# How are particles represnted in scalar field theories? And what determines their nodal mode?

In Quantum Field Theory, is a single elementary particle represented like a single normal mode of oscillation or a sum or superposition of more than one normal modes? And what determines the exact mode (like type and number of nodes) of a specific particle? Is it the principle (energy) quantum number, all quantum numbers or something else?

## 1 Answer

In QFT, a single elementary particle can be a wave packet which is a sum over the normal modes $k$. An arbitrary single particle state is, $$|\psi\rangle=\int d^{3}k \ \langle k|\psi\rangle a^{\dagger}_{k}|0\rangle$$ where $|0\rangle$ is the vacuum state and $a^\dagger_{k}$ is the creation operator for a particle in the normal mode $k$. A creation operator for a single particle in state $|\psi\rangle$ is, $$a^{\dagger}(\psi)=\int d^{3}k \ \langle k|\psi\rangle a^{\dagger}_{k}$$ so that $|\psi\rangle=a^{\dagger}(\psi)|0\rangle$.

• Thank you, I've seen these representations and I understand them and some other basic concepts of QFT.That will imply that a particle is (abstractly) always a superposition of normal modes.But I didn't ask the question completely correctly.What determines the exact mode of a specific particle? Is it the principle (energy) quantum number, all quantum numbers or something else?For example here electron wave packets are experimentally imaged like resembling a normal mode with zero or one nodal diameters and nine or ten nodal circles (last four pages): arxiv.org/ftp/arxiv/papers/0708/0708.1060.pdf – Georgi Pavlov May 29 '17 at 7:16