The Category of bounded self adjoint operators


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The Category of bounded self adjoint operators

Definition 1.10   [2] [3] the Set $ \mathcal{O}$ of bounded self-adjoint operators is a category such that
  • the objects of $ \mathcal{O}$ are the self-adjoint operators
  • given a function $ f:\sigma(\hat{A})\rightarrow{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ (from the spectrum of $ \hat{A}$ to the Reals) such that $ \hat{B}=f(\hat{A})$ then there exists a morphism $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$ in $ \mathcal{O}$ between operators $ \hat{B}$ and $ \hat{A}$

To show that the category $ \mathcal{O}$, so defined, is a category (see Definition 1.1), we need to show that it satisfies the identity law and composition law. This can be shown in the following way:
  • Identity Law: given any $ \mathcal{O}$-object $ \hat{A}$, the identity arrow is defined as the arrow $ id_{\mathcal{O}_A}:\hat{A}\rightarrow\hat{A}$ that corresponds to the arrow $ id:{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}\rightarrow{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ in $ {\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$.
  • Composition Condition: given two $ \mathcal{O}$-arrows $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$ and $ g_{\mathcal{O}}:\hat{C}\rightarrow\hat{B}$ such that $ \hat{B}=f(\hat{A})$ and $ \hat{C}= g(\hat{B})$, then, the composite function $ f_{\mathcal{O}}o g_{\mathcal{O}}$ in $ \mathcal{O}$ corresponds to the composite function $ f o g:{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}\rightarrow{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ in $ {\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$.
The category $ \mathcal{O}$, as defined above, represents a pre-ordered set (see 2.1). In fact, the function $ f:\sigma(\hat{A})\rightarrow{\mathchoice { \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} { \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ is unique up to isomorphism, therefore it follows that for any two objects in $ \mathcal{O}$ there exists, at most, one morphism between them, i.e. $ \mathcal{O}$ is a pre-ordered set (Definition 2.1). However, $ \mathcal{O}$ fails to be a poset (see 2.1) since it lacks the antisymmetry property . In fact, it can be the case that two operators $ \hat{B}$ and $ \hat{A}$ in $ \mathcal{O}$ are such that $ \hat{A}\neq\hat{B}$, but they are related by $ \mathcal{O}$-arrows $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$ and $ g_{\mathcal{O}}:\hat{A}\rightarrow\hat{B}$ in such a way that:

$\displaystyle g_{\mathcal{O}}o f_{\mathcal{O}}=id_B\hspace{.1in}and\hspace{.1in}f_{\mathcal{O}}og_{\mathcal{O}} =id_A$ (1.1)

(note that if $ \hat{B}$ and $ \hat{A}$ are related in such a way then $ W_A=W_B$ since
$ \hat{B}=f(\hat{A})\hspace{.1in}\Longrightarrow\hspace{.1in} W_B\subseteq W_A$ and $ \hat{A}=f(\hat{B})\hspace{.1in}\Longrightarrow\hspace{.1in} W_A\subseteq W_B$ ) It is possible to transform the set of self-adjoint operators into a poset by defining a new category $ [\mathcal{O}]$ in which the objects are taken to be equivalence classes of operators, whereby two operators are considered to be equivalent if the $ \mathcal{O}$-morphisms relating them satisfies equation 1.1.

Next: Category of Boolean subalgebras Up: Categories in Quantum Mechanics Previous: Categories in Quantum Mechanics
Cecilia Flori 2007-01-04