Dirac membrane

A model of a charged membrane introduced by Paul Dirac in 1962. Dirac's original motivation was to explain the mass of the muon as an excitation of the ground state corresponding to an electron. Anticipating the birth of string theory by almost a decade, he was the first to introduce what is now called a type of Nambu–Goto action for membranes.

In the Dirac membrane model the repulsive electromagnetic forces on the membrane are balanced by the contracting ones coming from the positive tension. In the case of the spherical membrane, classical equations of motion imply that the balance is met for the radius , where is the classical electron radius. Using Bohr–Sommerfeld quantisation condition for the Hamiltonian of the spherically symmetric membrane, Dirac finds the approximation of the mass corresponding to the first excitation as , where is the mass of the electron, which is about a quarter of the observed muon mass.

Action principleEdit

Dirac chose a non-standard way to formulate the action principle for the membrane. Because closed membranes in   provide a natural split of space into the interior and the exterior there exists a special curvilinear system of coordinates   in spacetime and a function   such that

-   defines a membrane

-  ,   describe a region outside or inside the membrane

Choosing   and the following gauge  ,  ,   where  , (  ) is the internal parametrization of the membrane world-volume, the membrane action proposed by Dirac is

 
 

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

 
 

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

Equations of motionEdit

There are 3 equations of motion following from the variation with respect to   and  . They are: - variation w.r.t.   for   - this results in sourceless Maxwell equations - variation w.r.t.   for   - this gives a consequence of Maxwell equations - variation w.r.t.   for  

 

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

 

Therefore, the balance condition   implies   where   is the radius of the balanced membrane. The total energy for the spherical membrane with radius   is

 

and it is minimal in the equilibrium for  , hence  . On the other hand, the total energy in the equilibrium should be   (in   units) and so we obtain  .

Hamiltonian formulationEdit

Small oscillations about the equilibrium in the spherically symmetric case imply frequencies -  . Therefore, going to quantum theory, the energy of one quantum would be  . 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  :   and   - momenta conjugate to   and   respectively ( , coordinate choice  )

- for  :   - momenta conjugate to  

Then one notices the following constraints

- for the Maxwell field

 

- for membrane momenta

 

where   - reciprocal of  ,  .

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

 
 

where   is the Hamiltonian for the electromagnetic field written in the curvilinear system.

QuantisationEdit

For spherically symmetric motion the Hamiltonian is

 

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:

 

and finds   for  .

See alsoEdit

ReferencesEdit

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