt;\(\alpha, \tau\)&lt;/span&gt; indicates whether they are the product of an even number of &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt;, indicated by &lt;span class=&quot;math inline&quot;&gt;\(+\)&lt;/span&gt;, or by an odd number of &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt;, indicated by a &lt;span class=&quot;math inline&quot;&gt;\(-\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;To show the usefulness of this rather abstract concept, a brief account of a well studied example is that of the &lt;em&gt;fuzzy sphere&lt;/em&gt; is included below.&lt;/p&gt;
&lt;h2 id=&quot;the-fuzzy-sphere&quot;&gt;The Fuzzy Sphere&lt;/h2&gt;
&lt;p&gt;For an appropriate approximation to the sphere, which gets closer and closer to model a sphere as we increase the matrix size, &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt;, we utilise the fact that the sphere has &lt;span class=&quot;math inline&quot;&gt;\(SO(3)\)&lt;/span&gt; symmetry. If we take the Lie algebra of &lt;span class=&quot;math inline&quot;&gt;\(SO(3)\)&lt;/span&gt;, namely &lt;span class=&quot;math inline&quot;&gt;\(so(3)\)&lt;/span&gt; and take an &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt;-dimensional representation of it. Its generators are to be denoted by &lt;span class=&quot;math inline&quot;&gt;\(L_{ij}\)&lt;/span&gt;, with &lt;span class=&quot;math inline&quot;&gt;\(i&amp;lt;j=1,2,3\)&lt;/span&gt; such that they satisfy: &lt;span class=&quot;math display&quot;&gt;\[\left[L_{ij}, L_{kl} \right] = \delta_{jk} L_{il} + \delta_{il} L_{jk} - \delta_{jl} L_{ik} - \delta_{ik} L_{jl}\]&lt;/span&gt; with the fact that &lt;span class=&quot;math inline&quot;&gt;\(L_{ij} = - L_{jk}\)&lt;/span&gt;. These generators, represented into the algebra, &lt;span class=&quot;math inline&quot;&gt;\(M_n(\mathbb{C})\)&lt;/span&gt;, with be the Hermitian/anti-Hermitian matrices that go into the Dirac operator. So the precise set up for the fuzzy sphere has undergone some changes in the recent past. One of the most noteworthy approaches is that by Grosse and Prešnajder &lt;span class=&quot;citation&quot; data-cites=&quot;Grosse:1995gq&quot;&gt;&lt;/span&gt;, however they choose a type &lt;span class=&quot;math inline&quot;&gt;\((0,3)\)&lt;/span&gt; fuzzy space, but this yields a KO dimension of &lt;span class=&quot;math inline&quot;&gt;\(s=3\)&lt;/span&gt; and as we want it to be the fuzzy analogue to the spheres dimensions which is &lt;span class=&quot;math inline&quot;&gt;\(s=2\)&lt;/span&gt;. However, the method implemented by Barrett &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;, enriches the &lt;span class=&quot;math inline&quot;&gt;\((0,3)\)&lt;/span&gt; geometry to a &lt;span class=&quot;math inline&quot;&gt;\((1,3)\)&lt;/span&gt; geometry, where the new Hermitian generator is denoted by &lt;span class=&quot;math inline&quot;&gt;\(\gamma_0\)&lt;/span&gt;. In which the KO dimension would be &lt;span class=&quot;math inline&quot;&gt;\(s=3-1 = 2\)&lt;/span&gt;, as desired for a fuzzy sphere.&lt;/p&gt;
&lt;p&gt;The Dirac operator for the fuzzy sphere is found to be:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math display&quot;&gt;\[D = \gamma_0 + \sum\limits_{i&amp;lt;j=1}^3 \gamma_0 \gamma_i \gamma_j \otimes \left[L_{ij}, \cdot \right]
\label{FuzzSphDirac}\]&lt;/span&gt; we note that the product &lt;span class=&quot;math inline&quot;&gt;\(\gamma_0 \gamma_i\gamma_j\)&lt;/span&gt; is anti-Hermitian and an odd product so this definition of the Dirac operator agrees with the Dirac operator definitions in &lt;a href=&quot;#D1&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;D1&quot;&gt;[D1]&lt;/a&gt; and &lt;a href=&quot;#D-1&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;D-1&quot;&gt;[D-1]&lt;/a&gt;. It is possible to arrive back at the Grosse-Prešnajder Dirac operator as done by Barrett in &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;, and for reference later on the form of the Grosse-Prešnajder Dirac operator is: &lt;span class=&quot;math display&quot;&gt;\[D_{\text{G-P}} = 1 + \sum\limits_{i&amp;lt;j=1}^{3}\sigma_i \sigma_j \otimes \left[L_{ij},\cdot \right]
\label{GPDirac}\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The Dirac operators both yields the spectrum &lt;span class=&quot;math display&quot;&gt;\[Spec(D) = \pm 1, \pm 2, \pm 3, \dots \pm n-1, +n\]&lt;/span&gt; So as we let the matrix size tend to infinity we retrieve the spectrum &lt;span class=&quot;math inline&quot;&gt;\(\pm\mathbb{Z}\)&lt;/span&gt;, which is the spectrum for the Dirac operator on the commutative 2-sphere.&lt;/p&gt;
&lt;p&gt;The link between these Dirac operators and the commutative 2-sphere’s Dirac operator is more clearly seen when the commutative 2-spheres Dirac operator is expressed as embedded from &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^3\)&lt;/span&gt; as opposed to intrinsically in terms of local coordinates. A method to get a general hyper-surface is reviewed below, and an attempt to look at the fuzzy analogue for the ellipsoid is pursued.&lt;/p&gt;
&lt;p&gt;However, finally, the reason current research into noncommutative geometry as a model for quantum gravity is due to the interpretation the ‘fuzziness’ has on the small scale of the fuzzy sphere. A great paper for further details about the following is that by Madore &lt;span class=&quot;citation&quot; data-cites=&quot;Madore:1992ij&quot;&gt;&lt;/span&gt;. In brief, it explains the reasons why noncommutativity is necessary for a finite model of a sphere, but that is still &lt;span class=&quot;math inline&quot;&gt;\(SO(3)\)&lt;/span&gt; invariant. The premise is that if we consider the commutative 2-sphere, and look at the algebra of coordinates, &lt;span class=&quot;math inline&quot;&gt;\(C(\mathbb{S}^2)\)&lt;/span&gt;. This is the algebra spanned by the products of the coordinates &lt;span class=&quot;math inline&quot;&gt;\(x^1, x^2, x^3\)&lt;/span&gt;, so elements have polynomial expansions such as: &lt;span class=&quot;math display&quot;&gt;\[f(x^i) = f_0 + f_a x^a + f_{ab}x^{a}x^b \dots\]&lt;/span&gt; for some constants &lt;span class=&quot;math inline&quot;&gt;\(f_0, f_a, f_{ab}, \dots\)&lt;/span&gt; and where Einstein summation is assumed. This algebra has the ability to separate the points of a sphere and is also dense is the algebra of smooth functions &lt;span class=&quot;math inline&quot;&gt;\(C^{\infty}(\mathbb{S}^2)\)&lt;/span&gt;, so it is good enough to deal with just &lt;span class=&quot;math inline&quot;&gt;\(C(\mathbb{S}^2)\)&lt;/span&gt; in this setting.&lt;/p&gt;
&lt;p&gt;Now if we approximate the functions by truncating their series expansion after the &lt;span class=&quot;math inline&quot;&gt;\(n^{\text{th}}\)&lt;/span&gt; products, i.e. &lt;span class=&quot;math inline&quot;&gt;\(f(x^i) = f_0 + f_a x^a + f_{ab}x^{a}x^b \dots + f_{a_1 a_2 \dots a_n} x^{a_1} x^{a_2} \dots x^{a_n}\)&lt;/span&gt; and take all the functions of this type, we would end up with a n-dimensional vector space of the entries &lt;span class=&quot;math inline&quot;&gt;\(f_0, \dots, f_{a_1 a_2 \dots a_n}\)&lt;/span&gt;. In order to turn this vector space into an algebra requires some thought. The normal product of two elements &lt;span class=&quot;math inline&quot;&gt;\(f,g\)&lt;/span&gt;, we would obviously be taken out of this set of objects as it would require terms such as &lt;span class=&quot;math inline&quot;&gt;\(x^{a_1} x^{a_2} \dots x^{a_n} x^{a_{n+1}}\)&lt;/span&gt; etc. Looking at the case when &lt;span class=&quot;math inline&quot;&gt;\(n =1\)&lt;/span&gt;, we could define a product to be &lt;span class=&quot;math inline&quot;&gt;\(f\cdot g = f_0g_0 + \sum_{a=1}^3f_a g_a x^a\)&lt;/span&gt;, this would turn the vector space into an algebra. However, we can identify this algebra with four copies of &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{C}\)&lt;/span&gt;, which is exactly the algebra of functions at four discrete points. This product can be extended to any value for &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt; in the straight forward manner and we still only get an algebra which is the functions on a set of discrete points. This is unappealing because these point will not be invariant under the action of &lt;span class=&quot;math inline&quot;&gt;\(SO(3)\)&lt;/span&gt; and thus we have no notion of a fuzzy sphere.&lt;/p&gt;
&lt;p&gt;A way to preserve this invariance under the action of &lt;span class=&quot;math inline&quot;&gt;\(SO(3)\)&lt;/span&gt; is to make the algebra we desire noncommutative. We start by taking the &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt;-dimensional generators of &lt;span class=&quot;math inline&quot;&gt;\(su(2)\simeq so(3)\)&lt;/span&gt;, and denote them by &lt;span class=&quot;math inline&quot;&gt;\(L_{a}\)&lt;/span&gt;, the we set &lt;span class=&quot;math inline&quot;&gt;\(x^a = \kappa L_a\)&lt;/span&gt;, where &lt;span class=&quot;math inline&quot;&gt;\(\kappa\)&lt;/span&gt; is a constant set by requiring &lt;span class=&quot;math inline&quot;&gt;\(\sum_i (x^i)^2 = 1\)&lt;/span&gt; (&lt;span class=&quot;math inline&quot;&gt;\(\kappa\)&lt;/span&gt;’s value will vary with the value for &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt;). Firstly we notice that the coordinates no longer commute, we have &lt;span class=&quot;math inline&quot;&gt;\(\left[x^1,x^1 \right] = i \kappa x^3\)&lt;/span&gt;, and the cyclic permutations. So the notion of a point &lt;span class=&quot;math inline&quot;&gt;\((x^1,x^2,x^3)\)&lt;/span&gt; vanishes as we can never know all three entries at the same time. However, if we take a look at the 2-dimensional representations; the Pauli matrices, we notice that we get 2 eigenvalues for each generator, namely &lt;span class=&quot;math inline&quot;&gt;\(\pm 1\)&lt;/span&gt;. This is interpreted as being able to only distinguish which hemisphere of the sphere we lie in. As we increase the dimension of the representation we get more eigenvalues for each of the generators, which means we can narrow down the zone we lie. Which gives credence in the name fuzzy sphere as the spaces we end up with are like the points of a sphere have been smeared together in a certain fashion. Also as for a given &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt;, the value of &lt;span class=&quot;math inline&quot;&gt;\(\kappa = \frac{1}{n^2 - 1}\)&lt;/span&gt; we have that as &lt;span class=&quot;math inline&quot;&gt;\(n \to \infty\)&lt;/span&gt; then &lt;span class=&quot;math inline&quot;&gt;\(\kappa \to 0\)&lt;/span&gt;, so the coordinates in the limit commute and we retrieve the normal commutative sphere. This interpretation gives us a natural picture of what we mean by a ‘quantum geometry’. However, there is another interpretation which is arguably more enticing as it imposes a cutoff on the spherical harmonics and in turn the energy states. For a full treatment of this view point see &lt;span class=&quot;citation&quot; data-cites=&quot;DAndrea:2012dh&quot;&gt;&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;Going back to the algebra of coordinates, we can decompose &lt;span class=&quot;math inline&quot;&gt;\(C(\mathbb{S}^2)\)&lt;/span&gt; in to a direct sum of irreducible representations of &lt;span class=&quot;math inline&quot;&gt;\(su(2)\)&lt;/span&gt;: &lt;span class=&quot;math display&quot;&gt;\[C(\mathbb{S}^2) \simeq \bigoplus\limits_{l=0}^\infty V_l\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(V_l\)&lt;/span&gt; is the vector space underlying the irreducible representation of &lt;span class=&quot;math inline&quot;&gt;\(su(2)\)&lt;/span&gt; with the highest weight &lt;span class=&quot;math inline&quot;&gt;\(l\in \mathbb{N}\)&lt;/span&gt; which is spanned by the spherical harmonics &lt;span class=&quot;math inline&quot;&gt;\(Y_{l,m}\)&lt;/span&gt;. We can then impose a cutoff in the energy spectrum by ignore all but the first &lt;span class=&quot;math inline&quot;&gt;\(n+1\)&lt;/span&gt; representations in the decomposition of &lt;span class=&quot;math inline&quot;&gt;\(C(\mathbb{S}^2)\)&lt;/span&gt;. Thus, in the fuzzy sphere’s spectral triple we can take the &lt;em&gt;fuzzy spherical harmonics&lt;/em&gt;&lt;a href=&quot;#fn9&quot; class=&quot;footnote-ref&quot; id=&quot;fnref9&quot;&gt;&lt;sup&gt;9&lt;/sup&gt;&lt;/a&gt; &lt;span class=&quot;math inline&quot;&gt;\(\hat{Y}_{l,m}\)&lt;/span&gt; to be the generators of &lt;span class=&quot;math inline&quot;&gt;\(M_{n+1}(\mathbb{C})\)&lt;/span&gt;, where&lt;a href=&quot;#fn10&quot; class=&quot;footnote-ref&quot; id=&quot;fnref10&quot;&gt;&lt;sup&gt;10&lt;/sup&gt;&lt;/a&gt; &lt;span class=&quot;math inline&quot;&gt;\(l&amp;lt;n\)&lt;/span&gt;. We can then decompose the algebra into &lt;span class=&quot;math display&quot;&gt;\[\mathcal{A}_n \simeq \bigoplus\limits_{l=0}^{n}V_l\]&lt;/span&gt; Thus so a fuzzy sphere can be viewed as having an energy cutoff, which can be recast as a minimal renderable distance, i.e. a Planck length. The implications of a planck length being a natural outcome of requiring the underlying space to be noncommutative is a very appealing property and the idea is that a noncommutative analogue to a spacetime will provide a good model for quantum gravity.&lt;/p&gt;
&lt;p&gt;A brief note on the standard model in noncommutative geometry. This is possibly the most talked about triumph of noncommutative geometry is that the standard model presents itself as a noncommutative internal structure attached to a commutative Minkowski spacetime. There are many book walking through the details on how you formulate integrals and quantum field theory on these &lt;em&gt;almost-commutative manifolds&lt;/em&gt;. The book by Suijlekom &lt;span class=&quot;citation&quot; data-cites=&quot;vanSuijlekom:2014kl&quot;&gt;&lt;/span&gt; is an excellent introduction to these topics however the book by Connes and Marcolli &lt;span class=&quot;citation&quot; data-cites=&quot;Connes:wl&quot;&gt;&lt;/span&gt; is much more involved and the finer details and subtleties can be found there.&lt;/p&gt;
&lt;h1 id=&quot;present-and-future-work&quot;&gt;Present and Future Work&lt;/h1&gt;
&lt;h2 id=&quot;the-fuzzy-ellipsoid&quot;&gt;The Fuzzy Ellipsoid&lt;/h2&gt;
&lt;p&gt;Not many examples of fuzzy spaces exist that are deemed approximations of commutative manifolds, so an investigation into constructing a fuzzy ellipsoid was made. This would be a very useful fuzzy space as it would be the first non-symmetric fuzzy space and may provide vital insight into how to generalise the procedure of ‘fuzzification’ to other manifolds. Firstly, the Dirac operator for an ellipsoid was found as an embedding in &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^3\)&lt;/span&gt;, as was suggested by the fuzzy sphere example.&lt;/p&gt;
&lt;h3 id=&quot;dirac-operators-for-hypersurfaces-of-mathbbrn&quot;&gt;Dirac operators for Hypersurfaces of &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^n\)&lt;/span&gt;&lt;/h3&gt;
&lt;p&gt;The following method for getting the induced Dirac operator on a hypersurface is a very brief and to the point method taken from &lt;span class=&quot;citation&quot; data-cites=&quot;Trautman:1995td&quot;&gt;&lt;/span&gt; and calculations for the sphere and ellipsoids are given. For full rigour of the results and steps to arrive at the formula, please refer to &lt;span class=&quot;citation&quot; data-cites=&quot;Trautman:1995td&quot;&gt;&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;The method of constructing the Dirac operator begins with taking the polynomial that defines the hypersurface, &lt;span class=&quot;math inline&quot;&gt;\(f(x^i)=0\)&lt;/span&gt;. First, for an embedding in &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^n\)&lt;/span&gt; we take &lt;span class=&quot;math inline&quot;&gt;\(n\)&lt;/span&gt; Dirac matrices, &lt;span class=&quot;math inline&quot;&gt;\(\{\gamma_a\}\)&lt;/span&gt; that satisfy the relation &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\gamma_b + \gamma_b\gamma_a = - 2\delta_{ab}\)&lt;/span&gt;. Specifically, we have that &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a^2 = -1\)&lt;/span&gt;. Then we find the unit normals to the surface &lt;span class=&quot;math inline&quot;&gt;\(\bm{n}\)&lt;/span&gt;. This can be done by using the formula &lt;span class=&quot;math inline&quot;&gt;\(\bm{n} = \frac{\nabla f}{|\nabla f|}\)&lt;/span&gt;. Then the formula for the Dirac operator, taken from &lt;span class=&quot;citation&quot; data-cites=&quot;1998hep.th...10018T&quot;&gt;&lt;/span&gt;, is as follows: &lt;span class=&quot;math display&quot;&gt;\[D = \sum\limits_{i,j=1}^3 (\gamma_i \gamma_j + \delta_{ij})n_i \partial_j + \frac{1}{2} \text{div}(n)
\label{diracembed}\]&lt;/span&gt; Where &lt;span class=&quot;math inline&quot;&gt;\(n_i\)&lt;/span&gt; are the &lt;span class=&quot;math inline&quot;&gt;\(i\)&lt;/span&gt;th components of the normal vector and &lt;span class=&quot;math inline&quot;&gt;\(\text{div}(n) = \sum\limits_{i,j=1}^3 \left(\delta_{ij} - n_i n_j \right)\partial_i n_j\)&lt;/span&gt;.&lt;/p&gt;
&lt;h3 id=&quot;the-dirac-operator-for-mathbbs2-embedded-in-mathbbrn&quot;&gt;The Dirac operator for &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{S}^2\)&lt;/span&gt; embedded in &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^n\)&lt;/span&gt;&lt;/h3&gt;
&lt;p&gt;For the sphere defined as: &lt;span class=&quot;math inline&quot;&gt;\(\sum\limits_{i=1}^{3} (x^i)^2 = 1\)&lt;/span&gt;, we find the normals by the method outine above and get: &lt;span class=&quot;math inline&quot;&gt;\(\bm{n}^i = x^i\)&lt;/span&gt;. Thus we can calculate: We note that: &lt;span class=&quot;math inline&quot;&gt;\(\partial_i n_j = \delta_{ij}\)&lt;/span&gt; and therefore: &lt;span class=&quot;math display&quot;&gt;\[\text{div}(n) = \sum\limits_{i,j=1}^3 \delta_{ij}\delta_{ij} - n_i n_j \delta_{ij}= \underbrace{\sum\limits_{i=1}^3 \delta_{ii}}_{=3} - \underbrace{\sum\limits_{i=1}^{3} x_i^2}_{=1}  = 2\]&lt;/span&gt; Making use of this result in &lt;a href=&quot;#diracembed&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;diracembed&quot;&gt;[diracembed]&lt;/a&gt; we can rewrite the Dirac operator as: &lt;span class=&quot;math display&quot;&gt;\[D  = \sum\limits_{i,j=1}^3 (\gamma_i \gamma_j + \delta_{ij})x_i \partial_j + 1 = \sum\limits_{i\neq j = 1}^{3} \gamma_i \gamma_j x_i \partial_j + \sum\limits_{i=1}^3 (\gamma_i^2 + 1)x_i \partial_j + 1\]&lt;/span&gt; Here we use the fact that &lt;span class=&quot;math inline&quot;&gt;\(\gamma_i^2 = -1\)&lt;/span&gt; and thus the middle term vanishes in general to give: &lt;span class=&quot;math display&quot;&gt;\[D = \sum\limits_{i\neq j = 1}^{3} \gamma_i \gamma_j x_i \partial_j  + 1\]&lt;/span&gt; Now using that &lt;span class=&quot;math inline&quot;&gt;\(\gamma_i \gamma_j = -\gamma_j \gamma_i\)&lt;/span&gt; we get the final form for the Dirac operator on &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{S}^2\)&lt;/span&gt;: &lt;span class=&quot;math display&quot;&gt;\[D_{\mathbb{S}^2} = \sum\limits_{i &amp;lt; j = 1}^3 \gamma_i \gamma_j (x_i \partial_j - x_j \partial_i) + 1\]&lt;/span&gt; Comparing this Dirac operator to the Dirac operators for the fuzzy sphere in Eq &lt;a href=&quot;#FuzzSphDirac&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;FuzzSphDirac&quot;&gt;[FuzzSphDirac]&lt;/a&gt; and &lt;a href=&quot;#GPDirac&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;GPDirac&quot;&gt;[GPDirac]&lt;/a&gt;, we can clearly see a relation. The basic premise is that the vector fields &lt;span class=&quot;math inline&quot;&gt;\(X_{ij} = x_i \partial_j - x_j \partial_i\)&lt;/span&gt; form are the &lt;em&gt;commutative limits&lt;/em&gt; of the commutators with the generators of the Lie algebra, &lt;span class=&quot;math inline&quot;&gt;\(\left[L_{ij}, \cdot \right]\)&lt;/span&gt;. This interpretation is discussed in further detail in &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;An attempt to extend this relationship to the case for an Ellipsoid was made as outlines below:&lt;/p&gt;
&lt;h3 id=&quot;the-dirac-operator-on-the-ellipsoid&quot;&gt;The Dirac operator on the Ellipsoid&lt;/h3&gt;
&lt;p&gt;Now for the Ellipsoid defined by: &lt;span class=&quot;math inline&quot;&gt;\(\sum\limits_{i=1}^3 (\frac{x_i}{\alpha_i})^2 = 1\)&lt;/span&gt;, the calculation follows the same steps however, the normals now have a more complicated form due the require them that have unit magnitude. The normals here are: &lt;span class=&quot;math inline&quot;&gt;\(n_i = N \frac{x^i}{\alpha_i^2}\)&lt;/span&gt;, where &lt;span class=&quot;math display&quot;&gt;\[\frac{1}{N}= \sqrt{\sum\limits_{k=1}^3 \frac{x_k^2}{\alpha_k^4}}\]&lt;/span&gt; The complexities lie in the &lt;span class=&quot;math inline&quot;&gt;\(\text{div}(n)\)&lt;/span&gt; terms as that requires use to take derivatives of &lt;span class=&quot;math inline&quot;&gt;\(N\)&lt;/span&gt;, which brings up extra terms. However the first term in Eqn &lt;a href=&quot;#diracembed&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;diracembed&quot;&gt;[diracembed]&lt;/a&gt; can be computed in a similar fashion to sphere case and we arrive at: &lt;span class=&quot;math display&quot;&gt;\[D_{\mathbb{E}^2} = \sum\limits_{i &amp;lt; j = 1}^3 \gamma_i \gamma_j N (\frac{x_i}{\alpha_i^2} \partial_j - \frac{x_j}{\alpha_j^2} \partial_i) + \frac{1}{2} \text{div}(n)\]&lt;/span&gt; The &lt;span class=&quot;math inline&quot;&gt;\(\text{div}(n)\)&lt;/span&gt; can also be computed, and is found to be: &lt;span class=&quot;math display&quot;&gt;\[\text{div}(n) = N \sum\limits_{j=1}^3  \left( \frac{1}{\alpha_j^2} - N^2 \frac{x_j^2}{\alpha_j^6}  \right)\]&lt;/span&gt; So the total Dirac Operator for the Ellipsoid is: &lt;span class=&quot;math display&quot;&gt;\[D_{\mathbb{E}^2} = N \left( \sum\limits_{i &amp;lt; j = 1}^3 \gamma_i \gamma_j  (\frac{x_i}{\alpha_i^2} \partial_j - \frac{x_j}{\alpha_j^2} \partial_i) + \frac{1}{2} \sum\limits_{j=1}^3  \left( \frac{1}{\alpha_j^2} - N^2 \frac{x_j^2}{\alpha_j^6}  \right) \right)
\label{ellipdirac}\]&lt;/span&gt; This is a much more complicated Dirac operator as the presence of &lt;span class=&quot;math inline&quot;&gt;\(N\)&lt;/span&gt; involves fractions with the coordinates in the denominator, which the ‘fuzzy’ analogue is currently unknown. I have included the calculation to explicitly show that this is the correct Dirac operator for an ellipsoid in Appendix &lt;a href=&quot;#sanity&quot; data-reference-type=&quot;ref&quot; data-reference=&quot;sanity&quot;&gt;8&lt;/a&gt;. Several attempts have been made to find make a sensible correspondence but with minimal success.&lt;/p&gt;
&lt;h2 id=&quot;future-work&quot;&gt;Future Work&lt;/h2&gt;
&lt;p&gt;The next port of call is to look at Dirac operators on general coadjoint orbits. As the sphere is a coadjoint orbit of &lt;span class=&quot;math inline&quot;&gt;\(SU(2)\)&lt;/span&gt;, the idea is to construct more examples using this correspondence as a guide line. More generally, investigations in to the &lt;em&gt;commutative limit&lt;/em&gt; of a fuzzy space is to be investigated. However, more examples of fuzzy spaces is required to make fruitful progress in this topic. Also investigations into looking for the noncommutative analogue to a Lorentzian manifold is planned. Some headway has been made in this topic, see for example &lt;span class=&quot;citation&quot; data-cites=&quot;Franco:2013bw&quot;&gt;&lt;/span&gt; &lt;span class=&quot;citation&quot; data-cites=&quot;DAndrea:2016vk&quot;&gt;&lt;/span&gt; amongst other. However, there is still much to uncover in this topic.&lt;/p&gt;


&lt;h1 id=&quot;alg&quot;&gt;Algebra&lt;/h1&gt;
&lt;p&gt;A vector space, &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt;, over a field &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{F}\)&lt;/span&gt;, is a set of objects that satisfy the following conditions:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;p&gt;It is closed under finite vector addition: &lt;span class=&quot;math inline&quot;&gt;\(\forall v,w\in V \Rightarrow v+w \in V\)&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;It is closed under scalar multiplication: &lt;span class=&quot;math inline&quot;&gt;\(\forall \lambda \in \mathbb{F}, v \in V \Rightarrow \lambda v \in V\)&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If &lt;span class=&quot;math inline&quot;&gt;\(U, V\)&lt;/span&gt; are complex vector spaces the the map &lt;span class=&quot;math inline&quot;&gt;\(f\colon U \to V\)&lt;/span&gt; is said to be &lt;em&gt;antilinear&lt;/em&gt; if &lt;span class=&quot;math inline&quot;&gt;\(f(u + v) = f(u) + f(v)\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(f(\lambda u) = \bar{\lambda} f(u)\)&lt;/span&gt;, where &lt;span class=&quot;math inline&quot;&gt;\(\bar{\lambda}\)&lt;/span&gt; denotes complex conjugation.&lt;/p&gt;
&lt;p&gt;A &lt;em&gt;isometry&lt;/em&gt; between normed vector spaces, &lt;span class=&quot;math inline&quot;&gt;\(U, V\)&lt;/span&gt;, is a linear map &lt;span class=&quot;math inline&quot;&gt;\(f\colon U \to V\)&lt;/span&gt; that preserves the norm: &lt;span class=&quot;math inline&quot;&gt;\(\|f(u)\| = \| u \|\)&lt;/span&gt; for all &lt;span class=&quot;math inline&quot;&gt;\(u \in U\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;A Hilbert space, &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; is a vector space with an inner product &lt;span class=&quot;math inline&quot;&gt;\(\langle f,g \rangle\)&lt;/span&gt; such that the norm defined by &lt;span class=&quot;math display&quot;&gt;\[\|f \| =\sqrt{\langle f,f \rangle}\]&lt;/span&gt; turns &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; into a complete metric space.&lt;/p&gt;
&lt;p&gt;An algebra &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A}\)&lt;/span&gt; over field &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{F}\)&lt;/span&gt; is a vector space equipped with a bilinear associative product (multiplication). An algebra is said to be unital if it has an identity element with respects to the multiplication.&lt;/p&gt;
&lt;p&gt;Let &lt;span class=&quot;math inline&quot;&gt;\(R\)&lt;/span&gt; be a ring and &lt;span class=&quot;math inline&quot;&gt;\(1_R\)&lt;/span&gt; be the multiplication identity. A &lt;em&gt;left &lt;span class=&quot;math inline&quot;&gt;\(R\)&lt;/span&gt;-module&lt;/em&gt;, &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{M}\)&lt;/span&gt;, consists of an abelian group &lt;span class=&quot;math inline&quot;&gt;\((\mathcal{M},+)\)&lt;/span&gt; and an operation &lt;span class=&quot;math inline&quot;&gt;\(\cdot:R\times \mathcal{M} \to \mathcal{M}\)&lt;/span&gt; such that &lt;span class=&quot;math inline&quot;&gt;\(\forall r,s\in R\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\forall x,y\in \mathcal{M}\)&lt;/span&gt;:&lt;/p&gt;

&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(r \cdot (x+y) = r\cdot x + r\cdot y\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\((r+s) \cdot x = r\cdot x + s\cdot x\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\((rs)\cdot x = r \cdot (s\cdot x)\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(1_R \cdot x = x\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The only difference between an algebra and a module is that an algebra is a module over a ring, where the ring is also a field.&lt;/p&gt;
&lt;p&gt;A &lt;span class=&quot;math inline&quot;&gt;\(*\)&lt;/span&gt;-algebra or &lt;em&gt;involutive algebra&lt;/em&gt; is an algebra &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A}\)&lt;/span&gt; together with a conjugate linear map called the involution: &lt;span class=&quot;math inline&quot;&gt;\(*\colon\mathcal{A} \to \mathcal{A}\)&lt;/span&gt; such that &lt;span class=&quot;math inline&quot;&gt;\((ab)^* = b^*a^*\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\((a^*)^* = a\)&lt;/span&gt; for any &lt;span class=&quot;math inline&quot;&gt;\(a,b \in \mathcal{A}\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;A representation of a finite-dimensional &lt;span class=&quot;math inline&quot;&gt;\(*\)&lt;/span&gt;-algebra &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{A}\)&lt;/span&gt; is a pair &lt;span class=&quot;math inline&quot;&gt;\((\mathcal{H},\pi)\)&lt;/span&gt;. Where &lt;span class=&quot;math inline&quot;&gt;\(\mathcal{H}\)&lt;/span&gt; is a Hilbert space and &lt;span class=&quot;math inline&quot;&gt;\(\pi\)&lt;/span&gt; is a &lt;span class=&quot;math inline&quot;&gt;\(*\)&lt;/span&gt;-algebra map: &lt;span class=&quot;math display&quot;&gt;\[\pi\colon \mathcal{A} \to L(\mathcal{H})\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Given a vector space &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt; (over &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{F}\)&lt;/span&gt;) and a quadratic form, &lt;span class=&quot;math inline&quot;&gt;\(Q\)&lt;/span&gt; on &lt;span class=&quot;math inline&quot;&gt;\(V\)&lt;/span&gt;. We define the &lt;em&gt;Clifford Algebra&lt;/em&gt; &lt;span class=&quot;math inline&quot;&gt;\(\text{Cl}(V,Q)\)&lt;/span&gt; as the algebra generated (over &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{F}\)&lt;/span&gt;) by the vectors &lt;span class=&quot;math inline&quot;&gt;\(v\in V\)&lt;/span&gt; and the multiplicative unit &lt;span class=&quot;math inline&quot;&gt;\(1_F\)&lt;/span&gt; such that: &lt;span class=&quot;math display&quot;&gt;\[v \cdot v= v^2 = Q(v) 1_F\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;A Clifford algebra can also be defined by taking a bilinear form, &lt;span class=&quot;math inline&quot;&gt;\(B\)&lt;/span&gt;, instead of a quadratic form and defining the quadratic form by: &lt;span class=&quot;math inline&quot;&gt;\(Q(v) = B(v,v)\)&lt;/span&gt;. One can also construct a bilinear form from a quadratic form via &lt;em&gt;polarisation&lt;/em&gt;: &lt;span class=&quot;math inline&quot;&gt;\(B(u,v) = \frac{1}{2}\left( Q(u+v) - Q(u) - Q(v) \right)\)&lt;/span&gt;. Using this polarisation we can show that for &lt;span class=&quot;math inline&quot;&gt;\(u,v \in V\)&lt;/span&gt; we have that: &lt;span class=&quot;math display&quot;&gt;\[u v + v u = 2 B(u,v)\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;There is a natural grading of a clifford algebra by the grading &lt;span class=&quot;math inline&quot;&gt;\(\chi(v_1 v_2 \dots v_k) = (-1)^k v_1 v_2 \dots v_k\)&lt;/span&gt; which allows us to decompose a Clifford algebra into even (&lt;span class=&quot;math inline&quot;&gt;\(\chi = +1\)&lt;/span&gt;) and odd (&lt;span class=&quot;math inline&quot;&gt;\(\chi = -1\)&lt;/span&gt;) parts: &lt;span class=&quot;math display&quot;&gt;\[\text{Cl}(V,Q) =\vcentcolon Cl^0(V,Q) \oplus Cl^1(V,Q)\]&lt;/span&gt; This decomposition is often useful in technicalities.&lt;/p&gt;
&lt;h1 id=&quot;geom&quot;&gt;Geometry&lt;/h1&gt;
&lt;p&gt;A fibre bundle is a very general object and many subclasses are often used in physics such as vector bundle or principal bundles. The bundles of object in spin geometry include algebra bundles along side the usual bundles. A fibre bundle in layman’s terms is a way to describe that a space &lt;em&gt;locally looks like a product space&lt;/em&gt;. If there is are topological spaces &lt;span class=&quot;math inline&quot;&gt;\(E, B, F\)&lt;/span&gt; and a map &lt;span class=&quot;math inline&quot;&gt;\(f\colon E \to B\)&lt;/span&gt;, such that for some neighbourhood &lt;span class=&quot;math inline&quot;&gt;\(U\)&lt;/span&gt; of a point &lt;span class=&quot;math inline&quot;&gt;\(x\in B\)&lt;/span&gt;, such that &lt;span class=&quot;math inline&quot;&gt;\(f^{-1}(U)\)&lt;/span&gt; is homeomorphic to &lt;span class=&quot;math inline&quot;&gt;\(U\times F\)&lt;/span&gt; in a specific way, then we call &lt;span class=&quot;math inline&quot;&gt;\((E,B,F,f)\)&lt;/span&gt; a fibre bundle. I include the rigorous definition below for clarity:&lt;/p&gt;
&lt;p&gt;A &lt;em&gt;fibre bundle&lt;/em&gt; is a structure &lt;span class=&quot;math inline&quot;&gt;\((E,B,F,f)\)&lt;/span&gt;, where &lt;span class=&quot;math inline&quot;&gt;\(E,B,F\)&lt;/span&gt; are all topological spaces and &lt;span class=&quot;math inline&quot;&gt;\(f\colon E\to B\)&lt;/span&gt; is a continuous surjection that satisfies the following properties: &lt;span class=&quot;math inline&quot;&gt;\(\forall x\in E\)&lt;/span&gt; there exists a neighbourhood &lt;span class=&quot;math inline&quot;&gt;\(U\)&lt;/span&gt; of &lt;span class=&quot;math inline&quot;&gt;\(f(x)\in B\)&lt;/span&gt; such that there exists a homeomorphism &lt;span class=&quot;math inline&quot;&gt;\(\phi: f^{-1}(U) \to U\times F\)&lt;/span&gt; which, when composed with projection onto the first component, agrees with &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;The condition for &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt; can be summarised in the requiring that the following diagram commutes:&lt;/p&gt;
&lt;p&gt;f^&lt;span&gt;-1&lt;/span&gt;(U) &amp;amp; &amp;amp; U F&lt;br /&gt;
&amp;amp; U&lt;/p&gt;
&lt;p&gt;Given a fibre bundle &lt;span class=&quot;math inline&quot;&gt;\((E, B, F, f)\)&lt;/span&gt; and given a topological group &lt;span class=&quot;math inline&quot;&gt;\(G\)&lt;/span&gt; that has a left action on the fibres &lt;span class=&quot;math inline&quot;&gt;\(F\)&lt;/span&gt;, then we can specify a &lt;span class=&quot;math inline&quot;&gt;\(G\)&lt;/span&gt;-atlas for the bundle, which is a local trivialisation such that for any overlapping charts &lt;span class=&quot;math inline&quot;&gt;\((U_i, \phi_i)\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\((U_j, \phi_j)\)&lt;/span&gt; the function: &lt;span class=&quot;math display&quot;&gt;\[\phi_i \phi_j^{-1} \colon (U_i \cup U_j)\times F \to (U_i \cup U_j)\times F\]&lt;/span&gt; is given by &lt;span class=&quot;math display&quot;&gt;\[\phi_i \phi_j^{-1} \left(x, \alpha \right) = \left(x, t_{ij}(x) \alpha \right)\]&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(t_{ij}\colon U_i \cap U_j \to G\)&lt;/span&gt; is a continuous map. The maps &lt;span class=&quot;math inline&quot;&gt;\(t_{ij}\)&lt;/span&gt; are called transition functions and given just &lt;span class=&quot;math inline&quot;&gt;\((B, F, \{ t_{ij} \})\)&lt;/span&gt; we can reconstruct the fibre bundle if the transition functions satisfy the following: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
t_{ii}(x)&amp;amp;=x \\
t_{ij}(x)&amp;amp;=t_{ji}^{-1}(x)\\
t_{ik}(x)&amp;amp;= t_{ij}(x)t_{jk}(x) \quad \text{on} \quad U_i \cap U_j \cap U_k\end{aligned}\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;A &lt;em&gt;section&lt;/em&gt; of a fibre bundle &lt;span class=&quot;math inline&quot;&gt;\((E, B, F, f)\)&lt;/span&gt; is a continuous map &lt;span class=&quot;math inline&quot;&gt;\(\pi\colon B \to E\)&lt;/span&gt; such that &lt;span class=&quot;math inline&quot;&gt;\(f(\pi(x)) = x\)&lt;/span&gt; for all &lt;span class=&quot;math inline&quot;&gt;\(x\in B\)&lt;/span&gt;. The space of section of a fibre bundle &lt;span class=&quot;math inline&quot;&gt;\((E, B, F, f)\)&lt;/span&gt; is denoted by &lt;span class=&quot;math inline&quot;&gt;\(\Gamma (E)\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;A &lt;em&gt;fibre bundle morphism&lt;/em&gt; (bundle map) between two fibre bundles &lt;span class=&quot;math inline&quot;&gt;\((E_1, B_1, F_1, f_1)\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\((E_2, B_2, F_2, f_2)\)&lt;/span&gt; is defined as a pair continuous maps &lt;span class=&quot;math inline&quot;&gt;\(\phi \colon E_1 \to E_2\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\alpha\colon B_1 \to B_2\)&lt;/span&gt; such there exists we have: &lt;span class=&quot;math inline&quot;&gt;\(f_2 \circ \phi = \alpha \circ f_1\)&lt;/span&gt;. If we have a bundle map whose inverse is also a bundle map, then we have a bundle isomorphism.&lt;/p&gt;
&lt;p&gt;The &lt;em&gt;endomorphism&lt;/em&gt; ring over an abelian group &lt;span class=&quot;math inline&quot;&gt;\(A\)&lt;/span&gt;, is denoted by &lt;span class=&quot;math inline&quot;&gt;\(End(A)\)&lt;/span&gt; is the set of all homomorphism of &lt;span class=&quot;math inline&quot;&gt;\(A\)&lt;/span&gt; into itself. If we take &lt;span class=&quot;math inline&quot;&gt;\(A\)&lt;/span&gt; to be a field, and then the &lt;em&gt;endormorhpism ring&lt;/em&gt; over the space &lt;span class=&quot;math inline&quot;&gt;\(A^n\)&lt;/span&gt;, is an &lt;span class=&quot;math inline&quot;&gt;\(A\)&lt;/span&gt;-algebra and is the set of all linear maps form &lt;span class=&quot;math inline&quot;&gt;\(A^n\)&lt;/span&gt; into &lt;span class=&quot;math inline&quot;&gt;\(A^n\)&lt;/span&gt;. We refer to this endomorphism ring as an &lt;em&gt;endormorphism algebra&lt;/em&gt;.&lt;/p&gt;
&lt;h1 id=&quot;sanity&quot;&gt;Sanity Checking for the Dirac operator for the Ellipsoid&lt;/h1&gt;
&lt;p&gt;This is to check that Eq &lt;a href=&quot;#ellipdirac&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;ellipdirac&quot;&gt;[ellipdirac]&lt;/a&gt; does indeed give the correct Dirac operator we require. We do this by using the equation &lt;span class=&quot;math inline&quot;&gt;\(D = e^{\alpha a} \sigma_a \nabla_{\alpha}\)&lt;/span&gt;. For the ellipse we want to use &lt;span class=&quot;math inline&quot;&gt;\(\nabla_{\alpha} = \partial_{\alpha} + \frac{1}{8} \omega^{a b}_{\alpha} \left(\gamma_a \gamma_b - \gamma_b \gamma_a\right)\)&lt;/span&gt;. If we ignore the spin connection terms and concentrate solely on the derivative part we can calculate the &lt;span class=&quot;math inline&quot;&gt;\(e^{\alpha a}\)&lt;/span&gt;’s. We do this by looking at the case where the gamma matrices are &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a = i \sigma_a\)&lt;/span&gt;, where &lt;span class=&quot;math inline&quot;&gt;\(\sigma_a\)&lt;/span&gt; are the Pauli matrices. Thus we have the following properties for the gamma matrices: &lt;span class=&quot;math display&quot;&gt;\[\left[\gamma_a, \gamma_b \right] = - 2 \tensor{\epsilon}{_{ab}^c} \gamma_c \qquad
\left\{ \gamma_a, \gamma_b \right\} = - 2 \delta_{ab} \qquad
\gamma_a \gamma_b = -\tensor{\epsilon}{_{ab}^c} \gamma_c - \delta_{ab}\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Using these relations we can simplify Eqn &lt;a href=&quot;#EllipDirac&quot; data-reference-type=&quot;eqref&quot; data-reference=&quot;EllipDirac&quot;&gt;[EllipDirac]&lt;/a&gt; to: &lt;span class=&quot;math display&quot;&gt;\[D_{\mathbb{E}^2} = N \left( \sigma_3 \left(\frac{x_1}{\alpha_1^2} \partial_2 - \frac{x_2}{\alpha_2^2}\partial_1 \right)- \sigma_2 \left(\frac{x_1}{\alpha_1^2} \partial_3 - \frac{x_3}{\alpha_3^2}\partial_1 \right) + \sigma_1 \left(\frac{x_2}{\alpha_2^2} \partial_3 - \frac{x_3}{\alpha_3^2}\partial_2 \right)  \right) + \frac{1}{2} \text{div}(n)\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;And thus we can read off the inverse tetrad: &lt;span class=&quot;math inline&quot;&gt;\(e^{\alpha a}\)&lt;/span&gt;:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math display&quot;&gt;\[e^{\alpha a} = N
\begin{pmatrix}
0 &amp;amp; \frac{x_3}{\alpha_3^2} &amp;amp; - \frac{x_2}{\alpha_2^2} \\
-\frac{x_3}{\alpha_3^2}  &amp;amp; 0 &amp;amp;  \frac{x_1}{\alpha_1^2} \\
\frac{x_2}{\alpha_2^2}  &amp;amp;- \frac{x_1}{\alpha_1^2} &amp;amp; 0
\end{pmatrix}\]&lt;/span&gt; Which produces the inverse metric via the formula: &lt;span class=&quot;math inline&quot;&gt;\(g^{\alpha \beta} = e^{\alpha a} e^{\beta b} \delta_{ab}\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math display&quot;&gt;\[g^{\alpha \beta} = N^2
\begin{pmatrix}
\frac{x_2^2}{\alpha_2^4} + \frac{x_3^2}{\alpha_3^4} &amp;amp; - \frac{x_1}{\alpha_1^2} \frac{x_2}{\alpha_2^2}&amp;amp; - \frac{x_1}{\alpha_1^2} \frac{x_3}{\alpha_3^2}\\
- \frac{x_2}{\alpha_2^2} \frac{x_1}{\alpha_1^2} &amp;amp; \frac{x_1^2}{\alpha_1^4} + \frac{x_3^2}{\alpha_3^4}  &amp;amp; \frac{x_2}{\alpha_2^2} \frac{x_3}{\alpha_3^2} \\
\frac{x_3}{\alpha_3^2} \frac{x_1}{\alpha_1^2} &amp;amp;\frac{x_3}{\alpha_3^2}\frac{x_2}{\alpha_2^2} &amp;amp; \frac{x_1^2}{\alpha_1^4} + \frac{x_2^2}{\alpha_2^4}
\end{pmatrix}\]&lt;/span&gt; This metric is degenerate and is rank 2. The vector which annihilates this metric is &lt;span class=&quot;math inline&quot;&gt;\(\left(\frac{x_1}{\alpha_1^2} ,\frac{x_2}{\alpha_2^2} ,\frac{x_3}{\alpha_3^2} \right)\)&lt;/span&gt;. Which can be rewritten as: &lt;span class=&quot;math inline&quot;&gt;\(df = d \left(\frac{1}{2} \left( \frac{x_1^2}{\alpha_1^2}+ \frac{x_2^2}{\alpha_2^2}+ \frac{x_3^2}{\alpha_3^2} \right)\right)\)&lt;/span&gt;. Using the fact that when this functiion &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt; is constant we can construct a unit normal to surface &lt;span class=&quot;math inline&quot;&gt;\(f= \text{const}\)&lt;/span&gt; by &lt;span class=&quot;math inline&quot;&gt;\(n = \frac{\nabla f}{|\nabla f |}\)&lt;/span&gt;. This is what makes us pursue that we have the metric on an ellipsoid in &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^3\)&lt;/span&gt;.&lt;/p&gt;
&lt;p&gt;So using the parametrisation: &lt;span class=&quot;math display&quot;&gt;\[\begin{aligned}
x_1 &amp;amp;= \alpha_1 \cos(u) \sin(v) \\
x_2 &amp;amp;= \alpha_2 \sin(u) \sin(v) \\
x_3 &amp;amp;= \alpha_3 \cos(v)\end{aligned}\]&lt;/span&gt; We get the following change of basis matrix (Jacobian)&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math display&quot;&gt;\[Jac(u,v) = D\left(\frac{u,v}{x_1,x_2,x_3}\right)=
\begin{pmatrix}
 -\frac{\csc(v) \sin (u)}{\alpha_1} &amp;amp; \frac{\cos (u) \csc (v)}{\alpha_2} &amp;amp; 0 \\
 0 &amp;amp; 0 &amp;amp; -\frac{1}{\alpha_3 \sqrt{\sin ^2(v)}} \\
\end{pmatrix}\]&lt;/span&gt; We note that as &lt;span class=&quot;math inline&quot;&gt;\(v\)&lt;/span&gt; runs between &lt;span class=&quot;math inline&quot;&gt;\(0\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\pi\)&lt;/span&gt; then &lt;span class=&quot;math inline&quot;&gt;\(\sin(v)&amp;gt;0\)&lt;/span&gt; and therefore &lt;span class=&quot;math inline&quot;&gt;\(\sqrt{\sin(v)^2} = \sin(v)\)&lt;/span&gt;. So we can write the Jacobian as follows:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math display&quot;&gt;\[Jac(u,v) =
\begin{pmatrix}
 -\frac{\csc(v) \sin (u)}{\alpha_1} &amp;amp; \frac{\cos (u) \csc (v)}{\alpha_2} &amp;amp; 0 \\
 0 &amp;amp; 0 &amp;amp; -\frac{\csc(v)}{\alpha_3} \\
\end{pmatrix}\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;We then change the inverse metric &lt;span class=&quot;math inline&quot;&gt;\(g^{-1}(x_i)\)&lt;/span&gt; by the formula: &lt;span class=&quot;math display&quot;&gt;\[g^{-1}(u,v) = Jac(u,v)\cdot g^{-1}(x_i(u,v)) \cdot Jac(u,v)^T\]&lt;/span&gt; Which gives the following formula:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math display&quot;&gt;\[\begin{pmatrix}
 \frac{\csc ^2(v) \left(\alpha_3^2 \cos ^4(u)+\left(\alpha_1^2 \cot ^2(v)+2 \alpha_3^2 \sin ^2(u)\right) \cos ^2(u)+\alpha_3^2 \sin ^4(u)+\alpha_2^2 \cot ^2(v) \sin ^2(u)\right)}{\left(\alpha_2^2 \cot ^2(v)+\alpha_3^2 \sin ^2(u)\right) \alpha_1^2+\alpha_2^2 \alpha_3^2 \cos ^2(u)} &amp;amp;
 \frac{(\alpha_1-\alpha_2) (\alpha_1+\alpha_2) \cos (u) \cot (v) \csc ^2(v) \sin (u)}{\left(\alpha_2^2 \cot ^2(v)+\alpha_3^2 \sin ^2(u)\right) \alpha_1^2+\alpha_2^2 \alpha_3^2 \cos ^2(u)} \\
 \frac{(\alpha_1-\alpha_2) (\alpha_1+\alpha_2) \cos (u) \cot (v) \sin (u)}{\alpha_1^2 \alpha_2^2 \cos ^2(v)+\alpha_3^2 \left(\alpha_2^2 \cos ^2(u)+\alpha_1^2 \sin ^2(u)\right) \sin ^2(v)} &amp;amp; \frac{\csc ^2(v)
 \left(\alpha_2^2 \cos ^2(u)+\alpha_1^2 \sin ^2(u)\right)}{\left(\alpha_2^2 \cot ^2(v)+\alpha_3^2 \sin ^2(u)\right) \alpha_1^2+\alpha_2^2 \alpha_3^2 \cos ^2(u)} \\
\end{pmatrix}\]&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;This ugly however, if we calculate the inverse of this matrix (as it now has full rank), we should be able to recognise the normal metric for the ellipsoid. And indeed after some trigonometry juggling we arrive at the following metric: &lt;span class=&quot;math display&quot;&gt;\[g(u,v) =
\begin{pmatrix}
 \left( \alpha_1^2 \sin^2(u)  \alpha_2^2 \cos^2(u) \right) \sin ^2(v) &amp;amp; \left(\alpha_2^2 - \alpha_1^2 \right) \cos(u) \cos(v) \sin(u) \sin(v) \\
 \left(\alpha_2^2 - \alpha_1^2 \right) \cos(u) \cos(v) \sin(u) \sin(v) &amp;amp;  \alpha_3 \sin^2(v) + \cos^2(v)\left( \alpha_1^2 \cos^2(u) + \alpha_2^2 \sin^2(u) \right)  \\
\end{pmatrix}\]&lt;/span&gt; Huzza!&lt;/p&gt;

&lt;section class=&quot;footnotes&quot;&gt;
&lt;hr /&gt;
&lt;ol&gt;
&lt;li id=&quot;fn1&quot;&gt;&lt;p&gt;Sometimes this is referred to as Noncommutative Differential geometry to distinguish it from Noncommutative Algebraic Geometry.&lt;a href=&quot;#fnref1&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn2&quot;&gt;&lt;p&gt;For details on clifford algebras, see the Appendix A&lt;a href=&quot;#fnref2&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn3&quot;&gt;&lt;p&gt;For more details on the construction of fibre bundles see Appendix B.s&lt;a href=&quot;#fnref3&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn4&quot;&gt;&lt;p&gt;This definition looks a little contrived without knowing the motivation for it, which arises when we consider the Euclidean space &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{R}^n\)&lt;/span&gt; with the standard metric, &lt;span class=&quot;math inline&quot;&gt;\(\delta\)&lt;/span&gt;. It can be shown that we have &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{C}\text{l}(\mathbb{R}^{2n}) \simeq M_n(\mathbb{C})\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(\mathbb{C}\text{l}^0(\mathbb{R}^{2n+1}) \simeq M_n(\mathbb{C})\)&lt;/span&gt;. And this definition is the generalisation to arbitrary Riemannian manifolds.&lt;a href=&quot;#fnref4&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn5&quot;&gt;&lt;p&gt;So that &lt;span class=&quot;math inline&quot;&gt;\(Q(e_i) = \pm 1\)&lt;/span&gt; and &lt;span class=&quot;math inline&quot;&gt;\(B(e_i, e_j) = 0\)&lt;/span&gt; if &lt;span class=&quot;math inline&quot;&gt;\(i \neq j\)&lt;/span&gt;. Where &lt;span class=&quot;math inline&quot;&gt;\(B\)&lt;/span&gt; is defined as in the Appendix &lt;a href=&quot;#alg&quot; data-reference-type=&quot;ref&quot; data-reference=&quot;alg&quot;&gt;6&lt;/a&gt;.&lt;a href=&quot;#fnref5&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn6&quot;&gt;&lt;p&gt;This is for Riemannian metrics, for pseudo-Riemannian metrics the &lt;span class=&quot;math inline&quot;&gt;\(\delta\)&lt;/span&gt; in the relation is replaced with the corresponding pseudo-Riemannian equivalent.&lt;a href=&quot;#fnref6&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn7&quot;&gt;&lt;p&gt;A Clifford module can be viewed as just representing a Clifford algebra as matrices. Hence the colloquial term for the &lt;span class=&quot;math inline&quot;&gt;\(\gamma_a\)&lt;/span&gt; discussed above as &lt;em&gt;gamma matrices&lt;/em&gt;.&lt;a href=&quot;#fnref7&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn8&quot;&gt;&lt;p&gt;The term matrix geometries arises from examples of this definiton being made by matrix contstructions. To see examples of how to construct some examples, see &lt;span class=&quot;citation&quot; data-cites=&quot;2015JMP....56h2301B&quot;&gt;&lt;/span&gt;.&lt;a href=&quot;#fnref8&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn9&quot;&gt;&lt;p&gt;These are matrix versions of the spherical harmonics, there precise constructure can be found in &lt;span class=&quot;citation&quot; data-cites=&quot;Chan:2002kt&quot;&gt;&lt;/span&gt;.&lt;a href=&quot;#fnref9&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;li id=&quot;fn10&quot;&gt;&lt;p&gt;Note that the reason &lt;span class=&quot;math inline&quot;&gt;\(l&amp;lt;n\)&lt;/span&gt; not &lt;span class=&quot;math inline&quot;&gt;\(l&amp;lt;n+1\)&lt;/span&gt; is because the index &lt;span class=&quot;math inline&quot;&gt;\(l\)&lt;/span&gt; starts at zero. So there are still &lt;span class=&quot;math inline&quot;&gt;\(n+1\)&lt;/span&gt; generators for &lt;span class=&quot;math inline&quot;&gt;\(M_{n+1}(\mathbb{C})\)&lt;/span&gt;&lt;a href=&quot;#fnref10&quot; class=&quot;footnote-back&quot;&gt;↩&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;
&lt;/ol&gt;
&lt;/section&gt;
&lt;/body&gt;
&lt;/html&gt;</content><author><name></name></author><category term="Tag 1" /><category term="Tag 2" /><summary type="html">First Year Report - Noncommutative Geometry and Quantum Gravity First Year Report - Noncommutative Geometry and Quantum Gravity Paul Druce</summary></entry><entry><title type="html">Welcome to Jekyll!</title><link href="http://localhost:4000/jekyll/update/2018/11/10/welcome-to-jekyll.html" rel="alternate" type="text/html" title="Welcome to Jekyll!" /><published>2018-11-10T18:19:24+00:00</published><updated>2018-11-10T18:19:24+00:00</updated><id>http://localhost:4000/jekyll/update/2018/11/10/welcome-to-jekyll</id><content type="html" xml:base="http://localhost:4000/jekyll/update/2018/11/10/welcome-to-jekyll.html">&lt;p&gt;Hello World!&lt;/p&gt;
&lt;p&gt;You’ll find this post in your &lt;code&gt;_posts&lt;/code&gt; directory. Go ahead and edit it and re-build the site to see your changes. You can rebuild the site in many different ways, but the most common way is to run &lt;code&gt;jekyll serve&lt;/code&gt;, which launches a web server and auto-regenerates your site when a file is updated.&lt;/p&gt;
&lt;p&gt;To add new posts, simply add a file in the &lt;code&gt;_posts&lt;/code&gt; directory that follows the convention &lt;code&gt;YYYY-MM-DD-name-of-post.ext&lt;/code&gt; and includes the necessary front matter. Take a look at the source for this post to get an idea about how it works.&lt;/p&gt;
&lt;p&gt;Jekyll also offers powerful support for code snippets:&lt;/p&gt;
&lt;figure class=&quot;highlight&quot;&gt;
&lt;pre&gt;&lt;code class=&quot;language-ruby&quot; data-lang=&quot;ruby&quot;&gt;&lt;span class=&quot;k&quot;&gt;def&lt;/span&gt; &lt;span class=&quot;nf&quot;&gt;print_hi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;
  &lt;span class=&quot;nb&quot;&gt;puts&lt;/span&gt; &lt;span class=&quot;s2&quot;&gt;&quot;Hi, &lt;/span&gt;&lt;span class=&quot;si&quot;&gt;#{&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;name&lt;/span&gt;&lt;span class=&quot;si&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;print_hi&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;'Tom'&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;c1&quot;&gt;#=&amp;gt; prints 'Hi, Tom' to STDOUT.&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;/figure&gt;
&lt;p&gt;Check out the &lt;a href=&quot;https://jekyllrb.com/docs/home&quot;&gt;Jekyll docs&lt;/a&gt; for more info on how to get the most out of Jekyll. File all bugs/feature requests at &lt;a href=&quot;https://github.com/jekyll/jekyll&quot;&gt;Jekyll’s GitHub repo&lt;/a&gt;. If you have questions, you can ask them on &lt;a href=&quot;https://talk.jekyllrb.com/&quot;&gt;Jekyll Talk&lt;/a&gt;.&lt;/p&gt;</content><author><name></name></author><summary type="html">Hello World!</summary></entry><entry><title type="html">Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York)</title><link href="http://localhost:4000/catagory%201/catagory%202/2018/11/10/Strong_Quantum_Energy_Conditions_and_Hawking_Singularity_Theorems_by_Eleni_Kontou-(University-of-York).html" rel="alternate" type="text/html" title="Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York)" /><published>2018-11-10T00:00:00+00:00</published><updated>2018-11-10T00:00:00+00:00</updated><id>http://localhost:4000/catagory%201/catagory%202/2018/11/10/Strong_Quantum_Energy_Conditions_and_Hawking_Singularity_Theorems_by_Eleni_Kontou%20(University%20of%20York)</id><content type="html" xml:base="http://localhost:4000/catagory%201/catagory%202/2018/11/10/Strong_Quantum_Energy_Conditions_and_Hawking_Singularity_Theorems_by_Eleni_Kontou-(University-of-York).html">&lt;!DOCTYPE html&gt;
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&lt;body&gt;
&lt;p&gt;These are my notes from this seminar. They are not a complete recount of the talk and are mainly the motivation and main ideas behind the approach.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;The overall aim and result(?) is a singularity theorem that is obeyed by quantum fields.&lt;/strong&gt; - this means that the normal singularity theorems are not obeyed by quantum fields? Something I was not aware of, but probably should be aware of.&lt;/p&gt;
&lt;p&gt;First up, we need to know what we are dealing with, what is a singularity? Naïve definition is “place where the curvature diverges”. This is a bad definition, because we don’t know how to properly talk about a &lt;em&gt;place&lt;/em&gt; when curvature diverse and the metric tensor is not defined there. Or maybe just in component form, not sure. My relativity isn’t all that great anymore.&lt;/p&gt;
&lt;p&gt;A better definition&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;A spacetime is singular if it has at least one complete geodesic&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Jorma Louko spoke up at this point, asking about alternative definitions of a singularity such as saying a spacetime is singular if it has an incomplete geodesic that cannot be completed by adding points to the manifold. To get rid of cases such as &lt;span class=&quot;math inline&quot;&gt;\(M/\{*\}\)&lt;/span&gt; and other pathological examples. But Eleni said that the theorems they have shown are using the above definition and it its a less restrictive condition and therefore makes the theorems stronger.&lt;/p&gt;
&lt;h4 id=&quot;general-structure-of-a-singularity-theorem&quot;&gt;General Structure of a singularity theorem&lt;/h4&gt;
&lt;ol type=&quot;1&quot;&gt;
&lt;li&gt;Causality conditions - this is a statement relating to the existence s of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy_surface&quot;&gt;Cauchy surface&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;Initial or Boundary conditions - this is a statement about the existence of a &lt;em&gt;trapper surface&lt;/em&gt; - this was a new concept to me, Eleni explained it as a space-like hyper surface for which 2 null normals have negative expansion. But this didn’t embellish the concept to a useable degree. Not for me anyways. Here is the wiki article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Trapped_surface&quot;&gt;trapped surfaces&lt;/a&gt; but even that isn’t too detailed. I think I should go find the explanation given in Hawking and Ellis.&lt;/li&gt;
&lt;li&gt;The energy conditions. There are two of them
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Penrose&lt;/strong&gt; - Null energy conditions, which is a statement: &lt;span class=&quot;math inline&quot;&gt;\(R_{ab} l^a l^b \geq 0\)&lt;/span&gt; where &lt;span class=&quot;math inline&quot;&gt;\(l^a\)&lt;/span&gt; is null. So I guess we just take vector fields &lt;span class=&quot;math inline&quot;&gt;\(l\)&lt;/span&gt; that are null. I’m not sure how general a vector field we take or whether there are any symmetry requirements etc.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Hawking&lt;/strong&gt; Strong energy condition, which is a similar looking statement &lt;span class=&quot;math inline&quot;&gt;\(R_{ab} U^a U^b \geq 0\)&lt;/span&gt; where now &lt;span class=&quot;math inline&quot;&gt;\(U^a\)&lt;/span&gt; is a timelike vector field.&lt;/li&gt;
&lt;/ul&gt;&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;Then the proofs requires the &lt;a href=&quot;https://en.wikipedia.org/wiki/Raychaudhuri_equation&quot;&gt;Raychaudhuri equation&lt;/a&gt;. This is not something I’d heard of before. But its a &lt;em&gt;related article&lt;/em&gt; on wikipedia so much be fairly well known. Most of this stuff looks like it is mentioned in Hawking and Ellis, which I am slowly making my way through. Too many books to read.&lt;/p&gt;
&lt;p&gt;Eleni presented the Raychaudhuri equation in the following way:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\dot{\theta} = \frac{1}{n-1} \theta^2 - 2 \sigma^2 - R_{ab}U^aU^b\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;where &lt;span class=&quot;math inline&quot;&gt;\(\theta\)&lt;/span&gt; is the expansion and &lt;span class=&quot;math inline&quot;&gt;\(\sigma\)&lt;/span&gt; is the shear. But I’m not familiar with what this really mean. (From wikipedia on &lt;a href=&quot;https://en.wikipedia.org/wiki/Congruence_(general_relativity)&quot;&gt;congruence&lt;/a&gt;, you can find that $&lt;em&gt;{ab} $ is the expansion tensor which is given in terms of some metric on a hyper surface which has vectors which are perpendicular to the vector field associated to the geodesic (a vague guess at this point). and the symmetrised covariant derivative of a vector field. And the expansion tensor contains both the &lt;span class=&quot;math inline&quot;&gt;\(\theta =\)&lt;/span&gt; trace and $&lt;/em&gt;{ab} = $ the antisymmetric part.&lt;/p&gt;
&lt;h4 id=&quot;proof-structure&quot;&gt;Proof structure&lt;/h4&gt;
&lt;ol type=&quot;1&quot;&gt;
&lt;li&gt;Initial conditions imply that the geodesics start focusing (I’m guessing there is a precise way to talk about focusing)&lt;/li&gt;
&lt;li&gt;Energy conditions implies that the focussing continues? I’m guessing this is the snowballing feature you need for a singularity to occur.&lt;/li&gt;
&lt;li&gt;Causal conditions imply that there are no focal points. (Not sure what this meant, but it didn’t seem super relevant to the talk).&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;These three steps give us geodesic incompleteness.&lt;/p&gt;
&lt;h3 id=&quot;problems-to-be-fixed&quot;&gt;Problems (to be fixed)&lt;/h3&gt;
&lt;p&gt;The strong energy condition is too strong and matter models that we know don’t satisfy it’s property.&lt;/p&gt;
&lt;p&gt;An example of a minimally coupled real scalar field violates the Strong Energy Condition, whenever the mass is high or if its derivatives vanished &lt;strong&gt;at any point in the manifold&lt;/strong&gt;. I think all quantum fields violate this property as well so it’s an issue.&lt;/p&gt;
&lt;h4 id=&quot;developments&quot;&gt;Developments&lt;/h4&gt;
&lt;p&gt;There are a few ideas floating around the academic world. The first is too take averaged energy conditions, rather than a condition that has to hold every where. An example formula is:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\displaystyle \int_\gamma \rho f^2(\tau) d\tau \geq 0\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;this wasn’t really elaborated on, but I’m guessing &lt;span class=&quot;math inline&quot;&gt;\(\gamma\)&lt;/span&gt; is the geodesic and &lt;span class=&quot;math inline&quot;&gt;\(\rho\)&lt;/span&gt; is the energy density and &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt; is some random function, not too sure what conditions it needs to satisfy.&lt;/p&gt;
&lt;p&gt;Another idea is that of quantum inequalities.&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;math inline&quot;&gt;\(\displaystyle \int_\gamma f^2(\tau) \langle\rho^\text{quant} \rangle \omega(\gamma(\tau)) \geq -A\)&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;This again wasn’t really delved into (or I just can’t remember and didn’t write anything down in my notes). But the value &lt;span class=&quot;math inline&quot;&gt;\(A\)&lt;/span&gt; depends on the function &lt;span class=&quot;math inline&quot;&gt;\(f\)&lt;/span&gt;. And this is a restriction of the magnitude and duration of any negative energy densities.&lt;/p&gt;
&lt;h5 id=&quot;the-rest-of-the-talk-was-about-algebraic-quantum-field-theory-and-elenis-actual-research-but-i-was-lost-somewhere-in-the-wealth-of-formulas-and-failed-to-gain-anything-else-from-this-talk.&quot;&gt;The rest of the talk was about algebraic quantum field theory and Eleni’s actual research but I was lost somewhere in the wealth of formula’s and failed to gain anything else from this talk.&lt;/h5&gt;
&lt;h3 id=&quot;take-aways-and-final-comments&quot;&gt;Take aways and final comments&lt;/h3&gt;
&lt;p&gt;So I think the aim is to show that semi-classical results are not enough to protect us from singularities forming before the quantum gravity regime. This is what I understood from the discussions that occurred. That the investigation into more realistic singularity theorems will still allow singularities to form, and therefore the semi-classical picture will need to be replaced by a theory of quantum gravity (in whatever form that may occur in).&lt;/p&gt;
&lt;p&gt;Semiclassical approach are also some of the most interesting results of modern day physics. Which blackhole entropy and Hawking radiation etc all coming from this sort of approach. So I definitely think this sort of research is needed for providing food for thought and providing some thought pathways for quantum gravity research.&lt;/p&gt;
&lt;p&gt;This whole approach becomes much harder when you move from vector fields to spinor fields or anything else. Which is not a surprise to me. And there are not many results in this area. It would be interesting to see more active research into fermions, however there are currently no massless fermions that have been detected, so that would add complications to the concept, where as massless fields provide an already fruitful and complex playground.&lt;/p&gt;
&lt;/body&gt;
&lt;/html&gt;</content><author><name></name></author><category term="Tag 1" /><category term="Tag 2" /><summary type="html">These are my notes from this seminar. They are not a complete recount of the talk and are mainly the motivation and main ideas behind the approach. The overall aim and result(?) is a singularity theorem that is obeyed by quantum fields. - this means that the normal singularity theorems are not obeyed by quantum fields? Something I was not aware of, but probably should be aware of. First up, we need to know what we are dealing with, what is a singularity? Naïve definition is “place where the curvature diverges”. This is a bad definition, because we don’t know how to properly talk about a place when curvature diverse and the metric tensor is not defined there. Or maybe just in component form, not sure. My relativity isn’t all that great anymore. A better definition A spacetime is singular if it has at least one complete geodesic Jorma Louko spoke up at this point, asking about alternative definitions of a singularity such as saying a spacetime is singular if it has an incomplete geodesic that cannot be completed by adding points to the manifold. To get rid of cases such as \(M/\{*\}\) and other pathological examples. But Eleni said that the theorems they have shown are using the above definition and it its a less restrictive condition and therefore makes the theorems stronger. General Structure of a singularity theorem Causality conditions - this is a statement relating to the existence s of a Cauchy surface Initial or Boundary conditions - this is a statement about the existence of a trapper surface - this was a new concept to me, Eleni explained it as a space-like hyper surface for which 2 null normals have negative expansion. But this didn’t embellish the concept to a useable degree. Not for me anyways. Here is the wiki article on trapped surfaces but even that isn’t too detailed. I think I should go find the explanation given in Hawking and Ellis. The energy conditions. There are two of them Penrose - Null energy conditions, which is a statement: \(R_{ab} l^a l^b \geq 0\) where \(l^a\) is null. So I guess we just take vector fields \(l\) that are null. I’m not sure how general a vector field we take or whether there are any symmetry requirements etc. Hawking Strong energy condition, which is a similar looking statement \(R_{ab} U^a U^b \geq 0\) where now \(U^a\) is a timelike vector field. Then the proofs requires the Raychaudhuri equation. This is not something I’d heard of before. But its a related article on wikipedia so much be fairly well known. Most of this stuff looks like it is mentioned in Hawking and Ellis, which I am slowly making my way through. Too many books to read. Eleni presented the Raychaudhuri equation in the following way: \(\dot{\theta} = \frac{1}{n-1} \theta^2 - 2 \sigma^2 - R_{ab}U^aU^b\) where \(\theta\) is the expansion and \(\sigma\) is the shear. But I’m not familiar with what this really mean. (From wikipedia on congruence, you can find that ${ab} $ is the expansion tensor which is given in terms of some metric on a hyper surface which has vectors which are perpendicular to the vector field associated to the geodesic (a vague guess at this point). and the symmetrised covariant derivative of a vector field. And the expansion tensor contains both the \(\theta =\) trace and ${ab} = $ the antisymmetric part. Proof structure Initial conditions imply that the geodesics start focusing (I’m guessing there is a precise way to talk about focusing) Energy conditions implies that the focussing continues? I’m guessing this is the snowballing feature you need for a singularity to occur. Causal conditions imply that there are no focal points. (Not sure what this meant, but it didn’t seem super relevant to the talk). These three steps give us geodesic incompleteness. Problems (to be fixed) The strong energy condition is too strong and matter models that we know don’t satisfy it’s property. An example of a minimally coupled real scalar field violates the Strong Energy Condition, whenever the mass is high or if its derivatives vanished at any point in the manifold. I think all quantum fields violate this property as well so it’s an issue. Developments There are a few ideas floating around the academic world. The first is too take averaged energy conditions, rather than a condition that has to hold every where. An example formula is: \(\displaystyle \int_\gamma \rho f^2(\tau) d\tau \geq 0\) this wasn’t really elaborated on, but I’m guessing \(\gamma\) is the geodesic and \(\rho\) is the energy density and \(f\) is some random function, not too sure what conditions it needs to satisfy. Another idea is that of quantum inequalities. \(\displaystyle \int_\gamma f^2(\tau) \langle\rho^\text{quant} \rangle \omega(\gamma(\tau)) \geq -A\) This again wasn’t really delved into (or I just can’t remember and didn’t write anything down in my notes). But the value \(A\) depends on the function \(f\). And this is a restriction of the magnitude and duration of any negative energy densities. The rest of the talk was about algebraic quantum field theory and Eleni’s actual research but I was lost somewhere in the wealth of formula’s and failed to gain anything else from this talk. Take aways and final comments So I think the aim is to show that semi-classical results are not enough to protect us from singularities forming before the quantum gravity regime. This is what I understood from the discussions that occurred. That the investigation into more realistic singularity theorems will still allow singularities to form, and therefore the semi-classical picture will need to be replaced by a theory of quantum gravity (in whatever form that may occur in). Semiclassical approach are also some of the most interesting results of modern day physics. Which blackhole entropy and Hawking radiation etc all coming from this sort of approach. So I definitely think this sort of research is needed for providing food for thought and providing some thought pathways for quantum gravity research. This whole approach becomes much harder when you move from vector fields to spinor fields or anything else. Which is not a surprise to me. And there are not many results in this area. It would be interesting to see more active research into fermions, however there are currently no massless fermions that have been detected, so that would add complications to the concept, where as massless fields provide an already fruitful and complex playground.</summary></entry><entry><title type="html">Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York)</title><link href="http://localhost:4000/catagory%201/catagory%202/2018/11/10/TestStrongQuant.html" rel="alternate" type="text/html" title="Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York)" /><published>2018-11-10T00:00:00+00:00</published><updated>2018-11-10T00:00:00+00:00</updated><id>http://localhost:4000/catagory%201/catagory%202/2018/11/10/TestStrongQuant</id><content type="html" xml:base="http://localhost:4000/catagory%201/catagory%202/2018/11/10/TestStrongQuant.html">&lt;!DOCTYPE html&gt;
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&lt;title&gt;Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York)&lt;/title&gt;&lt;/head&gt;
&lt;body&gt;&lt;h1&gt;Seminar: 7/11/2018&lt;/h1&gt;
&lt;h3&gt;Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York&lt;/h3&gt;
&lt;p&gt;These are my notes from this seminar. They are not a complete recount of the talk and are mainly the motivation and main ideas behind the approach. &lt;/p&gt;
&lt;p&gt;&lt;strong&gt;The overall aim and result(?) is a singularity theorem that is obeyed by quantum fields.&lt;/strong&gt; - this means that the normal singularity theorems are not obeyed by quantum fields? Something I was not aware of, but probably should be aware of. &lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;First up, we need to know what we are dealing with, what is a singularity? Naïve definition is “place where the curvature diverges”. This is a bad definition, because we don&amp;#39;t know how to properly talk about a &lt;em&gt;place&lt;/em&gt; when curvature diverse and the metric tensor is not defined there. Or maybe just in component form, not sure. My relativity isn&amp;#39;t all that great anymore. &lt;/p&gt;
&lt;p&gt;A better definition&lt;/p&gt;
&lt;blockquote&gt;&lt;p&gt;A spacetime is singular if it has at least one complete geodesic&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Jorma Louko spoke up at this point, asking about alternative definitions of a singularity such as saying a spacetime is singular if it has an incomplete geodesic that cannot be completed by adding points to the manifold. To get rid of cases such as $M/\{*\}$ and other pathological examples. But Eleni said that the theorems they have shown are using the above definition and it its a less restrictive condition and therefore makes the theorems stronger. &lt;/p&gt;
&lt;h4&gt;General Structure of a singularity theorem &lt;/h4&gt;
&lt;ol start='' &gt;
&lt;li&gt;&lt;p&gt;Causality conditions - this is a statement relating to the existence s of a &lt;a href='https://en.wikipedia.org/wiki/Cauchy_surface'&gt;Cauchy surface&lt;/a&gt; &lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;Initial or Boundary conditions - this is a statement about the existence of a &lt;em&gt;trapper surface&lt;/em&gt; - this was a new concept to me, Eleni explained it as a space-like hyper surface for which 2 null normals have negative expansion. But this didn&amp;#39;t embellish the concept to a useable degree. Not for me anyways. Here is the wiki article on &lt;a href='https://en.wikipedia.org/wiki/Trapped_surface'&gt;trapped surfaces&lt;/a&gt; but even that isn&amp;#39;t too detailed. I think I should go find the explanation given in Hawking and Ellis. &lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;The energy conditions. There are two of them &lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Penrose&lt;/strong&gt; - Null energy conditions, which is a statement: $R_{ab} l^a l^b \geq 0$ where $l^a$ is null. So I guess we just take vector fields $l$ that are null. I&amp;#39;m not sure how general a vector field we take or whether there are any symmetry requirements etc. &lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Hawking&lt;/strong&gt; Strong energy condition, which is a similar looking statement $R_{ab} U^a U^b \geq 0$ where now $U^a$ is a timelike vector field. &lt;/li&gt;

&lt;/ul&gt;
&lt;/li&gt;

&lt;/ol&gt;
&lt;p&gt;Then the proofs requires the &lt;a href='https://en.wikipedia.org/wiki/Raychaudhuri_equation'&gt;Raychaudhuri equation&lt;/a&gt;. This is not something I&amp;#39;d heard of before. But its a &lt;em&gt;related article&lt;/em&gt; on wikipedia so much be fairly well known. Most of this stuff looks like it is mentioned in Hawking and Ellis, which I am slowly making my way through. Too many books to read. &lt;/p&gt;
&lt;p&gt;Eleni presented the Raychaudhuri equation in the following way:&lt;/p&gt;
&lt;p&gt;$\dot{\theta}  = \frac{1}{n-1} \theta^2 - 2 \sigma^2 - R_{ab}U^aU^b$ &lt;/p&gt;
&lt;p&gt;where $\theta$ is the expansion and $\sigma$ is the shear. But I&amp;#39;m not familiar with what this really mean. (From wikipedia on &lt;a href='https://en.wikipedia.org/wiki/Congruence_(general_relativity)'&gt;congruence&lt;/a&gt;, you can find that $\theta_{ab}$ is the expansion tensor which is given in terms of some metric on a hyper surface which has vectors which are perpendicular to the vector field associated to the geodesic (a vague guess at this point). and the symmetrised covariant derivative of a vector field. And the expansion tensor contains both the $\theta =$ trace and $\sigma_{ab} =$ the antisymmetric part. &lt;/p&gt;
&lt;h4&gt;Proof structure&lt;/h4&gt;
&lt;ol start='' &gt;
&lt;li&gt;Initial conditions imply that the geodesics start focusing (I&amp;#39;m guessing there is a precise way to talk about focusing)&lt;/li&gt;
&lt;li&gt;Energy conditions implies that the focussing continues? I&amp;#39;m guessing this is the snowballing feature you need for a singularity to occur. &lt;/li&gt;
&lt;li&gt;Causal conditions imply that there are no focal points. (Not sure what this meant, but it didn&amp;#39;t seem super relevant to the talk). &lt;/li&gt;

&lt;/ol&gt;
&lt;p&gt;These three steps give us geodesic incompleteness. &lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;h3&gt;Problems (to be fixed)&lt;/h3&gt;
&lt;p&gt;The strong energy condition is too strong and matter models that we know don&amp;#39;t satisfy it&amp;#39;s property. &lt;/p&gt;
&lt;p&gt;An example of a minimally coupled real scalar field violates the Strong Energy Condition, whenever the mass is high or if its derivatives vanished &lt;strong&gt;at any point in the manifold&lt;/strong&gt;. I think all quantum fields violate this property as well so it&amp;#39;s an issue. &lt;/p&gt;
&lt;h4&gt;Developments&lt;/h4&gt;
&lt;p&gt;There are a few ideas floating around the academic world. The first is too take averaged energy conditions, rather than a condition that has to hold every where. An example formula is:&lt;/p&gt;
&lt;p&gt;$\displaystyle \int_\gamma \rho f^2(\tau) d\tau \geq 0$&lt;/p&gt;
&lt;p&gt;this wasn&amp;#39;t really elaborated on, but I&amp;#39;m guessing $\gamma$ is the geodesic and $\rho$ is the energy density and $f$ is some random function, not too sure what conditions it needs to satisfy. &lt;/p&gt;
&lt;p&gt;Another idea is that of quantum inequalities. &lt;/p&gt;
&lt;p&gt;$\displaystyle \int_\gamma f^2(\tau) \langle\rho^\text{quant} \rangle \omega(\gamma(\tau)) \geq -A$ &lt;/p&gt;
&lt;p&gt;This again wasn&amp;#39;t really delved into (or I just can&amp;#39;t remember and didn&amp;#39;t write anything down in my notes). But the value $A$ depends on the function $f$. And this is a restriction of the magnitude and duration of any negative energy densities. &lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;h5&gt;The rest of the talk was about algebraic quantum field theory and Eleni&amp;#39;s actual research but I was lost somewhere in the wealth of formula&amp;#39;s and failed to gain anything else from this talk. &lt;/h5&gt;
&lt;h3&gt;Take aways and final comments&lt;/h3&gt;
&lt;p&gt;So I think the aim is to show that semi-classical results are not enough to protect us from singularities forming before the quantum gravity regime. This is what I understood from the discussions that occurred. That the investigation into more realistic singularity theorems will still allow singularities to form, and therefore the semi-classical picture will need to be replaced by a theory of quantum gravity (in whatever form that may occur in). &lt;/p&gt;
&lt;p&gt;Semiclassical approach are also some of the most interesting results of modern day physics. Which blackhole entropy and Hawking radiation etc all coming from this sort of approach. So I definitely think this sort of research is needed for providing food for thought and providing some thought pathways for quantum gravity research.&lt;/p&gt;
&lt;p&gt;This whole approach becomes much harder when you move from vector fields to spinor fields or anything else. Which is not a surprise to me. And there are not many results in this area. It would be interesting to see more active research into fermions, however there are currently no massless fermions  that have been detected, so that would add complications to the concept, where as massless fields provide an already fruitful and complex playground. &lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;/body&gt;
&lt;/html&gt;</content><author><name></name></author><category term="Tag 1" /><category term="Tag 2" /><summary type="html">Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York) Seminar: 7/11/2018 Strong Quantum Energy Conditions and Hawking Singularity Theorems by Eleni Kontou (University of York These are my notes from this seminar. They are not a complete recount of the talk and are mainly the motivation and main ideas behind the approach. The overall aim and result(?) is a singularity theorem that is obeyed by quantum fields. - this means that the normal singularity theorems are not obeyed by quantum fields? Something I was not aware of, but probably should be aware of. &amp;nbsp; First up, we need to know what we are dealing with, what is a singularity? Naïve definition is “place where the curvature diverges”. This is a bad definition, because we don&amp;#39;t know how to properly talk about a place when curvature diverse and the metric tensor is not defined there. Or maybe just in component form, not sure. My relativity isn&amp;#39;t all that great anymore. A better definition A spacetime is singular if it has at least one complete geodesic Jorma Louko spoke up at this point, asking about alternative definitions of a singularity such as saying a spacetime is singular if it has an incomplete geodesic that cannot be completed by adding points to the manifold. To get rid of cases such as $M/\{*\}$ and other pathological examples. But Eleni said that the theorems they have shown are using the above definition and it its a less restrictive condition and therefore makes the theorems stronger. General Structure of a singularity theorem Causality conditions - this is a statement relating to the existence s of a Cauchy surface Initial or Boundary conditions - this is a statement about the existence of a trapper surface - this was a new concept to me, Eleni explained it as a space-like hyper surface for which 2 null normals have negative expansion. But this didn&amp;#39;t embellish the concept to a useable degree. Not for me anyways. Here is the wiki article on trapped surfaces but even that isn&amp;#39;t too detailed. I think I should go find the explanation given in Hawking and Ellis. The energy conditions. There are two of them Penrose - Null energy conditions, which is a statement: $R_{ab} l^a l^b \geq 0$ where $l^a$ is null. So I guess we just take vector fields $l$ that are null. I&amp;#39;m not sure how general a vector field we take or whether there are any symmetry requirements etc. Hawking Strong energy condition, which is a similar looking statement $R_{ab} U^a U^b \geq 0$ where now $U^a$ is a timelike vector field.</summary></entry></feed>