When the infinite sum of 1/2 quanta energy in (3-59) was first found, physicists wanted desperately to make it go away. The amount of energy involved should, via general relativity, curve the universe to such an enormous degree that the light emanating from your finger would be bent so much that it would never reach your eyes. But that isn't what happens in our world, so something isn't correct. In fact, the difference in mass-energy level of the vacuum, between what is predicted by theory and what is observed, is on the order of a factor of 1 0 120, the biggest discrepancy between theory and experiment in the history of science.
One approach to solving ("hiding" may be a better word) this problem is something called normal ordering. Normal ordering, in any term, consists of moving all destruction operators to the right hand side of that term.
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Although use of normal ordering became quite widespread, it suffers from a pretty fundamental probkm. It violates the foundational postulate of non-commutation of certain operators, upon which all of QFT stands. Invoking normal ordering means assuming, in this one area of QFT, that $a(k)$ and $a^\dagger( k )$ (as well as $b(k)$ and $b^\dagger ( k )$) commute! But they don't. And the fact that they don't is fundamental to every other part of QFT I . In normal ordering, we simply suspend commutation long enough to get a zero energy for the vacuum, then bring it back for the rest of the theory. It is not unreasonable to conclude that use of normal ordering for this purpose is questionable, at best.
page 60 in Student Friendly QFT by Klauber
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