deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 694b48a..08abc3c 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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\end{equation}  This sets apart the behaviour behavior  of the two operators, the manner they go about extracting the information is different. Perhaps, if we have a system of quanta and set the condition if at any time I SEE the system empty it must be destroyed, then $r$ would correspond to looking into the system and taking half a quantum out, and $s$ to not looking and taking half a quantum out. $r$ carries with it the act of observation. However, if one operates further \begin{equation} ss|0>=\\  rr|0>=\\ 
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s^+r=r^+s, Herm...  \end{equation}  So we have mixed hermitian eigenoperators of state |n>. $|n>$.  Which means a double application of $r^+r$ is equal to the number operator $N=a^+a$, a double application of $ss^+$ is equivalent to the operator $aa^+$ and it implies that $s^+s \equiv rr^+$. This is interesting, we can measure the number of particles in an oscillator by only taking half of one out and putting it back ($r$ then $r^+$ is the operation $r^+r$), but there are two ways to take the particle out and two ways to put it back and each appear as different actions with different 'compensations'.  These sub-operations work in line with the conjugate operation. if we introduce a basic ladder opperator operator  with no coefficients \begin{equation} \nu^+|n>=|n+1> \\  \nu|n>=|n-1> 
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\end{equation}  If we assume there is an inverse of r^+ $r^+$  we can eliminate $\nu$ and come to the identity $s^+s \equiv rr^+$ which was true from definition before. We have also \begin{equation} s\nu^+s=s^+\nu s^+ \\  r\nu^+r=r^+\nu r^+ 
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[s^+_k,s^+_l]=f(n)\delta_{kl}   \end{equation}  where $f(n)$ is some relationship between the |n> $|n>$  state the commutator is applied to. It would be nicer still if f(n) $f(n)$  was a constant, however this does not appear to be true. In some respects, with the correct left and right divisions \begin{equation} 
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H=\sum_{k} \hbar \omega a^+_ka_k + \sum_{k,l} a^+_la_k + \sum_{k,l,m,n} a^+_ma^+_na_ka_l  \end{equation}  Which would correspond to a Feynman diagram with incoming states k,l $k,l$  and outgoing states n,m $n,m$  respectively. When we expand the normal self interaction terms in the fractional operators \begin{equation} H= \sum_{k} \hbar \omega a^+_ka_k \\  H= \sum_{k} \hbar \omega r^+_ks^+_ks_kr_k \\