Supermembranes are hypothesized objects that live in the 11-dimensional theory called
M-Theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's ...
and should also exist in 11-dimensional
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
. Supermembranes are a generalisation of
superstrings
Superstring theory is an theory of everything, attempt to explain all of the Elementary particle, particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetry, supersymmetric String (physics), st ...
to another dimension. Supermembranes are 2-dimensional surfaces. For example, they can be spherical or shaped like a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
. As in
superstring
Superstring theory is an theory of everything, attempt to explain all of the Elementary particle, particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetry, supersymmetric String (physics), st ...
theory the vibrations of the supermembranes correspond to different particles. Supermembranes also exhibit a symmetry called
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
without which the vibrations would only correspond to
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
and not
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
.
Energy
The energy of a classical supermembrane is given by its surface area. One consequence of this is that there is no difference between one or two membranes since two membranes can be connected by a long 1 dimensional string of zero area. Hence, the idea of 'membrane-number' has no meaning. A second consequence is that unlike strings a supermembrane's vibrations can represent several particles at once. In technical terms this means it is already 'second-quantized'. All the particles in the Universe can be thought to arise as vibrations of a single membrane.
Spectrum
When going from the classical theory to the quantum theory of supermembranes it is found that they can only exist in 11 dimensions, just as superstrings can only exist in 10 dimensions. When examining the energy spectrum (the allowed frequencies that a string can vibrate in) it was found that they can only be in discrete values corresponding to the masses of different particles.
It has been shown:
* The energy spectrum for the classical bosonic membrane is continuous.
* The energy spectrum for the quantum bosonic membrane is discrete.
* The energy spectrum for the quantum supermembrane is continuous.
At first the discovery that the spectrum was continuous was thought to mean the theory didn't make sense. But it was realised that it meant that supermembranes actually correspond to multiple particles. (The continuous degrees of freedom corresponding to the coordinates/momenta of the additional particles).
Action
The action for a classical membrane is simply the surface area of the world sheet. The quantum version is harder to write down, is non-linear and very difficult to solve. Unlike the superstring action which is quadratic, the supermembrane action is quartic which makes it exponentially harder. Adding to this the fact that a membrane can represent many particles at once not much progress has been made on supermembranes.
Low energy sector
It has been proven that the low energy vibrations of the supermembrane correspond to the particles in 11 dimensional supergravity.
Topology
A supermembrane can have multiple thing tubes or strings coming out of it with little or no extra energy cost since strings, for example, have no area. This means that all orientable topologies of membranes are physically the same. Also, joined and disjointed supermembranes are physically the same. Thus the topology of a supermembrane has no physical meaning.
Mathematics
The infinite supermembrane can be described in terms of an infinite number of patches. The coordinates of (each patch of) a supermembrane at any casual slice of time are 11 dimensional and depend on two continuous parameters
and a third integer parameter (k) denoting the patch number:
:
Therefore the super membrane can describe an infinite number of particles if we associate somehow the coordinate of each particle with some topological property of the patches - perhaps holes in the membrane or closed loops.
Supermembrane Field Theory
Since supermembranes correspond to multiple particles the field theory of membranes correspond to a
Fock space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
. Informally, let a(x) denote the continuous degrees of freedom in the energy spectrum:
:
The action can be written as
: