LOG#141. Basic QFT in curved ST(I).

This post begins a thread about Quantum Field Theory (QFT) in curved spacetime (ST). They are based on some of my notes about the subject. Enjoy it!

I. LIMITS FROM QUANTUM MECHANICS AND GENERAL RELATIVITY.

The Einstein-Hilbert action in General Relativity (GR) reads

(1) $\begin{equation*} \displaystyle{S_{EH}=\dfrac{\hbar}{16\pi L_p^2}\int d^4x \sqrt{-g}R}\end{equation*}$

The Planck length is

(2) $\begin{equation*} L_P^2=\dfrac{G_N\hbar}{c^3}\end{equation*}$

Therefore, the classical GR action is proportional to

(3) $\begin{equation*} S_{EH}\sim \hbar \left(\dfrac{L}{L_p}\right)^2\end{equation*}$

Quantum curvature will become important when $L\leq L_p$ , and quantum corrections will be important if

$S\lesssim\hbar$

and the path integral will read

$\displaystyle{\langle g_2\vert g_1\rangle=\int\left[Dg\right]e^{\frac{iS}{\hbar}}}$

Semiclassical gravity approach treats gravity as a classical field and the other fields as pure quantum fields. Of course, it is only an effective theory or approximation to the realm of pure quantum gravity. However, it is well defined in the limit $S_{EH}/\hbar$ if $L>>L_p$ .

The Heisenberg Uncertainty Principle (HUP) reads

$\Delta E\Delta t\approx \hbar$

$\Delta E\left(\dfrac{R}{c}\right)\approx \hbar$

Due to the energy-mass equivalente $\Delta E=\Delta Mc^2$ , then we write

(4) $\begin{equation*} \Delta M=\dfrac{\Delta E}{c^2}\sim\dfrac{\hbar}{c r}\end{equation*}$

and

(5) $\begin{equation*} E_G\sim \dfrac{G_N\hbar^2}{r^3c^2}\end{equation*}$

Whenever $\dfrac{G_NM^2}{r}\sim \dfrac{\hbar c}{r}$ , then $\Delta E\sim E_G$ , that is, when $r\sim L_p$ , Quantum Gravity becomes inevitable!

Heisenberg Uncertainty Principle is sometimes extended to the Spacetime Uncertainty Principle (non-commutative geometry based, pioneered by Snyder and others), or SUP:

(6) $\begin{equation*} \Delta x\Delta t=\Delta x\Delta y\sim L_P^2\end{equation*}$

There are some interesting extensions of the HUP and SUP I am not going to cover in this thread. However, let me remark that an interesting relationship and link between IR (infra-red, long distances) and UV (ultra-violet, short distances) or IR/UV entanglement/interplay enters here when we combine QFT or QM with GR:

(7) $\begin{equation*} \Delta x \Delta t=\dfrac{2G_N\Delta E}{c^4}\dfrac{\hbar c}{\Delta E}\sim L_p^2\end{equation*}$

Remember the notion of Schwarzschild radius, related to the scape velocity of light in a Black Hole (BH):

(8) $\begin{equation*} \dfrac{1}{2}mv^2=\dfrac{G_NMm}{r}\end{equation*}$

so, when $v=c$ we get

$R_S=\dfrac{2G_NM}{c^2}$

Collider tests allow us to test quantum dimensions with spectrum

$\Delta E\sim \dfrac{\hbar c}{r}$

If $r<L_p$

then

$R_S(r)>L_p>r$

II. INTERLUDE: BACKGROUND IN QM AND CLASSICAL MECHANICS.

Firstly, we can compare Quantum Mechanics (QM) with Classical Mechanics (CM). Any physical theory answers some basic questions:

1st. What are the states or degrees of freedome?

2nd. What can we observe or measure?

3rd. How does the system evolve in time?

CM answers in the following way:

1) States are generalized coordinates $q$ and generalized momenta $p$ , plus some initial conditions or boundary conditions.

2) Observers “see” functions of generalized coordinates and momenta $f(q,p)$ , i.e., observers “see” the phase space!

3) Systems evolve with equations of motion (EOM) provided by some lagrangian or hamiltonian function ( $L(q,\dot{q},t), H(q,p,t)$ ). The EOM plus initial (boundary) conditions are a well posed problem in lagrangian or hamiltonian dynamics (excepting some “degenerated” systems that deserve special attention).

By the other hand, QM answers a bit differently as follows:

1) States are “rays” in a complex (projective) Hilbert space, $\vert \Psi\rangle\in\mathcal{C}$ .

2) Observables “are” (by definition in standard quantum mechanics) self-adjoint operators having real spectra and acting on some Hilbert space. Since $A=A^+$ , then we have

$\langle \Psi_2\vert A\Psi_1\rangle=\langle A^+\Psi_2\vert \Psi_1\rangle$

3) Evolution follows according certain unitary operator via the Schrödinger’s equation

$\vert \Psi(t)\rangle=U(t)\vert\Psi (0)\rangle$

This is the Schrödinger picture. OR we can leave fixed the state and then operators evolve with the EOM according to the so-called Heisenberg picture, the analogue of the classical mechanics approach with Poisson brackets extended to quantum commutators.

The canonical approach to quantization is review now:

A) Take a lagrangian with a harmonic oscillator term with unit mass (m=1):

$L=\dfrac{\dot{x}^2}{2}-\dfrac{\omega^2x^2}{2}$

The EOM reads $\ddot{x}+\omega^2 x=0$

The hamiltonian reads:

$H=p\dot{x}-L=\dfrac{p^2}{2}+\dfrac{1}{2}\omega^2x^2$

The canonical momentum is

$p=\dfrac{\partial L}{\partial \dot{x}}$

B) Canonical equations:

$\dot{x}=\partial_p H=p$

$\dot{p}=-\partial_x H=-\omega^2x$

$x(t)=x_0e^{i(\omega t+\phi_0)}$

C) Canonical commutators ( $\hbar=1$ ):

$\left[x,p\right]=i$

$p=-i\partial_x$

$H=-\dfrac{1}{2}\partial_x^2+\dfrac{1}{2}\omega^2x^2$

$H\Psi=i\partial_t\Psi$

$\Psi_n(x,t)=A_ne^{-\frac{1}{2}\omega x^2}H_n(\sqrt{\omega}x)e^{-iE_nt}$

and where $H_n(x)$ are Hermite polynomials, $E_n=\omega(n+\dfrac{1}{2})$ and $A_n$ is a normalization constant. Indeed, the general state for this system is a linear superposition

$\displaystyle{\Psi(x,t)=\sum_{n=0}^\infty c_n\Psi_n(x,t)}$

Remember that $n=0, 1, \ldots, \infty$ for the harmonic oscillator.

D) Creation/Annihilation representation for the standard harmonic oscillator (SHO):

$a=\dfrac{1}{\sqrt{2\omega}}\left(\omega x+ip\right)$

$a^+=\dfrac{1}{\sqrt{2\omega}}\left(\omega x-ip\right)$

In terms of these operators, we have

$x=\dfrac{1}{\sqrt{2\omega}}\left(a+a^+\right)$

$p=\dfrac{1}{\sqrt{2\omega}}\left(a-a^+\right)$

$\left[a,a^+\right]=1$

$H=\omega\left(a^+a+\dfrac{1}{2}\right)$

$N=a^+a$

$\left[H,a\right]=-\omega a$

$\left[H,a^+\right]=+\omega a^+$

$N\vert n\rangle=n\vert n\rangle$

$a\vert 0\rangle=0$ (vacuum definition)

$\vert 0\rangle=\dfrac{(a^+)^n}{\sqrt{n!}}\vert 0\rangle$

$a\vert n\rangle =\sqrt{n}\vert n\rangle$

$a^+\vert n\rangle=\sqrt{n+1}\vert n+1\rangle$

$\Psi (t)=\displaystyle{\sum_{n=0}^{\infty}}e^{-iE_n(t)}\vert n\rangle$

$\Psi (t)=U(t)\vert \Psi (0)\rangle$

$U(t)=\exp \left(-i\int H(t')dt'\right)$

$\langle \Psi_2(t)\vert A\vert \Psi_1(t)\rangle=\langle \Psi_2 (0)\vert U^+(t)AU(t)\vert\Psi_1(0)\rangle$

$\langle \Psi_2(t)\vert A\vert \Psi_1(t)\rangle=\langle \Psi_2 \vert A(t)\vert\Psi_1\rangle$

$A(t)=U^+(t)AU(t)$

$\dot{a}=-i\omega a$

$\dot{a}^+=i\omega a^+$

$a(t)=e^{-i\omega t}a(0)$

$a^+(t)=e^{i\omega t}a^+(0)$

$N(t)=a^+(t)a(t)=a^+(0)a(0)$

III. PARTICLE CREATION IN CURVED SPACETIME.

The general idea of QFT in curved spacetime and the phenomenon of pair creation is the following. Consider some function or observable F(t) during some time interval $t_1<t<t_2$ . Then, we have

$H=\dfrac{p^2}{2}+\dfrac{1}{2}\omega^2x+F(t)=\dfrac{p^2}{2}+\omega^2(t)x$

with

$\omega (t)=\omega+\dfrac{F(t)}{\omega}$

The consequences are the definition of two set of “states” (in, out) in the asymptotic limit:

$n(t<t_1)\vert n_{in}\rangle=n\vert n_{in}\rangle$

$n(t>t_2)\vert n_{out}\rangle=n\vert n_{out}\rangle$

$\vert n_{in}\rangle, \vert n_{out}\rangle$ are aymptotic states in the past and future where the notion of particle or more generally “quasiparticle” is well defined. In general, we also write

$\vert n_{out}\rangle=\displaystyle{\sum_m}\langle m_{in}\vert n_{out}\rangle \vert m_{in}\rangle$

We define the scattering matrix (popularly known as S-matrix) as

$S=\langle m_{in}\vert n_{out}\rangle$

It is generally a complex number (it has complex components!). Then, $F(t)$ represents the effect of particle physics interactions between asymptotic states. In curved spacetimes, the curvature acts as certain external field $F(t)$ .

See you in my next QFT post!

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The Spectrum Of Riemannium

Mobilis in mobili: Physmatics Chronicles!

LOG#141. Basic QFT in curved ST(I).

I. LIMITS FROM QUANTUM MECHANICS AND GENERAL RELATIVITY.

II. INTERLUDE: BACKGROUND IN QM AND CLASSICAL MECHANICS.

III. PARTICLE CREATION IN CURVED SPACETIME.

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I. LIMITS FROM QUANTUM MECHANICS AND GENERAL RELATIVITY.

II. INTERLUDE: BACKGROUND IN QM AND CLASSICAL MECHANICS.

III. PARTICLE CREATION IN CURVED SPACETIME.

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