5_twopointspace.tex (10364B)
1 \subsection{Almost-commutative Manifold\label{sec:5}} 2 \subsubsection{Two-Point Space} 3 One of the basics forms of noncommutative space is the Two-Point space $X 4 := \{x, y\}$. The Two-Point space can be represented by the following spectral triple 5 \begin{align} 6 F_X := \left(C(X) = \mathbb{C}^2, H_F, D_F; J_F, \gamma _f\right). 7 \end{align} 8 Three properties of $F_X$ stand out. First of all the action of $C(X)$ on 9 $H_F$ is faithful for $dim(H_F) \geq 2$, thus a simple choice for the 10 Hilbertspace can be made, for instance $H_F = \mathbb{C}^2$. Furthermore 11 $\gamma_F$ is the $\mathbb{Z}_2$ grading, which allows for a decomposition of 12 $H_F$ into 13 \begin{align} 14 H_F = H_F^+ \otimes H_F^- = \mathbb{C} \otimes \mathbb{C}, 15 \end{align} 16 where 17 \begin{align} 18 H_F^\pm = \left\{\psi \in H_F |\; \gamma_F\psi = \pm \psi\right\}, 19 \end{align} 20 are two eigenspaces. And lastly the Dirac operator $D_F$ lets us 21 interchange between the two eigenspaces $H_F^\pm$, 22 \begin{align} 23 D_F = 24 \begin{pmatrix}0 & t \\ \bar{t} & 0\end{pmatrix}, \;\;\;\;\; 25 \text{with} \;\; t\in\mathbb{C}. 26 \end{align} 27 28 The Two-Point space $F_X$ can only have a real structure if the Dirac 29 operator vanishes, i.e. $D_F = 0$. In that case the KO-dimension is 0, 30 2 or 6. To elaborate further, we draw the only two diagram representations of 31 $F_X$ at $\underbrace{\mathbb{C} \oplus \mathbb{C}}_{C(X)}$ on 32 $\underbrace{\mathbb{C} \oplus\mathbb{C}}_{H_F}$, which are 33 \begin{figure}[h!] \centering 34 \begin{tikzpicture}[ 35 dot/.style = {draw, circle, inner sep=0.06cm}, 36 no/.style = {}, 37 ] 38 \node[no](a) at (0,0) [label=left:$\textbf{1}^\circ$] {}; 39 \node[no](b) at (0, -1) [label=left:$\textbf{1}^\circ$] {}; 40 \node[no](c) at (1, 0.5) [label=above:$\textbf{1}$] {}; 41 \node[no](d) at (2, 0.5) [label=above:$\textbf{1}$] {}; 42 \node[dot](d0) at (2,0) [] {}; 43 \node[dot](d0) at (1,-1) [] {}; 44 45 \node[no](a1) at (6,0) [label=left:$\textbf{1}^\circ$] {}; 46 \node[no](b2) at (6, -1) [label=left:$\textbf{1}^\circ$] {}; 47 \node[no](c2) at (7, 0.5) [label=above:$\textbf{1}$] {}; 48 \node[no](d2) at (8, 0.5) [label=above:$\textbf{1}$] {}; 49 \node[dot](d0) at (7,0) [] {}; 50 \node[dot](d0) at (8,-1) [] {}; 51 \end{tikzpicture} 52 \caption{Two diagram representations of $F_X$} 53 \end{figure}\newline 54 If the Two-Point space $F_X$ would be a real spectral triple then $D_F$ can 55 only go vertically or horizontally. This would mean that $D_F$ vanishes. 56 As for the KO-dimension The diagram on the left has KO-dimension 2 and 6, the diagram on the 57 right 0 and 4. Yet KO-dimension 4 is ruled out because 58 $dim(H_F^\pm) = 1$ (Lemma 3.8 in \cite{ncgwalter}) , which ultimately means $J_F^2 = -1$ is 59 not allowed. 60 \subsubsection{Product Space} 61 By Extending the Two-Point space with a four dimensional Riemannian spin 62 manifold, we get an almost commutative manifold $M\times F_X$, given by 63 \begin{align} 64 M\times F_X = \big(C^\infty(M, \mathbb{C}^2), L^2(S)\otimes \mathbb{C}^2, 65 D_M\otimes 1 ; J_M\otimes J_F, \gamma_M \otimes \gamma_F\big), 66 \end{align} 67 where 68 \begin{align} 69 C^\infty(M, \mathbb{C}^2) \simeq C^\infty(M) \oplus C^\infty(M). 70 \end{align} 71 According to Gelfand duality the algebra $C^\infty(M, \mathbb{C}^2)$ of the 72 spectral triple corresponds to the space 73 \begin{align} 74 N:= M\otimes X. 75 \end{align} 76 Keep in mind that we still need to find an appropriate real structure on the 77 Riemannian spin manifold, $J_M$. Furthermore the total Hilbertspace can be 78 decomposed into $H = L^2(S) \oplus L^2(S)$, such that for $\underbrace{a,b\in 79 C^\infty(M)}_{(a, b) \in C^\infty(N)}$ and $\underbrace{\psi, \phi \in 80 L^2(S)}_{(\psi, \phi) \in H}$ we have 81 \begin{align} 82 (a, b)(\psi, \phi) = (a\psi, b\phi). 83 \end{align} 84 Along with the decomposition of the total Hilbertspace a 85 distance formula on $M\times F_X$ can be considered with 86 \begin{align}\label{eq:commutator inequality} 87 d_{D_F}(x,y) = \sup\left\{ |a(x) - a(y)|:a\in A_F, ||[D_F, a]|| \leq 88 1 \right\}. 89 \end{align} 90 To calculate the distance between two points on the Two-Point space $X= \{x, 91 y\}$, between $x$ and $y$, we consider an $a \in \mathbb{C}^2 = C(X)$, which is 92 specified by two complex numbers $a(x)$ and $a(y)$. Then we simplify the 93 commutator inequality in \eqref{eq:commutator inequality} 94 \begin{align} 95 &||[D_F , a]|| = ||(a(y) - a(x))\begin{pmatrix}0 &t\\\bar{t} &0 96 \end{pmatrix}|| \leq 1,\\ 97 &\Leftrightarrow |a(y) - a(x)|\leq \frac{1}{|t|}. 98 \end{align} 99 The supremum then gives us the distance 100 \begin{align} 101 d_{D_F} (x,y) = \frac{1}{|t|}. 102 \end{align} 103 An interesting observation here is that, if the Riemannian spin manifold can be 104 represented by a real spectral triple then a real structure $J_M$ exists, 105 along the lines it follows that $t=0$ and the distance becomes infinite. This is a 106 purely mathematical observation and has no physical meaning. 107 108 We can also construct a distance formula on $N$ (in reference to a point $p 109 \in M$) between two points on $N=M\times X$, $(p, x)$ and $(p,y)$. Then an $a 110 \in C^\infty(N)$ is determined by $a_x(p):=a(p, x)$ and $a_y(p):=a(p, y)$. 111 The distance between these two points is 112 \begin{align} 113 d_{D_F\otimes 1}(n_1, n_2) = \sup \left\{ |a(n_1) - a(n_2)|: a\in 114 A, ||[D\otimes 1, a]||\right\}. 115 \end{align} 116 On the other hand if we consider $n_1 = (p,x)$ and $n_2 = (q, x)$ 117 for $p,q \in M$ then 118 \begin{align} 119 d_{D_M \otimes 1} (n_1, n_2) = |a_x(p) - a_x(q)| \;\;\;\text{for}\;\; 120 a_x\in 121 C^\infty(M) \;\; \text{with} \;\; ||[D\otimes 1, a_x]|| \leq 1. 122 \end{align} 123 The distance formula turns to out to be the geodesic distance formula 124 \begin{align} 125 d_{D_M\otimes1}(n_1, n_2) = d_g(p, q), 126 \end{align} 127 which is to be expected since we are only looking at the manifold. 128 However if $n_1 = (p, x)$ and $n_2 = (q, y)$ then the two conditions are 129 \begin{align} 130 &||[D_M, a_x]|| \leq 1, \;\;\; \text{and}\\ 131 &||[D_M, a_y|| \leq 1. 132 \end{align} 133 These conditions have no restriction which results in the distance being 134 infinite! And $N = M\times X$ is given by two disjoint copies of M which are 135 separated by infinite distance 136 137 The distance is only finite if $[D_F, a] < 1$. In this case the commutator 138 generates a scalar field and the finiteness of the distance is 139 related to the existence of scalar fields. 140 141 \subsubsection{$U(1)$ Gauge Group} 142 To get a insight into the physical properties of the almost commutative 143 manifold $M\times F_X$, that is to calculate the spectral action, we need to 144 determine the corresponding Gauge group. 145 For this we set of with simple definitions and important propositions to 146 help us break down and search for the gauge group of the Two-Point $F_X$ 147 space which we then extend to $M\times F_X$. We will only be diving 148 superficially into this chapter, for further reading we refer to 149 \cite{ncgwalter}. 150 \begin{mydefinition} 151 Gauge Group of a real spectral triple is given by 152 \begin{align} 153 \mathfrak{B}(A, H; J) := \{ U = uJuJ^{-1} | u\in U(A)\}. 154 \end{align} 155 \end{mydefinition} 156 \begin{mydefinition} 157 A *-automorphism of a *-algebra $A$ is a linear invertible 158 map 159 \begin{align} 160 &\alpha:A \rightarrow A,\;\;\; \text{with}\\ 161 \nonumber\\ 162 &\alpha(ab) = \alpha(a)\alpha(b),\\ 163 &\alpha(a)^* = \alpha(a^*). 164 \end{align} 165 The \textbf{Group of automorphisms of the *-Algebra $A$} is denoted by 166 $(A)$.\newline 167 The automorphism $\alpha$ is called \textbf{inner} if 168 \begin{align} 169 \alpha(a) = u a u^* \;\;\; \text{for} \;\; U(A), 170 \end{align} 171 where $U(A)$ is 172 \begin{align} 173 U(A) = \{ u\in A|\;\; uu^* = u^*u=1\}. \;\;\; 174 \text{(unitary)} 175 \end{align} 176 \end{mydefinition} 177 The Gauge group of $F_X$ is given by the quotient $U(A)/U(A_J)$. 178 To get a nontrivial Gauge group so we need to choose a $U(A_J) \neq 179 U(A)$ and $U((A_F)_{J_F}) \neq U(A_F)$. 180 We consider our Two-Point space $F_X$ to be equipped with a real structure, 181 which means the operator vanishes, and the spectral triple representation is 182 \begin{align} 183 F_X := \left(\mathbb{C}^2,\mathbb{C}^2, D_F =\begin{pmatrix} 184 0&0\\0&0\end{pmatrix}; J_f =\begin{pmatrix} 185 0&C\\C&0\end{pmatrix}, 186 \gamma_F = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\right). 187 \end{align} 188 Here $C$ is the complex conjugation, and $F_X$ is a real even finite 189 spectral triple (space) of KO-dimension 6. 190 191 \begin{myproposition} 192 The Gauge group of the Two-Point space $\mathfrak{B}(F_X)$ is $U(1)$. 193 \end{myproposition} 194 \begin{proof} 195 Note that $U(A_F) = U(1) \times U(1)$. We need to show that $U(A_F) \cap 196 U(A_F)_{J_F}) \simeq U(1)$, such that $\mathfrak{B}(F) \simeq U(1)$. So 197 for an element $a \in \mathbb{C}^2$ to be in $(A_F)_{J_F}$, it has to 198 satisfy $J_F a^* J_F = a$, 199 \begin{align} 200 J_F a^* J^{-1} = 201 \begin{pmatrix}0&C\\C&0\end{pmatrix} 202 \begin{pmatrix}\bar{a}_1&0\\0&\bar{a}_2\end{pmatrix} 203 \begin{pmatrix}0&C\\C&0\end{pmatrix} 204 = 205 \begin{pmatrix}a_2&0\\0&a_1\end{pmatrix}. 206 \end{align} 207 This can only be the case if $a_1 = a_2$. So we have 208 $(A_F)_{J_F} \simeq \mathbb{C}$, whose unitary elements 209 from $U(1)$ are contained in the diagonal subgroup of 210 $U(A_F)$. 211 \end{proof} 212 213 An arbitrary hermitian field $A_\mu = -ia\partial _\mu b$ is given by 214 two $U(1)$ Gauge fields $X_\mu^1, X_\mu^2 \in C^\infty(M, \mathbb{R})$. 215 However $A_\mu$ appears in combination $A_\mu - J_F A_\mu J_F^{-1}$: 216 \begin{align} 217 A_\mu - J_F A_\mu J_F^{-1} = 218 \begin{pmatrix}X_\mu^1&0\\0&X_\mu^2 \end{pmatrix} 219 - 220 \begin{pmatrix}X_\mu^2&0\\0&X_\mu^1 \end{pmatrix} 221 =: 222 \begin{pmatrix}Y_\mu&0\\0&-Y_\mu \end{pmatrix} 223 = Y_\mu \otimes \gamma _F, 224 \end{align} 225 where $Y_\mu$ the $U(1)$ Gauge field is defined as 226 \begin{align} 227 Y_\mu := X_\mu^1 - X_\mu^2 \in C^\infty(M, \mathbb{R}) = C^\infty(M, 228 i\ u(1)). 229 \end{align} 230 231 \begin{myproposition} 232 The inner fluctuations of the almost-commutative manifold $M\times 233 F_X$ are parameterized by a $U(1)$-gauge field $Y_\mu$ as 234 \begin{align} 235 D \mapsto D' = D + \gamma ^\mu Y_\mu \otimes \gamma_F. 236 \end{align} 237 The action of the gauge group $\mathfrak{B}(M\times F_X) \simeq 238 C^\infty (M, U(1))$ on $D'$ is implemented by 239 \begin{align} 240 Y_\mu \mapsto Y_\mu - i\ u\partial_\mu u^*; \;\;\;\;\; (u\in 241 \mathfrak{B}(M\times F_X)). 242 \end{align} 243 \end{myproposition} 244