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By José Ignacio Illana

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127) tiene la misma forma que el lagrangiano original. Adem´as el a´ lgebra de Clifford perma´ que se usa con frecuencia nece invariante, {γ µ γ ν + γ ν γ µ } = 2gµν . Una representacion es la representaci´on est´andar o representaci´on de Dirac, que se obtiene a partir de la quiral mediante 1 U= √ 2 1 1 . 129) 0 σi , −σi 0 γ5 = iγ0 γ1 γ2 γ3 = 0 1 . 130) ´ de Dirac resulta comoda ´ La representacion en el l´ımite no relativista, mientras que la quiral es m´as conveniente en el l´ımite ultrarrelativista.

La carga U(1) conservada es ˆ Q=i ↔ d3 x : φ† ∂0 φ : = i ˆ ˆ d3 x d3 p (2π )3 2E p ˆ d3 q (2π )3 2Eq a†p eipx + b p e−ipx ∂0 aq e−iqx + bq† eiqx − ∂0 a†p eipx + b p e−ipx ˆ ˆ ˆ d3 p d3 q 3 = d x (2π )3 2E p (2π )3 2Eq ×: a†p eipx + b p e−ipx Eq aq e−iqx − bq† eiqx + E p a†p eipx − b p e−ipx ˆ ˆ ˆ d3 p d3 q 3 = d x : a†p aq ei(q− p)x − b p bq† e−i(q− p)x : (2π )3 (2π )3 ×: aq e−iqx + bq† eiqx aq e−iqx + bq† eiqx : : Tema 3: Cuantizaci´on de campos libres 48 ˆ = d3 p † ( a a p − b†p b p ). 34) Por tanto, el estado a†p |0 tiene carga Q = +1 y b†p |0 tiene carga Q = −1.

19) vemos que δ3 ( p − q ) = δ3 ( p − q ) ∂E γ β +1 ∂pz = Eδ3 ( p − q) E = δ3 ( p − q ) γ( βpz + E) E ⇒ E p δ3 ( p − q ) = E p δ3 ( p − q ) . 21) y en el segundo, dE pz = , dpz E pues E = m2 + p2 . 22) Veamos ahora cu´al es la energ´ıa de los estados multipart´ıcula. Para ello expresaremos ´ y destruccion ´ (hacemos el primero el hamiltoniano en t´erminos de operadores creacion c´alculo en t = 0 por simplicidad, pues el hamiltoniano es una constante del movimiento): ˆ H= × ˆ 3 d x H( x ) = 1 1 d x Π2 + (∇φ)2 + m2 φ2 = 2 2 3 ˆ ˆ 3 d x d3 p (2π )3 2E p − E p Eq a p aq ei( p+q)·x + a†p a†q e−i( p+q)·x − a p a†q ei( p−q)·x − a†p aq e−i( p−q)·x − p · q a p aq ei( p+q)·x + a†p a†q e−i( p+q)·x − a p a†q ei( p−q)·x − a†p aq e−i( p−q)·x + m2 a p aq ei( p+q)·x + a†p a†q e−i( p+q)·x + a p a†q ei( p−q)·x + a†p aq e−i( p−q)·x ˆ d3 q (2π )3 2Eq Tema 3: Cuantizaci´on de campos libres 46 1 = 2 ˆ d3 p E p ( a†p a p + a p a†p ) = (2π )3 ˆ d3 p Ep (2π )3 1 a†p a p + V .

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