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Here are the two most important tricks: Commutators via derivative If $A$ is an operator which has been written as a normal ordered$^{[1]}$ product of $a$ and $a^\dagger$, then the following is true $$[a, A] = \frac{\partial A}{\partial a^\dagger} \quad \textrm{and} \quad [a^\dagger, A] = -\frac{\partial A}{\partial a} \, .$$ Wick's theorem (boson ...


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What kind of 'tricks' are you thinking about? You can use commutation relations to arbitrarily reorder them and, for example, eventually evaluate in something in the vacuum state. Something that may be of interest in this context is Wick's theorem, which can be formulated in operator form.


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It seems that "fish" refers to $\propto$. It is means "proportional to". Since @Kyle Kanos has provided useful links about this, I won't repeat them. But I want to add a useful tool for checking unfamilar symbols: http://detexify.kirelabs.org/classify.html You can draw the symbol and get its latex code, and then it's easy to find its meaning (usually the ...


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As stated in the comment, the "fish" is the mathematical symbol for "proportional to". In the case of the amplitude and energy, $$ E\propto A^2 $$ means that $$ E=CA^2 $$ for some constant $C$.



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