- #1
Tan Tixuan
- 3
- 1
In quantum field theory, we have the following expansion on a scalar field (I follow the convention of Schwarz's book)
$$\phi(\vec{x},t)=\int d^3 p \frac{a_p exp(-ip_\mu x^\mu)+a_p^{\dagger}exp(ip_\mu x^\mu)}{(2\pi)^3 \sqrt{2\omega_p}} \quad p^{\mu}=(\omega_p,\vec{p})$$
With commutation relation
$$[a_q,a_p^{\dagger}]=(2\pi)^3 \delta^3 (p-q)$$
We can construct a coherent state of the field by the following, with $\beta_p\equiv \beta(p)$
$$|C\rangle=exp\{-\frac{1}{2}\int d^3p |\beta_p|^2\}exp\{\int \frac{d^3p}{(2\pi)^\frac{3}{2}} (\beta_p a_p^{\dagger})\}|0\rangle$$
It is then not hard to verify that the field expectation value is
$$\langle C|\phi|C\rangle=\int \frac{d^3 p}{(2\pi)^{3/2}}\frac{\beta_p e^{-ip^{\mu}x_{\mu}}}{\sqrt{2\omega_p}}+H.C.$$
**My question is the following**:
It seems to me that ##\phi## is only a simple addition of a bunch of independent harmonic oscillators, and the value of ##\beta(p)## can be determined totally arbitrarily,**i.e. there is no a priori way to determine the dependence of $\beta$ on p . I want to know what is the reasonable way to determine this dependence.** For example, we can make ##\beta(p)## be non-zero only for ##p=0##, and then we would only be left with one simple harmonic oscillator, and the resulting field expectation value is
$$\langle C|\phi|C\rangle\sim cos(mt+\beta_0)$$However,it is often said in the literature that classical field is produced by the coherent state, especially in the study of dark matter cosmology. For example, in [this paper][1], equation 2.3, it is assumed that because the occupation number is huge, the dark matter field is almost classic, and can assume the profile
$$\phi_1(\vec{x},t)=A(\vec{x})cos(mt+\alpha(\vec{x}))$$
In this case, how should I construct ##\beta_1(p)## corresponding to ##\phi_1##, and what is the justification for this kind of profile?(the profile of ##\beta##). i.e. how should I describe it in terms of quantum field theory?
[1]: https://arxiv.org/abs/1309.5888
$$\phi(\vec{x},t)=\int d^3 p \frac{a_p exp(-ip_\mu x^\mu)+a_p^{\dagger}exp(ip_\mu x^\mu)}{(2\pi)^3 \sqrt{2\omega_p}} \quad p^{\mu}=(\omega_p,\vec{p})$$
With commutation relation
$$[a_q,a_p^{\dagger}]=(2\pi)^3 \delta^3 (p-q)$$
We can construct a coherent state of the field by the following, with $\beta_p\equiv \beta(p)$
$$|C\rangle=exp\{-\frac{1}{2}\int d^3p |\beta_p|^2\}exp\{\int \frac{d^3p}{(2\pi)^\frac{3}{2}} (\beta_p a_p^{\dagger})\}|0\rangle$$
It is then not hard to verify that the field expectation value is
$$\langle C|\phi|C\rangle=\int \frac{d^3 p}{(2\pi)^{3/2}}\frac{\beta_p e^{-ip^{\mu}x_{\mu}}}{\sqrt{2\omega_p}}+H.C.$$
**My question is the following**:
It seems to me that ##\phi## is only a simple addition of a bunch of independent harmonic oscillators, and the value of ##\beta(p)## can be determined totally arbitrarily,**i.e. there is no a priori way to determine the dependence of $\beta$ on p . I want to know what is the reasonable way to determine this dependence.** For example, we can make ##\beta(p)## be non-zero only for ##p=0##, and then we would only be left with one simple harmonic oscillator, and the resulting field expectation value is
$$\langle C|\phi|C\rangle\sim cos(mt+\beta_0)$$However,it is often said in the literature that classical field is produced by the coherent state, especially in the study of dark matter cosmology. For example, in [this paper][1], equation 2.3, it is assumed that because the occupation number is huge, the dark matter field is almost classic, and can assume the profile
$$\phi_1(\vec{x},t)=A(\vec{x})cos(mt+\alpha(\vec{x}))$$
In this case, how should I construct ##\beta_1(p)## corresponding to ##\phi_1##, and what is the justification for this kind of profile?(the profile of ##\beta##). i.e. how should I describe it in terms of quantum field theory?
[1]: https://arxiv.org/abs/1309.5888
