10.5 Time-Independent Solutions Defining Physical Sectors
63
and
δ
(λ) H = δλ
∂ H λ
∂λ
= −(δλ)λ
−2
[|∇ϕ|
2
+ 3λ
−2 ˜
U (ϕ)]d
s x.
Hence, the condition of stationarity and the positivity of ˜
U yield
|∇ϕ|
2 d
s x = 0, i.e. ∇ϕ = 0, i.e. ϕ = ϕ ∞ ,
and ˜
U (ϕ ∞ ) = 0.
63
and
δ
(λ) H = δλ
∂ H λ
∂λ
= −(δλ)λ
−2
[|∇ϕ|
2
+ 3λ
−2 ˜
U (ϕ)]d
s x.
Hence, the condition of stationarity and the positivity of ˜
U yield
|∇ϕ|
2 d
s x = 0, i.e. ∇ϕ = 0, i.e. ϕ = ϕ ∞ ,
and ˜
U (ϕ ∞ ) = 0.