In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, a Dirac membrane is a model of a charged
membrane
A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. ...
introduced by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
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
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
. 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
The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...
. 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 principle
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 motion
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 formulation
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.
Quantisation
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 also
*
Brane
In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime accordin ...
References
P. A. M. Dirac, An Extensible Model of the Electron, Proc. Roy. Soc. A268, (1962) 57–67.
Quantum models
Electron