The Pfaffian of a skew-symmetric matrix is a square root of its determinant.
Let be a skew-symmetric -matrix with entries in some field (or ring) .
The Pfaffian is the element
In terms of Berezinian integrals
Let be the Grassmann algebra on generators , which we think of as a vector
Then the Pfaffian is the Berezinian integral
Pfaffians appear in the expression of certain multiparticle wave functions. Most notable is the pfaffian state of spinless electrons
where denotes the Pfaffian of the matrix whose labels are and is the filling fraction, which is an even integer. For Pfaffian state see
- Gregory Moore, N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. 360B(1991)362 pdf
There is also a deformed noncommutative version of Pfaffian related to quantum linear group?s:
- Naihuan Jing, Jian Zhang, Quantum Pfaffians and hyper-Pfaffians, arxiv/1309.5530
Pfaffian variety is subject of 4.4 in
- Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, arxiv/1404.3143
Revised on April 14, 2014 00:59:59
by Zoran Škoda