21.2. Bogoliubov transformation and boundary state Consider a free massive scalar field (x, t) in (1+1) dimension

21.2. Bogoliubov transformation and boundary state Consider a free massive scalar field ϕ(x, t) in (1+1) dimension with mass m = m− t < 0 m+ t ≥ 0.

Therefore it admits two different mode expansions, with two sets of annihilation and creation operators (A−,A†

−) and (A+,A†

+), which refer to t < 0 and t ≥ 0 cases respectively

ϕ(x, t) =  dk 2π

√

2Ea

Aa(k) ei(kx−Eat) +A†a (k) e−i(kx−Eat), a=±

where Ea =

√

m2 +k2.

a. Use the continuity of the field ϕ(x, t) at t = 0 to show that the two set of modes are related by a Bogoliubov transformation A+(k) = c(k)A−(k)+d(k)A†
−(−k), A†
+(k) = c(k)A†
−(k)+d(k)A−(−k).
and compute explicitly the coefficients c(k) and d(k).

b. In terms of themodes for t < 0, the vacuum of the theory is identified by A−(k)|0.
Use the Bogoliubov transformation to show that the boundary state |B which lives at t = 0 is expressed in terms of the modes (A+,A†
+) as |B = exp
∞
−∞
dkK(k)A†
+(k)A†
+(−k)|0
where K(k) = d(k)/c(k).