Spectral Algebras


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Spectral Algebras

Definition 3.8   : A spectral algebra $ W_A$ of an operator A is the Boolean algebra3.2 associated with those projectors that form the spectral decomposition of A, projectors that project onto the eigenspaces associated with the eigenvectors of A.

In order to explain the above definition, and the existing relations between different spectral algebras associated with different self-adjoint operators, we will consider an example of lattice of properties for a quantum system given by Bub in his book ``Interpreting the Quantum World".
If we consider two physical quantities A and B, whose representative operators have the following spectral decomposition:

$\displaystyle \hat{A}=a_1\hat{P}+a_2\hat{P}_{a_2}+a_3\hat{P}_{a_3}$ (3.1)

$\displaystyle \hat{B}=b_1\hat{P}+b_2\hat{P}_{b_2}+b_3\hat{P}_{b_3}$ (3.2)

such that the operators $ \hat{A}$ and $ \hat{B}$ have a common projector $ \hat{P}$, the lattice of properties would then be [7] \begin{diagram}$ $ \begin{displaymath}\xymatrix{ &&1\ar@{-}[rrd]\ar@{-}[rd]\ar@{-}[d]\ar@{-}[ld]\ar@{-}[lld]&&\\ P\vee P_{a_2}\ar@{-}[rdd]\ar@{-}[dd]&P\vee P_{a_3}\ar@{-}[rdd]\ar@{-}[ldd]&P^{\perp}\ar@{-}[rrdd] \ar@{-}[rdd]\ar@{-}[dd]\ar@{-}[ldd]&P\vee P_{b_2}\ar@{-}[dd]\ar@{-}[llldd]&P\vee P_{b_3}\ar@{-}[dd] \ar@{-}[lllldd]\\ &&&&\\ P\ar@{-}[rrd]&P_{a_2}\ar@{-}[rd]&P_{a_3}\ar@{-}[d]&P_{b_2}\ar@{-}[ld]&P_{b_3}\ar@{-}[lld]\\ &&0&&\\ }\end{displaymath}\end{diagram} where

$\displaystyle \hat{P}^\perp$ $\displaystyle =\hat{P}_{a_2}\vee\hat{P}_{a_3}=\hat{P}_{b_1}\vee\hat{P}_{b_3}$    
  $\displaystyle =\hat{P}_{a_2}\vee\hat{P}_{b_2}$    
  $\displaystyle =\hat{P}_{a_2}\vee\hat{P}_{b_3}$    
  $\displaystyle =\hat{P}_{a_3}\vee\hat{P}_{b_2}=\hat{P}_{a_3}\vee\hat{P}_{b_3}$    

Now, let us consider a Borel function $ f:\sigma(\hat{A})\rightarrow\mathds{R}$ such that $ \hat{B}=f(\hat{A})$ .
According to diagram % latex2html id marker 5360 $ \ref{dia:2}$ the spectral projectors of $ \hat{A}$ are $ \hat{P}, \hat{P}_{a_2}, \hat{P}_{a_3}$ with respective eigenvalues $ a_1, a_2, a_3$. Therefore, the Boolean sublattice of $ P(\mathcal{H})$, which represents the spectral algebra $ W_A$ of $ \hat{A}$ is composed by $ \hat{0}$, $ \hat{1}$, $ \hat{P}^\perp$, $ \hat{P}$ , $ \hat{P}_{a_2}$, and $ \hat{P}_{a_3}$. If we choose f such that the eigenvalues of $ \hat{B}$ are $ f(a_1),f(a_2)=f(a_3)$ then, the spectral projectors of $ \hat{B}$ are $ \hat{P}$ and $ \hat{P}^\perp$. This entails that the spectral algebra $ W_{f(A)}$ of the operator $ \hat{B}$ is a subalgebra of $ W_A$, i.e. $ W_{f(A)}\subseteq W_A$.
Since in Quantum Mechanics propositions are represented by spectral projectors the question that arises is the following: how do the relations between different spectral projectors determine the relations between the propositions that they represent? To answer this question let us consider, again, the above example where, the proposition $ A=a_1$ is represented by the projection operator $ \hat{P}_{A=a_1}$ which is equivalent to $ \hat{P}$ that, in turn, is equivalent to $ \hat{P}_{f(A)=f(a_1)}$ since f is one to one with respect to $ a_1$. Therefore, in this case, the two propositions $ A=a_1$ and $ f(A)=f(a_1)$ are equivalent. If, instead, we consider propositions $ A=a_2$ and $ f(A)=f(a_2)$ then, the situation changes since f is many to one with respect to $ a_2$. In fact, we would have the following: $ \hat{P}_{f(A)=f(a_2)}=\hat{P}^\perp=\hat{P}_{a_2}\vee\hat{P}_{a_3}=\hat{P}_{A=a_2}\vee\hat{P}_{A=a_3}$. Since $ \hat{P}_{a_2}\vee\hat{P}_{a_3}$ represents the join3.3 of $ \hat{P}_{a_2}$ and $ \hat{P}_{a_3}$ it follows, by definition of join, that $ \hat{P}_{A=a_2}\leq\hat{P}_{f(A)=f(a_2)}$ with respect to the partial ordering of the lattice. Therefore the proposition $ f(A)=f(a_2)$ is weaker than proposition $ A=a_2$ and so we say that $ f(A)=f(a_2)$ is the coarse graining of $ A=a_2$.

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Cecilia Flori 2007-01-02