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In response to your edit: The real part of $(A^*(S)A(T))$ is equal to the real part of $(A(S)A^*(T))$, since they are just complex conjugates of each other. So he could have written either one. Concrete example: let $(A^*(S)A(T))=a+ib$ for some $a,b$. Then $(A(S)A^*(T))=a-ib$, since it's just the complex conjugate. Then $(A^*(S)A(T))+(A(S)A^*(T))=2a$. We ...


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$\mathfrak{R}e$: real part. $A^*$: complex conjugate of probability amplitude $A$.


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I got the answer: if an index appears twice, it implies summation.


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Since you have not provided a direct reference, it's hard to be completely sure (and particularly to pin down the details), but there's really only one general idea that this can refer to. In general, any arbitrary isometry $S$ of euclidean space has the form $$ \mathbf x\mapsto R\mathbf x+\mathbf t, $$ where $\mathbf t$ is an arbitrary vector and ...


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I think I can see where this is going, but I don't understand your notation so I will use different, but I hope that this will make sense.... The point here is that if translation is a symmetry operation (call it $t$) and rotation is a symmetry operation (call it $C_n$ - where $n$ is the number of $C_n$ operations required for a full turn - i.e. an $n-$fold ...



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