Dirac membrane

Dirac chose a non-standard way to formulate the action principle for the membrane. Because closed membranes in ${\displaystyle \mathbb {R} ^{3}}$ provide a natural split of space into the interior and the exterior there exists a special curvilinear system of coordinates ${\displaystyle x^{\mu }}$ in spacetime and a function ${\displaystyle f(x)}$ such that

- ${\displaystyle f(x)=0}$ defines a membrane

- ${\displaystyle f(x)>0}$, ${\displaystyle f(x)<0}$ describe a region outside or inside the membrane

Choosing ${\displaystyle x^{1}=f(x)}$ and the following gauge ${\displaystyle \sigma ^{0}=x^{0}=:\tau }$, ${\displaystyle \sigma ^{1}=x^{2}}$, ${\displaystyle \sigma ^{2}=x^{3}}$ where ${\displaystyle \sigma ^{\alpha }}$, ( ${\displaystyle \alpha =0,1,2}$) is the internal parametrization of the membrane world-volume, the membrane action proposed by Dirac is

${\displaystyle S=S_{EM}+S_{membrane}}$

${\displaystyle S_{EM}=-{\frac {1}{16\pi }}\int _{x^{1}>0}Jg^{\mu \rho }g^{\nu \sigma }F_{\mu \nu }F_{\rho \sigma }d^{4}x,\ \ \ \ S_{mem.}=-{\frac {\omega }{4\pi }}\int _{x^{1}=0}Mdx^{0}dx^{2}dx^{3}}$

where the induced metric and the factors J and M are given by

${\displaystyle g_{\mu \nu }=\partial _{\mu }y^{\Lambda }\partial _{\nu }y_{\Lambda },\ \ \ \Lambda =0,1,2,3}$

${\displaystyle J={\sqrt {-\det g_{\mu \nu }}}.\ \ \ M=J{\sqrt {-g^{11}}}}$

In the above ${\displaystyle y^{\Lambda }}$ are rectilinear and orthogonal. The space-time signature used is (+,-,-,-). Note that ${\displaystyle S_{EM}}$ is just a usual action for the electromagnetic field in a curvilinear system while ${\displaystyle S_{membrane}}$is the integral over the membrane world-volume i.e. precisely the type of the action used later in string theory.

There are 3 equations of motion following from the variation with respect to ${\displaystyle A_{\mu }}$ and ${\displaystyle y^{\Lambda }}$. They are: - variation w.r.t. ${\displaystyle A_{\mu }}$ for ${\displaystyle x_{1}>0}$ - this results in sourceless Maxwell equations - variation w.r.t. ${\displaystyle y^{\Lambda }}$ for ${\displaystyle x_{1}>0}$ - this gives a consequence of Maxwell equations - variation w.r.t. ${\displaystyle y^{\Lambda }}$ for ${\displaystyle x_{1}=0}$

${\displaystyle {\frac {1}{2}}F_{\alpha 1}F^{\alpha 1}=\omega J^{-1}(Mg^{1\mu }/g^{11})_{,\mu }}$

The last equation has a geometric interpretation: the r.h.s. is proportional to the curvature of the membrane. For the spherically symmetric case we get

${\displaystyle {\frac {e^{2}}{2\rho ^{4}}}=\omega {\frac {d}{dt}}{\frac {\dot {\rho }}{\sqrt {1-{\dot {\rho }}^{2}}}}+{\frac {2\omega }{\rho {\sqrt {1-{\dot {\rho }}^{2}}}}}}$

Therefore, the balance condition ${\displaystyle {\dot {\rho }}=0}$ implies ${\displaystyle a^{3}=e^{2}/4\omega }$ where ${\displaystyle a}$ is the radius of the balanced membrane. The total energy for the spherical membrane with radius ${\displaystyle \rho }$ is

${\displaystyle E(\rho )=e^{2}/2\rho +\beta \rho ^{2}}$

and it is minimal in the equilibrium for ${\displaystyle \beta =\omega }$, hence ${\displaystyle E(a)=3e^{2}/4a}$. On the other hand, the total energy in the equilibrium should be ${\displaystyle m_{e}}$ (in ${\displaystyle c=1}$ units) and so we obtain ${\displaystyle a=0.75r_{e}}$.

Small oscillations about the equilibrium in the spherically symmetric case imply frequencies - ${\displaystyle {\sqrt {6}}/a}$. Therefore, going to quantum theory, the energy of one quantum would be ${\displaystyle h\nu ={\sqrt {6}}\hbar /a=448m_{e}}$. This is much more than the muon mass but the frequencies are by no means small so this approximation may not work properly. To get a better quantum theory one needs to work out the Hamiltonian of the system and solve the corresponding Schroedinger equation.

For the Hamiltonian formulation Dirac introduces generalised momenta

- for ${\displaystyle x^{1}>0}$: ${\displaystyle B^{\mu }}$ and ${\displaystyle w_{R}}$ - momenta conjugate to ${\displaystyle A_{\mu }}$ and ${\displaystyle y^{R}}$ respectively (${\displaystyle R=1,2,3}$, coordinate choice ${\displaystyle x^{0}=y^{0}}$)

- for ${\displaystyle x^{1}=0}$: ${\displaystyle W_{R}}$ - momenta conjugate to ${\displaystyle y^{R}}$

Then one notices the following constraints

- for the Maxwell field

${\displaystyle B^{0}=0,\ \ \ {B^{r}}_{,r}=0,\ \ \ w_{R}{y^{R}}_{,s}-B^{r}F_{rs}=0}$

- for membrane momenta

${\displaystyle W_{R}{y^{R}}_{,2}=W_{R}{y^{R}}_{,3}=0,\ \ \ 16\pi ^{2}W_{R}W_{R}=\omega ^{2}M^{2}c^{00}(c^{00}-1)}$

where ${\displaystyle c^{ab}}$ - reciprocal of ${\displaystyle g_{ab}}$, ${\displaystyle a,b=0,2,3}$.

These constraints need to be included when calculating the Hamiltonian, using the Dirac bracket method. The result of this calculation is the Hamiltonian of the form

${\displaystyle H=H_{EM}+H_{s}}$

${\displaystyle H_{s}={\frac {1}{4\pi }}\int {\sqrt {16\pi ^{2}W_{R}W_{R}+\omega ^{2}(g_{22}g_{33}-g_{23}^{2})}}dx^{2}dx^{3}}$

where ${\displaystyle H_{EM}}$ is the Hamiltonian for the electromagnetic field written in the curvilinear system.

For spherically symmetric motion the Hamiltonian is

${\displaystyle H={\sqrt {\eta ^{2}+\omega ^{2}\rho ^{4}}}+e^{2}/2\rho ,\ \ \ \{\rho ,\eta \}=1}$

however the direct quantisation is not clear due to the square-root of the differential operator. To get any further Dirac considers the Bohr - Sommerfeld method:

${\displaystyle 2\pi \hbar n=2\int _{\rho _{min}}^{\rho _{max}}\eta d\rho }$

and finds ${\displaystyle E_{1}\approx 53m_{e}}$ for ${\displaystyle n=1}$.

• Brane

P. A. M. Dirac, An Extensible Model of the Electron, Proc. Roy. Soc. A268, (1962) 57–67.