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- TL;DR Summary
- What can you do with a quantum field?
I'm learning some QFT from QFT for the Gifted Amateur. Chapter 11 develops the massive scalar quantum field but they don't seem subsequently to do anything with it. I've looked ahead at the next few chapters, which move on to other stuff, which leaves me wondering what we we actually do with this field? What we have is the field operator:
$$\hat \phi(x) = \int \frac{d^3p}{(2\pi)^{3/2}(2E_p)^{1/2}}(\hat{a}_{\vec p}exp(-ip\cdot x)+\hat{a}^{\dagger}_{\vec p}exp(ip\cdot x))$$
Where ##x, p## are four-vectors, ##E_p = \sqrt{\vec p^2 + m^2}## and ##d^3p## is the integral over all three-momentum states ##\vec p##.
This leads to the normally ordered Hamiltonian:
$$N(\hat H) = \int d^3p (E_p \hat{a}^{\dagger}_{\vec p} \hat{a}_{\vec p}) = \int d^3p (E_p \hat{n}_p)$$
That is all good, but we don't seem to do anything with this now. The only exercise they do is to show that:$$\langle p | \hat \phi(x) | 0 \rangle = \exp(i p \cdot x)$$
This leads me to interpret this as ##\hat \phi(x) | 0 \rangle = |x \rangle##. I.e. this operator acting on the vacuum state results in a single particle with four-position ##x##.
However, although they introduce the concept of a four-momentum state ##|p \rangle##, they don't say anything about four-position states.
My main question is what we do now we have this quantum field? I feel like I want to do something with it! But, what?
The other question is whether we generally continue to use three-momentum creation and annihilation operators (and where the occupation number representation relates to three-momentum states); or, is there going to be a wholesale shift towards using four-momentum states? Or, do we juggle between the two?
Thanks
$$\hat \phi(x) = \int \frac{d^3p}{(2\pi)^{3/2}(2E_p)^{1/2}}(\hat{a}_{\vec p}exp(-ip\cdot x)+\hat{a}^{\dagger}_{\vec p}exp(ip\cdot x))$$
Where ##x, p## are four-vectors, ##E_p = \sqrt{\vec p^2 + m^2}## and ##d^3p## is the integral over all three-momentum states ##\vec p##.
This leads to the normally ordered Hamiltonian:
$$N(\hat H) = \int d^3p (E_p \hat{a}^{\dagger}_{\vec p} \hat{a}_{\vec p}) = \int d^3p (E_p \hat{n}_p)$$
That is all good, but we don't seem to do anything with this now. The only exercise they do is to show that:$$\langle p | \hat \phi(x) | 0 \rangle = \exp(i p \cdot x)$$
This leads me to interpret this as ##\hat \phi(x) | 0 \rangle = |x \rangle##. I.e. this operator acting on the vacuum state results in a single particle with four-position ##x##.
However, although they introduce the concept of a four-momentum state ##|p \rangle##, they don't say anything about four-position states.
My main question is what we do now we have this quantum field? I feel like I want to do something with it! But, what?
The other question is whether we generally continue to use three-momentum creation and annihilation operators (and where the occupation number representation relates to three-momentum states); or, is there going to be a wholesale shift towards using four-momentum states? Or, do we juggle between the two?
Thanks
but it would need new lyrics.