16.3. Lax pair Consider the pair of first-order differential operators (called Lax pair):

L(x, t | θ) = d dx

+iβ

4

∂tφσ3 +msinhθ,cos

βφ

2

σ1 +mcoshθ sin

βφ

2

σ2 M(x, t | θ) = d dt

+iβ

4

∂xφσ3 +mcoshθ cos

βφ

2

σ1 +msinhθ sin

βφ

2

σ2 where σi are the Pauli matrices and θ the rapidity variable. If [L,M] = 0

a. show that the field φ satisfies the equation of motion of the Sine–Gordon model

φ + m2

β

sinβφ = 0.

b. Take a rectangle domain −L ≤ x ≤ L;0 ≤ t ≤ T and assume periodic boundary condition φ(−L) = φ(L).With the notation UL(θ, t) =

→

exp  L

−L U(x, t | θ)dx , VL(θ) =

→

exp  T 0

V(x, t | θ)dt for the ordered integrals, show that UL(θ,T) = V−1 L (θ)UL(θ,0)VL(θ)

so that TrUL(θ, t) is independent of t; conclude that, θ being arbitrary, there are an infinite number of conserved quantities.