Gravity in the 2+1 dimensions

In a previous post, we point out in the d+1 dimensions, the Coulomb's law reads, \[
\varphi(r) = \frac{\Gamma(\frac{d-2}{2})}{4\pi^{\frac{d}{2}}}\frac{1}{r^{d-2}}.
\] Apparently, in the 2+1 dimensions, this formula is divergent. To see this, first note that $x^\epsilon = 1 + \epsilon \log x$, and $\Gamma(\epsilon) = 1/\epsilon - \gamma$, the potential reads, \[
\varphi(r) \triangleq \lim_{\epsilon\to 0^+} \frac{\Gamma(\frac{\epsilon}{2})}{4\pi^{\frac{2+\epsilon}{2}}}\frac{1}{r^{\epsilon}} = \frac{1}{2\pi}\log\frac{1}{r} + \frac{1}{2\pi\epsilon} - \frac{1}{4\pi}\log e^\gamma\pi
\] We can regularize the potential by adding an infinitely large constant such that it obtains a finite value at a given distance $r_0$: \[
\varphi_r(r) = \frac{1}{2\pi}\log\frac{r_0}{r}.
\] But still, the potential divergences at infinity, which sort of violates the spirit of "locality".

The interpretation of this mathematical phenomenon is that in the 2+1 dimensions, long range correlation can be created with infinitely small energy. A consequence of this is the Coleman's theorem, stating that there is no spontaneous symmetry breaking of the continuous symmetry in the 2 dimensions.

In terms of the gravity, this infrared divergence is obviously non-physical. The potential becomes infinitely large near the source and far away from the source. A plausible interpretation is therefore $\varphi = \text{constant}$, at least for the type of gravity as we understand (We are not claiming the Poisson equation has no non-trivial solution in 2 dimensions.).

Of course, with a strong field strength, Newton's law should be replace by Einstein's equation: \[
R_{\mu\nu}  =  8\pi G \Big( T_{\mu\nu} - T g_{\mu\nu} \Big).
\] Here $T = g^{\mu\nu} T_{\mu\nu}$. In the vacuum, this equation reduces to $R_{\mu\nu} = 0$.  But this does not necessarily mean the space-time curvature is zero. The space time curvature is ultimately determined by the Riemann tensor $R^\mu_{\nu\sigma\rho}$. Non-vanishing curvature represents the propagation of the gravity from the source inside the vacuum.

Riemann tensor has $D^2(D^2-1)/12$ independent components, where $D=d+1$ is the space-time dimensionality. In the 2+1 case, it has 6 independent components, whereas the Ricci tensor $R_{\mu\nu}$ has $(3 \times 3 +3)/2 = 6$ independent components as well. Thus, Riemann tensor is expressible through the Ricci tensor: \[
R_{\mu\nu\rho\sigma} = \big( g_{\mu\rho}R_{\nu\sigma} - (\rho \leftrightarrow \sigma) \big) - (\nu \leftrightarrow \rho) - \frac{1}{2} (g_{\mu\rho}g_{\nu\sigma} - g_{\mu\sigma}g_{\nu\rho}) R
\] Here $R = g^{\mu\nu}R_{\mu\nu}$. As a result, Riemann tensor, hence the space-time curvature, is zero in the vacuum, in the 2+1 dimensions. This means gravity cannot propagator through the free space in the 2+1 dimensions! Einstein's equation is a full pledged dynamical theory of gravity with Newton's gravitational theory as its weak field approximation. Now, Einstein's theory actually agrees with Newton's theory that in the 2+1 dimensions there is no gravity!

Applying the gravitational theory in the 3+1 dimensions into the 2+1 dimension is somewhat unjustified. After all, non-body knows wether these are the correct modeling. However Newton's law is not an ad loc construction. It follows, as the Coulomb's law, the Poisson equation $\nabla^2 \varphi(x) = 0$. Einstein's equation has a stronger built-in beauty in it: it acquires a geometric interpretation. So these interesting observation is not a major consequence of our gravitational theory per se. It is more closely related to the special topology of the low dimensions, which is also well known. In fact, in the 2 dimensions, topology is more central than the local operators we cherish much in the 3 dimensions.

Everything Seems Linear in the Log Scale

Figure: the Standard Model fermion mass hierarchy shown in the log scale.

The Super Critical Charge

Schwinger Effect

According to quantum field theory, vacuum, the state of no nothingness, is actually filled with virtual particles. In particular, virtual electron and positron pairs are created and annihilated in short time and distance: \[
\Delta t \sim \frac{h}{2m_ec^2} \simeq 10^{-21} \,\mathrm{s}, \quad \Delta x \sim \frac{h}{2m_ec} \simeq 10^{-12} \,\mathrm{m}. \]

Fig. 1: Vacuum fluctuation into electron-positron pair.

Then, it is no surprise that if we exert a strong electric field $\vec E$ on the vacuum, the virtual electron and positron would be pulled in the opposite direction. As a net result, real electron and positron pairs can be created out of vacuum. Of course, the electric field has to be strong, to provide enough energy ($\gtrsim 2m_e c^2$) for the pair production. We can do an estimation as following. In order to create real particles, the electron and positron have to be pulled apart by at least one Compton length $\lambda_e = \frac{\hbar}{m_e c}$ (so that their wave packets do not overlap much). Over such a distance, the energy provided by the electric field is, $e E \lambda_e \ge 2 m_e c^2$. Therefore the threshold electric field is \[
E_\mathrm{th} \triangleq \frac{2 m_e c^3}{e\hbar} \simeq 10^{18} \, \mathrm{V/m}.
\]

Fig. 2: The Schwinger effect.

We can also look at the problem in another way (Fig. 2). In the Dirac theory, the vacuum is filled with negative energy electrons, known as the Dirac sea. In order to become physical, the electron has to get to the positive mass shell $E = \sqrt{\vec p^2 c^2 + m^2c^4}$. There is a $2 m_e c^2$ mass gap between them. (The Coulomb potential is negligible comparing to the mass gap.)

Fig. 3: The mass gap. The $x$-axis is the separation between the electron and its hole. Due to zitterbewegung, this distance can not be smaller than the Compton length.

Therefore, the creation of an electron (and the positron - the hole in the Dirac sea) can be viewed as a quantum tunneling process (Fig. 3). Then the external electric field turns the well to a barrier. The width of the barrier is proportional to $w \sim 2 m_e c^2/E$. According to WKB, the tunneling probability is \[
P \sim \exp\Big( - w \sqrt{2 m_e \times e (2m_e c^2) } \Big) \simeq \exp\Big( - E_\mathrm{th}/E \Big)
\]

Fig. 4: The electric field term the mass well to a barrier.

Super Critical Charge

We have seen in the last section that strong electric field can create electron-positron pairs out of vacuum. The Coulomb potential have a field strength proportional to $1/r^2$: $eE(r) = \frac{Z\alpha}{\hbar c r^2}$. Very close to the charge, the electric field would be extremely intense. So, it is natural to ask, if/why not such a phenomenon can be seen around a point charge. To achieve the threshold strength, the distance $r_\mathrm{th} = \sqrt{\frac{1}{2}Z\alpha} \lambda_e \simeq \sqrt{Z/2} \times 10^{-13}\,\mathrm{m}$ has to be large than the smallest distance between two particles.

For poin-like charged particles, such as electron, muons, the smallest distance is determined by 1). their Compton wavelength (zitterbewegung) $\lambda_\ell = \frac{m_e}{m_\ell}\lambda_e$; 2). the size of the bound state $\frac{m_\ell}{m\ell+m_e}\lambda_e/(Z\alpha)$. For electron/positron, this distance $r_\mathrm{th} = \sqrt{\alpha/2} \lambda_e$ is much smaller than their Compton wavelength. Therefore, even if such a phenomenon exists, it is included in the quantum fluctuation of the particle itself. Muon is about 207 times heavier than the electron. So its Compton wavelength is 207 times smaller than electron's. Then it is possible to produce a pair of real electron-positron. But as the pair were produced, the positron would be bound with the muon, forming a $\mu^-e^+$ atom.  However, the ground state radius of this atom $r \simeq a_0 = \lambda_e / \alpha \gg r_\mathrm{th}$. In other words, there is no enough energy to produce the electron-positron pair.  Similar cases happen for other leptons.

Fig. 5: The electron-positron pair production by a muon. This process is kinematically forbidden.

For composite systems, the charge factor $Z$ can be rather large. As a result, the possible bound state radius would decrease and the threshold radius increase (while the Compton wavelength remains the same). At some critical charge factor $Z_c \simeq 2/\alpha = 274$, the system would have enough energy to produce the electron-positron pair.

Again, Dirac Sea

To give a little bit quantitative touch, let's turn to the Dirac theory again (We don't turn to Schroedinger's theory because at the super critical charge, the system is relativistic.). The Dirac equation with a Coulomb potential produces the energy spectrum (Sommerfeld fine-structure formula): \[
E_{n,j} = mc^2 \left[ 1 + \frac{Z^2\alpha^2}{\left[n-j-\frac{1}{2} + \sqrt{(j+\frac{1}{2})^2-Z^2\alpha^2}\right]^2} \right]^{-1/2}
\] where $n = 1, 2, 3, \cdots$ is called the principal quantum number and $j=1/2, 3/2, 5/2, \cdots n-1/2$ is the total angular momentum. As we can see, for the S-state ($j=1/2$), the energy becomes imaginary if $Z > 1/\alpha \simeq 137$. This is half of the critical charge we found above using a crude estimate.

The existence of imaginary energy eigenvalues implies the Dirac Hamiltonian is no longer Hermitian as Dirac promised. Something is wrong. The solution of the Dirac equation does not represent the motion of an electron. Instead, given the time-dependent part $\exp\big(- |E|t \big)$, it represents the amplitude of some tunneling process.

Even though Dirac's theory is broken at $Z>137$, we can still mill some plausible physics out of it, invoking the Dirac sea -- as one may have noted, the creation of particles in the Dirac theory always involves the Dirac sea. It turns out, at the presence of the super critical charge, the Coulomb level dives into the Dirac sea. Therefore, the electrons at the Dirac sea jump down to Coulomb levels below and leave holes in the Dirac sea, manifested as the positron (see Fig. 6). This interpretation can be elaborated (including using other methods) to give the semi-quantitative dynamics of the process.

Fig. 6: The Dirac interpretation of the electron-positron pair production by a super critical charge.

QED with Strong External Fields

This problem can also be described by QED. Unlike the Dirac theory, the QED Hamiltonian is always Hermitian. Hence a self-consistent quantitative description of the problem has come from QED.

In QED, the vacuum is the ground state of the charge-zero sector:\[
H_\mathrm{QED} |\Omega\rangle = E_\Omega |\Omega\rangle
\] Here $H_\mathrm{QED}$ is the QED Hamiltonian. $E_\Omega$ is the vacuum energy. It can be renormalized to 0.

In our problem, there exists a charged heavy ion $Z^+$, which generates an strong electromagnetic field $\mathcal A$. With the presence of this field, the ground state may be different: \[
\big( H_\mathrm{QED} + J_\mu \mathcal A^\mu \big) |\Omega_{\mathcal A}\rangle = E_\mathcal{A} |\Omega_{\mathcal A}\rangle
\] Where $|\Omega_{\mathcal A}\rangle$ represents the polarized QED vacuum within field $\mathcal A^\mu$, $J^\mu = e\bar\psi \gamma^\mu \psi$ is the fermion charge current.

If the charge of the heavy ion is subcritical, the polarized vacuum only involves the photon loop-correction and electron loop-corrections (see Fig. 7), which can be taken into account by renormalization (UV renormalization and the bremsstrahlung) of the heavy ion. When the mass of the heavy ion is sufficiently large, the renormalization effects can be neglected. When the charge of the heavy ion $Z^+$ exceeds some critical value $Z_c$, a new vacuum state emerges. It consists a free positron and a $Z^+e^-$ bound-state. Such a vacuum state is called a charged vacuum. Dynamically, we see the spontaneous production of an electron-positron pair.

Fig. 7: The QED vacuum within the subcritical (top) and supercritical (bottom) external field. The double line represent the heavy ion with a positive subcritical charge. The single line loop represents the electron/positron loop correction. The wavy lines represents the photons. The dot represents the coupling between the photon and the super critical charge. The blue single line represents the emitted positron. The red single line represents the bound electron.

In practice, the matrix can be truncated to the first few sector to retain the minimal physical of the charged vacuum (See Fig. 9). The minimal sector should include at least one photon and one pair of electron and positron.

Fig. 9: The QED matrix within the first few Fock sectors. The vertex with a dot represents the super critical charge coupling $Ze$.

We can further suppress the heavy ion sector and treat the field it generates simply as a classical external field. In this way the charged vacuum is more qualified for its name. (Otherwise it is just the ground state of the supercritical charge sector.)

Fig. 10: (left panel) the vacuum polarization within a subcritical potential; (right panel) the pair production (the charged vacuum) within a supercritical potential.

The charged vacuum can also be studied within the path integral formalism. According to this formalism, the vacuum expectation value of an operator within the external field is, simply, \[
\langle \mathrm T\mathcal O_\psi \rangle_{\mathcal A} = \int \mathcal D_{\psi,\bar\psi, \mathcal A} \, \mathcal O_\psi \, \exp\Big[ i\int d^4x \, \bar \psi(x) \big( i\gamma^\mu D_\mu - m\big) \psi(x)\Big]
\] where $D_\mu = \partial_\mu - ie\mathcal A_\mu$.

Both the matrix diagonalization and the path integral methods are non-perturbative. But different approximation may arise for each formalism. We shall not delve into those lengthy technologies but only mention one famous result due to Schwinger. Let's consider the production rate, which is related to the vacuum decay probability: $P = 1 - |\langle\Omega_{\mathcal A}|\Omega_{\mathcal A}\rangle|^2 \approx 1 - \exp( -2 \mathrm{Im} S_\mathrm{eff})$. Here Schwinger applied an approximation: $\langle\Omega_{\mathcal A}|\Omega_{\mathcal A}\rangle = \det^{-1} \bar\psi \big( i\gamma^\mu D_\mu - m\big) \psi \approx \exp(i S_\mathrm{eff})$. $S_\mathrm{eff}=\int d^4x\,\mathcal L_\mathrm{eff}$ is called the effective action. The one-loop effective action was calculated obtained Heisenberg and Bonn. Plug into this effective action, and assuming a uniform electric field $E$, Schwinger's conclusion for the vacuum pair production rate is, \[
R \triangleq \frac{dN}{dVdt} = \frac{(eE)^2}{4\pi^3}\sum_{n=1}^\infty \frac{1}{n^2} \exp\big( -n\pi E_c/E\big)
\] where $E_c \triangleq m_e^2 c^3/(e\hbar) \sim 10^{18} \,\mathrm{V/m}$ is the critical electric field. This formulation is the famous Schwinger mechanism of vacuum pair production.


One can also solve the Schroedinger's equation directly, starting for example, from the normal QED vacuum state: \[
 i \frac{\partial}{\partial t} |\psi(t)\rangle = \big( H_\mathrm{QED} + J^\mu \mathcal A_\mu\big) | \psi(t)\rangle, \quad
 |\psi(0)\rangle = |\Omega\rangle.
\] In this way, the time evolution of the QED vacuum can be studied. This approach is also non-perturbative.

In the above description, we deliberately avoid the photon mediated electron-positron interaction. This can be included by the quantized electromagnetic field $A^\mu$. So the full covariant derivative can be written as $D_\mu = \partial_\mu - i e\mathcal A_\mu - ie A_\mu$. Note that this field is small comparing with the external field $\mathcal A$ generated by the supercritical heavy ion, $|\mathcal A| \sim Z |A|$. In practice, the correction due to quantized photon can be included through the usual perturbation theory.

Fig. 11: the same diagrams as Fig. 10, but taking into account the quantized photons. The external fields are represented as double wavy lines. The real photon lines are the single wavy lines.

Pair Production in the Relativistic Heavy Ion Collisions

Experiments

[ To Be Continued ... ]

---

References:

[1]: W. Greiner, B. Muller and J. Rafelski, Quantum electrodynamics of strong fields, (1985) Springer-Verlag, Berlin Heidelberg.

The Year of Sheep, or Goat?

In Feb. 18, 2015, East Asians celebrated the Lunar New Year, the year of yang (羊)。 Some English media suddenly found it hard to convey the message to the English audience, as the word yang (羊) in Chinese may refer to several types of animals in English, most commonly, sheep (called mian-yang, 绵羊, in Chinese) and goat (called shan-yang, 山羊 in Chinese). So what year is it, sheep or goat?

So problem is that English does not have a collective word for sheep and goat. Scientifically speaking, the name sheep or goat itself is also a notation that refers to several similar animals. This is, of course,  very common in linguistics and may happen between any two languages. But it is not exactly true. Because there actually exists a word for animals similar to sheeps and goats: Caprinae. This is a subfamily name, which includes both sheep or ovis and goat capra. So, next time, when asked, you can tell people, IT IS THE YEAR OF CAPRINAE, if you want to be scientifically accurate.