What is intuitive or physical meaning of wave functional and field configuration and field eigenfunction?

Question

what is the physical meaning of field configuration in quantum field theory. I have come across such terminologies in Schrodinger field theory and path integral field theory. What is the actual difference between quantum field operator and quantum field functional or wave functional of the given field configuration? Also is there any eigenfunction and eigenstate there for field operator? what is the physical meaning of the eigen of a field operator in quantum field theory? Also can someone give intuition for how functional derivatives defines and explains quantum field theory in the place of usually partial differential operator?

Answer 1

Yes, there are field eigenvalues. The difference between the field operator-valued-distribution $\hat\phi(\mathbf x)$, wave-functional $\Psi[\phi]$, and state $|\Psi(t)\rangle$ in quantum field theory is analogous to that between the position operator $\hat{\mathbf{x}}$, wave-function $\psi(\mathbf x)$, and state $|\psi(t)\rangle$ in particulate quantum mechanics.

Schematic comparison

In the Schrödinger picture of particulate quantum mechanics in 3+1D Euclidean space, the position eigenstates are the states $\hat{\mathbf x}\left|\mathbf{x}\right\rangle = \mathbf{x}\left|\mathbf{x}\right\rangle$, with "classical position" eigenvalues $\mathbf x \in\mathbb R^3$. These states are not stationary. The position eigenbasis $\{\left|\mathbf{x}\right\rangle\}$ defines the usual position-space wave-function for state $|\psi\rangle$ as $\psi(\mathbf x)=\left\langle \mathbf{x}\mid\psi\right\rangle$ (showing explicit time dependence, $\psi(t,\mathbf x)=\left\langle \mathbf{x}\mid\psi(t)\right\rangle$). The state evolves per the Schrödinger equation $i\hbar\partial_t \psi(t,\mathbf x)=\hat H \psi(t,\mathbf x)$.

In the Schrödinger picture of quantum field theory in 3+1D Minkowski space, for a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$, the field eigenstates are the states $\hat\phi(\mathbf x)|\Phi\rangle=\Phi(\mathbf x)|\Phi\rangle$, with the "classical field configuration" eigenvalues (capitalized here for clarity) $\Phi:\mathbb R^3\to\mathbb R$ (in other words, $\Phi\in\mathbb{R}^{\mathbb{R}^3}$). Like above, these states aren't stationary. The field eigenbasis $\{\left|\phi\right\rangle\}$ defines the wave-functional $\Psi[\phi]=\langle\phi|\Psi\rangle$, given state $|\Psi\rangle$. The state evolves per the functional Schrödinger equation $i\hbar\partial_t \Psi[t,\phi]=\hat H \Psi[t,\phi]$.

Bottom line

(Abusing lots of terminology and distinctions,) if the "intuitive meaning" of the wave-function is a superposition of positions, then the "intuitive meaning" of the wave-functional is a superposition of field configurations. Functional derivatives are intuitively mandatory because $\Psi[t,\phi]$ is a function of a function ($\phi:\mathbb{R}^3\to\mathbb{R}$), whereas $\psi(t,\mathbf x)$ was merely a function of a vector ($\mathbf x \in\mathbb R^3$).

For introductory information on QFT in the Schrödinger picture, you might try this recent European preprint. And note, the Schrödinger picture is completely equivalent to the Heisenberg picture (and the Interaction picture) in all cases.

Answer 2

The field operators $\psi^{\dagger}$ and $\psi$ are creation and annihilation operators for particles in the position eigenstate $\left|x\right>$. A wavefunctional is the analogue of a wavefunction in ordinary QM, it gives you the probability amplitude when applied to an entire classical field configuration, just like an ordinary wavefunction gives you an amplitude when you apply that function to a position of a particle. This book explains the formalism of QFT much better than standard textbooks, it allows you to understand what QFT is including advanced concepts, without getting bogged down in intricate mathematical details.