LOG#143. Basic QFT in curved ST(III).

We are now ready to go back to curved spacetime in the IN/OUT formalism!

We have

$\vert 0_{in}\rangle\neq\vert 0_{out}\rangle$

Indeed,

$\langle N_k\rangle=_{in}\langle 0\vert b^+b\vert 0\rangle_{in}=\displaystyle{\sum_j \vert \beta_{kj}\vert ^2\neq 0}$

where $\beta_{kj}\neq 0$ when we get particles from gravity! The particle density reads

$n=\dfrac{\displaystyle{\sum_k N_k}}{V}$

For instance, in a Friedmann-Robertson-Walker metric (FRW universe) we obtain

$ds^2=-dt^2+a(t)^2dx^2$

where $a(t)=C(t)$ is the scale factor and

$H=E_k=\sqrt{m^2+\frac{k^2}{a^2}}$

is the energy of the mode k. The Heisenberg Uncertainty Principle (HUP) implies (heuristically) that

$\int_t^{t+\Delta t}E(t')dt'\sim \hbar$

An interesting consequence of all this crazy stuff is that with massless particles ( $m\sim 0$ ) and a de Sitter (dS) spacetime we get virtual particles produced by this effect or alternatively $\Delta t$ is finite!

The IN/OUT asymptotically stationary phase implies gravitational particle creation! The Bogoliubov coefficients $\beta$ are understood in the following sense:

$\sum \vert \beta\vert^2\neq 0\leftrightarrow \vert \overline{0}\rangle =\vert 0\rangle_b=\vert 0\rangle_{out}$

contains “particles” (quasiparticles to be more precise) that are related to a set of normal modes. In fact, the complete set of Bogoliubov coefficients satisfy certain relations:

$\vert \alpha_k\vert^2-\vert\beta_k\vert^2=1$

for bosons and

$\vert \alpha_k\vert^2+\vert\beta_k\vert^2=1$

for fermions. The quantum analogue of this coefficients are the scattering problem over a potential barrier (they can be understood as “tunneling amplitudes” somehow).

In flat FRW universes, we get

$g_{\mu\nu}=\Omega^2\eta_{\mu\nu}$

$dt=\Omega d\eta$

$\displaystyle{\lim_{n\rightarrow -\infty}\Omega=\Omega_1}$

$\displaystyle{\lim_{n\rightarrow +\infty}\Omega=\Omega_2}$

The Klein-Gordon field has solutions (in curved spacetime!):

$u_k=\dfrac{1}{(2\pi)^{3/2}}\dfrac{1}{\Omega (\eta)}e^{ikx}\chi_k (\eta)$

with

$(u_k,u_l)=\delta^3(k-l)$ normalization. The time dependent simple harmonic oscillator with variable $\omega$ satisfy

$\chi_k''(\eta)+\omega_k^2(\eta)\chi_k(\eta)=0$

$\omega_k^2(\eta)=k^2+m^2\Omega^2+(6\xi-1)\dfrac{\ddot{\Omega}}{\Omega}$

and where $\omega=0$ is the minimal coupling case, and $\xi=\dfrac{1}{6}$ determines the conformal coupling case in D=4. Furthermore,

$\langle N_k\rangle=\langle \overline{0}\vert a^+_ka_k\vert \overline{0}\rangle=_{out}\langle 0\vert a^+_ka_k\vert 0\rangle_{out}=\vert \beta_k\vert^2$

and thus

$u_k\rightarrow \overline{u}_k=\alpha_k u_k+\beta_k u^*_{-k}$

The particle number uncertainty due to the particle creation can be also calculated:

$\Delta N\geq \dfrac{1}{m\Delta t}+\vert A\vert \Delta t$

It has a minimum when

$\Delta t=\sqrt{(\vert A\vert m)^{-1}}$

and

$\Delta N\Delta t\geq \dfrac{1}{m}+\vert A\vert (\Delta t)^2$

$\Delta N=2\sqrt{\vert A\vert/m}$

Fluctuations in the particle number are ALWAYS non zero if there is particle creation! The proof is easy to understand:

$\langle N_k^2\rangle-\langle N_k\rangle^2=\langle \overline{0}\vert a_k^+a_ka_k^+\vert \overline{0}\rangle =2\vert \alpha_k\vert^2\vert \beta_k\vert^ 2$

There is a nice relationship between the “squeezed” states and the vacuum in this formalism:

If $\vert 0\rangle$ is a squeezed state respect to $\vert \overline{0}\rangle$ , then

$\vert\overline{m_k},\overline{n_k}\rangle=\dfrac{1}{\sqrt{m!n!}}(\overline{a^+}_k)^m(\overline{a^+}_{-k})^n\vert 0_{k,-k}\rangle$

$\vert 0\rangle=\displaystyle{\prod_k}\dfrac{1}{\sqrt{\vert\alpha_k\vert}}e^{-\frac{\beta_k^*\overline{a_k^+}\overline{a_{-k}^+}}{2\alpha_k}}\vert\overline{0}\rangle$

The conformal coupling is special: massless case implies that there is no particle creation (indeed, conformal field theories has no notion of particle theirselves!).

When we have a smooth expansion, in the massive case $m\neq 0$ , then $\xi=\dfrac{1}{6}$ is the conformal case. Moreover, the solutions to the EOM can be easily found by common procedures of differential equations. Thus, we have

$\Omega^2(\eta)=A+B\tanh (\rho \eta)$ with $\rho=ct$

$\chi_k''(\eta)+(k^2+m^2\Omega^2)\chi_k(\eta)=0$ has two solutions with frequencies

$\Omega_1^2=A-B$ and $\Omega_2^2=A+B$ !

A well known example in the literature is when $\vert \beta_k\vert^2\neq 0$ corresponds to the case

$\langle N_k\rangle\rightarrow e^{-2\pi \omega_{in}/\rho}$

when

$\rho\rightarrow 0$

Thus,

$\dfrac{\dot{\Omega}}{\Omega}\sim \rho\rightarrow 0$

and it corresponds to the adiabatic expansion case.

With a step function expansion ( $\rho\rightarrow \infty$ ), we get

$\Omega (\eta)=\Omega_1\theta (\eta_0-\eta)+\Omega_2\theta(\eta-\eta_0)$

and it corresponds to the “sudden approximation”. The modes read then, in this “sudden” case,

$\chi_k (\eta)=e^{-i\omega_{in}\eta}$

$\eta<\eta_0$

and

$\chi_k (\eta)=\dfrac{\omega^+}{\omega_{out}}e^{-2i\omega^-\eta_0}e^{-i\omega_{out}\eta}-\dfrac{\omega^-}{\omega_{out}}e^{-2i\omega^+\eta_0}e^{+i\omega_{out}\eta}$

if $\eta>\eta_0$ . The particle number, the critical frequency and the variation of $\eta$ correspond respectively to

$N_k\sim e^{-\pi\omega \Delta \eta}$

$\omega_c\sim \dfrac{1}{\pi\Delta \eta}$

$\Delta \eta\leftrightarrow \dfrac{1}{\rho}$

After all these rudimentary facts, we can understand two of the most celebrated phenomena in curved spacetime QFT:

1) The Unruh effect (or the relativity of vacuum in curved spacetime QFT).

2) The Hawking effect (particle creation in the surrounding of black holes).

These two effects will be covered in my next posts from a very elementary viewpoint!

See you in the next TSOR posts!

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The Spectrum Of Riemannium

Mobilis in mobili: Physmatics Chronicles!

LOG#143. Basic QFT in curved ST(III).

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