ly let us study in?nitesimal transformations. The Lagrangian transforms to linear order in?(where λ= 1 +?) as L(x)→(1 +?) 4L ((1 +?)x ) (3) =L(x ) +?x μ?μL +4?L (4) =L(x ) +??μ(x μL ) (5) =L(x ) +??μJ μ, (6) where we de?neJ μ=x μL. The ?eldφ transforms as φ(x)→(1 +?)φ((1 +?)x ) (7) =φ(x ) +?φ+?x μ?μφ(8) =φ(x ) +??φ, (9) where we de?ne ?φ=φ+x μ?μφ. Noether’s theorem gives a conserved current as J μD= ?L ?(?μφ) ?φ?J μ. (10) In particular, for our case it is J μD = (?μφ) [φ+x ν?νφ]?x μ ?12 (?αφ) 2?ρφ 4 ?. (11) This leads to a conserved charge,D , by integratingJ 0Dover an equal time surface. Since J 0D= (? 0φ) hφ+t? 0φ+x· ~?φ i? 12 t ?(? 0φ) 2??~?φ? 2 ?+tρφ 4, (12) 1
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