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

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• ShayanJ
In summary, Hatfield discusses the Schrodinger representation of QFT in chapter 10 of his book "Quantum Field Theory of Point Particles and Strings". This includes introducing field operators and their eigenstates, as well as the wave-functional for the "coordinate" representation of the field state. There are two questions raised regarding this representation, including the relationship between Fock states and their superpositions created by field operators and the eigenstates of the field. Ticciati's book is recommended for further clarification on these topics.
ShayanJ
Gold Member
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

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... :-)).

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!

## 1. What are field operator eigenstates in Hatfield's Schrodinger representation?

Field operator eigenstates in Hatfield's Schrodinger representation are states that represent the eigenvalues of the field operator. They are used to describe the quantum states of a system in terms of the field variables, such as position and momentum.

## 2. How are field operator eigenstates related to Fock states?

Field operator eigenstates are used to construct Fock states, which are states that represent the number of particles in a given quantum system. Fock states are constructed by applying field operators to the vacuum state, creating a multi-particle state.

## 3. What is the significance of Fock states in quantum field theory?

Fock states are important in quantum field theory as they provide a way to describe the quantum states of a system with multiple particles. They also allow for the calculation of observable quantities, such as energy and momentum, in a quantum field theory framework.

## 4. How are Fock states related to the creation and annihilation operators?

Fock states are constructed using creation and annihilation operators, which act on the vacuum state to create and destroy particles. The number of creation and annihilation operators used in the construction of a Fock state determines the number of particles in that state.

## 5. Can Fock states be used to describe interacting quantum fields?

Yes, Fock states can be used to describe interacting quantum fields. In the presence of interactions, the creation and annihilation operators are modified, leading to changes in the Fock states. This allows for the description of interacting particles in a quantum field theory framework.

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