- #1

George Wu

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- TL;DR Summary
- In peskin P102 ，it mentions spatial wavefunction, I don't know what does it means exactly.

My understanding is:

$$\phi (\mathbf{k})=\int{d^3}\mathbf{x}\phi (\mathbf{x})e^{-i\mathbf{k}\cdot \mathbf{x}}$$

But what is ##\phi (\mathbf{x})## in Qft?

In quantum mechanics,

$$|\phi \rangle =\int{d^3}\mathbf{x}\phi (\mathbf{x})\left| \mathbf{x} \right> =\int{d^3}\mathbf{k}\phi (\mathbf{k})\left| \mathbf{k} \right> $$

where ##\left| \mathbf{x} \right> ## and ##\left| \mathbf{k} \right> ##are the eigenvectors of operater ##\mathbf{X}## and##\mathbf{K}##

In qft, ##\left| \mathbf{k} \right> ##is still the eigenvector of ##\mathbf{K}=-\int{d^3}x\pi (\mathbf{x})\nabla \phi (\mathbf{x})=\int{\frac{d^3k}{(2\pi )^3}}\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}##

However what about ##\left| \mathbf{x} \right> ##?

My question is:

Is there any proper definition of ##\left| \mathbf{x} \right> ##?

Can ##|\phi \rangle ## still be written as:

$$|\phi \rangle =\int{d^3}\mathbf{x}\phi (\mathbf{x})\left| \mathbf{x} \right> $$(maybe with some factors)?

If not, what does spatial wavefunction ##\phi (\mathbf{x})##mean?

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