12.5. Jacobi identity The aim of this exercise is to prove, by physical arguments, the Jacobi identity

∞



n=1

(1−qn)(1+qn−1/2w)(1+qn−1/2w−1) =

∞



n=−∞

qn2/2wn that holds for |q| < 1 and w = 0. Consider then the partition function of a free system of f fermions and ¯ f anti-fermions, with energy levels E = E0(n− 12

), n ∈ Z, and total fermion number N = Nf −N¯

f. Let q = e−E0/T and w = eμ/T.

a. Show that the grand canonical partition function is given by Z(w,q) =

f , ¯ f e−E/T+μN/T =

∞



N=−∞

wN ZN(q) (12.4.14)

=

∞



n=1

(1+qn−1/2w)(1+qn−1/2w−1)

where ZN(q) is the partition function at a given number N of the fermions.

b. Consider Z0. The lowest energy states that contribute to this quantity have all negative energy levels occupied (they form the Dirac sea, with a total energy normalized to the value E = 0) whereas the excited states are described by the integers k1 ≥ k2 ≥ k3 . . .kl > 0 with l

i ki =M. The energy of these states is E = ME0. Prove that Z0 is given by Z0 =

∞



M=0 P(M)qM =

∞



n=1 1

1−qn where P(M) is the combinatoric function that expresses in how many ways an integer M is expressed as a sum of numbers minor than it.

c. Consider now the sector with fermionic number N, where the first positive levels are occupied. Argue that this sector contributes with the factor q1/2 · · ·qN−3/2 qN−1/2 = q Nn

=1(j−1/2) = qN2/2 in their partition function, while the remaining excitation gives rise to the same partition function Z0, so that ZN = qN2/2 Z0.

d. Use now eqn. (12.4.14) to prove the Jacobi identity.