# Norm Ordering for a many electron system

1. Jul 6, 2010

### janakiraman

I'm confused with the definition of a norm ordering of operators. The basic definition of norm ordering as understood by me was "Place the annihilation operators to right and creation operators to the left". However I also read another definition "The motivation of norm ordering is to ensure that the expectation value of the normed operators is 0 for vacuum state"

I understand the relation between the above two definitions for a simple non relativistic harmonic oscillator as explained here http://arxiv.org/abs/physics/0212061" [Broken]. But I do not understand the relationship in a many particle system.

Assume a 'N' particle non interacting fermion system, where the ground state is provided by the Slater determinant of the 'N' lowest eigenfunctions. In this case how does placing the annihilation operators (a$$_{\alpha}$$) to the right of creation operators (a$$^{\beta}$$
ensures that the expectation w.r.t vacuum state to be 0. [Note $$\alpha$$ and $$\beta$$ can be different eigen states of the Hamiltonian]. Also I read in a book (Many Body theory Exposed, Chp 8) that placing the annihilator to right is valid only if $$\alpha$$ and $$\beta$$ are higher that the total ground state energy of the system. If they are both lower then the creation operator is placed to right of annihilation. If one of them is lower while other is higher than ground state energy, then the ordering doesn't matter. I'm really confused with this issue. It will be great if some one can throw more light into this problem

Last edited by a moderator: May 4, 2017
2. Jul 7, 2010

### alxm

Did you try the wikipedia article on http://en.wikipedia.org/wiki/Wick_ordering" [Broken]?

Last edited by a moderator: May 4, 2017
3. Jul 8, 2010

### janakiraman

Yes I had a look at wikipedia. But its confusing. For example If I have 3 operators (2 annihilation (A) and one creation (C)). How must i proceed? Can i just use CAA or ACA? I believe both will give a zero expectation for vacuum state? Which of them must be used for the norm order in this case