# Connection between first and second quantization

This is my question: In a book on many body quantum theory I came across equality between antisymmetrized many-particle state vector which, as you know, includes sum over permutations of product states of one particle states AND collection of creation operators acting upon vacuum state. Now, I know that this product of creation operators is defined to be
anti- symmetric but how can you get it to look like a sum, which is implied when antisymmetrization operator acts on product space? Is it i mistake? Did they maybe wanted to write a product of field operators? Thank you.

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This is a strange question. If you already write a state in the second-quantized Hilbert space as a polynomial in creation operators, you no longer have to sum over permutations - you no longer have to antisymmetrize - because the creation operators already anticommute with each other and do the antisymmetrization automatically. The sum over permutations is only necessary to do the antisymmetrization which is needed when it's not done automatically, i.e. in a multi-particle formalism that a priori admits non-antisymmetric wave functions. –  Luboš Motl Sep 22 '12 at 7:08
but you did not write it as a polynomial. you wrote a product of creation operators acting on a vacuum state, then put equality sign and then usual sum over antisymmetric products of one particle states. yes, you have your anticomm. relation but you dont have a sum...at least i dont see it. –  Marko Sep 24 '12 at 23:32