# A Field operator eigenstates & Fock states(Hatfield's Sch rep)

1. Dec 30, 2016

### ShayanJ

In chapter 10 of his book "Quantum Field Theory of Point Particles and Strings", Hatfield treats what he calls the Schrodinger representation of QFT. He starts with a free scalar field and introduces field operators $\hat \varphi(\vec x)$ and its eigenstates $\hat \varphi(\vec x)|\phi\rangle=\phi(\vec x)|\phi \rangle$. Then he says that the "coordinate" representation of the field state $|\Psi\rangle$ is given by the wave-functional $\Psi[\phi]=\langle \phi |\Psi\rangle$. I have two questions about this:

1) Is the ground state of the field, one of the eigenstates of $\hat \varphi(\vec x)$, i.e. is $\hat \varphi(\vec x)|0\rangle=0$ correct?( At the bottom of page 224 of his book "A modern introduction to quantum field theory", Maggiore calls $\phi(\vec x)=0$ the vacuum. It seems to me its related to my question. Is he correct or just being sloppy? )

2) What is the relationship between the Fock states that are created by $\hat a_k$ and $\hat a_k^\dagger$ and their superpositions that are created by $\hat \varphi(\vec x)$ and $\hat \varphi^\dagger(\vec x)$ and the eigenstates of $\hat \varphi(\vec x)$? Can we say that the eigenstates of $\hat \varphi(\vec x)$ are some kind of a coherent state because $\hat \varphi(\vec x)$ is an anihilation operator for Fock states?

Thanks

2. Dec 30, 2016

### MathematicalPhysicist

I refer you to the book by Robin Ticciati "QFT for Mathematicians" chapter 2, hopefully after reading this chapter you'll understand your misconceptions. (I don't have time to copy the answer here, I am studyding for Stat Mech... :-)).

3. Dec 30, 2016

### ShayanJ

Thanks. Ticciati's formula for the ground state was surprising for me. But there was no answer to my second question. Or at least I couldn't find it!