Block Spin Transformations of 2D Sigma Model,
Toward Solving a Millennium Problem
K.R.Ito
Abstract: We analyze 2D O(N) sigma model by block spin transformation:
[
]
∑
1
g
0
exp − ⟨ϕ, G−1
(ϕ2 (x) − N β)2
0 ϕ⟩ −
2
2N x
2
2
where ϕ(x) = (ϕ1 (x), · · · , ϕN (x)), G−1
0 = −∆ + m0 and −∆ is the lattice laplacian on Z .
Put ϕ0 = ϕ and recursively define block spins
ϕn+1 (x) =
1 ∑
ϕn (Lx + ζ)
L2
|ζ|<L/2
We represnt ϕn in terms of block-spin ϕn+1 of next order and fluctuations zn . zn is a
massive Gaussian but gets strong effects of back ground field ϕn+1 . We define a new
notion of domain walls in the sigma model which has O(N) symmetry. The domain wall
regions Dw have high energies, and on (Dw )c we can safely implement the block spin
transformations. Thus we obtain the renormalization group flow of the 2D sigma model,
which enables us to prove our long-standing claim.
1