In mathematics, the hafnian is a scalar function of a symmetric matrix that generalizes the permanent. The hafnian was named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhagen (Hafnia in Latin)."
Definition The hafnian of a 2 n × 2 n {\displaystyle 2n\times 2n} symmetric matrix A {\displaystyle A} is defined as
haf ( A ) = ∑ ρ ∈ P 2 n 2 ∏ { i , j } ∈ ρ A i , j , {\displaystyle \operatorname {haf} (A)=\sum _{\rho \in P_{2n}^{2}}\prod _{\{i,j\}\in \rho }A_{i,j},}
where P 2 n 2 {\displaystyle P_{2n}^{2}} is the set of all partitions of the set { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} into subsets of size 2 {\displaystyle 2} . This definition is similar to that of the Pfaffian, but differs in that the signatures of the permutations are not taken into account. Thus the relationship of the hafnian to the Pfaffian is the same as relationship of the permanent to the determinant.
Basic properties Besides its definition as a sum over perfect pairings, the hafnian of a 2 n × 2 n {\displaystyle 2n\times 2n} symmetric matrix A {\displaystyle A} can equivalently be written as
haf ( A ) = 1 n ! 2 n ∑ σ ∈ S 2 n ∏ i = 1 n A σ ( 2 i − 1 ) , σ ( 2 i ) , {\displaystyle \operatorname {haf} (A)={\frac {1}{n!2^{n}}}\sum _{\sigma \in S_{2n}}\prod _{i=1}^{n}A_{\sigma (2i-1),\sigma (2i)},}
where S 2 n {\displaystyle S_{2n}} is the symmetric group on { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} . An equivalent Levi-Civita representation holds for any even-dimensional symmetric matrix V = ( V i j ) i , j = 1 N {\displaystyle V=(V_{ij})_{i,j=1}^{N}} :
haf ( V ) = 1 2 N / 2 ( N / 2 ) ! ∑ i 1 , … , i N = 1 N | ϵ i 1 ⋯ i N | V i 1 i 2 ⋯ V i N − 1 i N , {\displaystyle \operatorname {haf} (V)={\frac {1}{2^{N/2}(N/2)!}}\sum _{i_{1},\dots ,i_{N}=1}^{N}|\epsilon ^{i_{1}\cdots i_{N}}|V_{i_{1}i_{2}}\cdots V_{i_{N-1}i_{N}},}
where N {\displaystyle N} is even. The hafnian also admits a fermionic (Berezin integral) representation. If V {\displaystyle V} is a symmetric 2 L × 2 L {\displaystyle 2L\times 2L} matrix and χ i , χ ¯ i {\displaystyle \chi _{i},{\bar {\chi }}_{i}} are Grassmann variables, then
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