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I've recently stumbled upon something that looked kind of silly, but I still find myself a bit confused by it. Namely in quantum field theory, when we quantize a scalar field, we impose commutation relations on creation and annihilation operators that correspond to momenta in their mode expansion.

$$[a_{\textbf{k}},a^\dagger_{\textbf{k}'}] = \delta^{(3)}(\textbf{k} - \textbf{k}')$$

This continuum case of the usual creation/annihilation commutator makes a ##\delta^{(3)}(0)## factor when we act on a certain state vector with its annihilation operator. When we work with expected values of operators and this factor appears, we often treat it as the volume of space enclosing the system, and then draw conclusions in terms of densities corresponding to this volume. However, what if we're interested in looking exactly at a state vector, not bilinear forms that would be operators, nor their expected values.

For example, let's say we define a vector ##\lvert 1_{\textbf{k}}>##, as a vector which has occupation number 1 for 3-momentum ##\textbf{k}##, that is, it has one particle of this momentum. We act on it with annihilation operator:

$$a_{\textbf{k}}\lvert 1_{\textbf{k}}> = a_{\textbf{k}}a^\dagger_{\textbf{k}}\lvert 0> = (a^\dagger_{\textbf{k}}a_{\textbf{k}} + \delta^{(3)}(0))\lvert 0> = \delta^{(3)}(0)\lvert 0>$$

What sense do we make of this resulting state, if it has this diverging factor in front of it? I assume it may have to do with our assumption that the momentum of the particle is innitially completely defined, but this looked like the treatment that was usual so far in QFT textbooks, just they never consider this case as far as I've noticed.

$$[a_{\textbf{k}},a^\dagger_{\textbf{k}'}] = \delta^{(3)}(\textbf{k} - \textbf{k}')$$

This continuum case of the usual creation/annihilation commutator makes a ##\delta^{(3)}(0)## factor when we act on a certain state vector with its annihilation operator. When we work with expected values of operators and this factor appears, we often treat it as the volume of space enclosing the system, and then draw conclusions in terms of densities corresponding to this volume. However, what if we're interested in looking exactly at a state vector, not bilinear forms that would be operators, nor their expected values.

For example, let's say we define a vector ##\lvert 1_{\textbf{k}}>##, as a vector which has occupation number 1 for 3-momentum ##\textbf{k}##, that is, it has one particle of this momentum. We act on it with annihilation operator:

$$a_{\textbf{k}}\lvert 1_{\textbf{k}}> = a_{\textbf{k}}a^\dagger_{\textbf{k}}\lvert 0> = (a^\dagger_{\textbf{k}}a_{\textbf{k}} + \delta^{(3)}(0))\lvert 0> = \delta^{(3)}(0)\lvert 0>$$

What sense do we make of this resulting state, if it has this diverging factor in front of it? I assume it may have to do with our assumption that the momentum of the particle is innitially completely defined, but this looked like the treatment that was usual so far in QFT textbooks, just they never consider this case as far as I've noticed.

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