Question

The classical Lagrangian density for the free complex scalar field is given by: $mathcal{L} = (partial_{mu}phi^)(partial^{mu}phi) - m^2phi^phi$ 1. Show that: a) it is invariant under the discrete transformation: $phi(x) o phi_c(x) = eta phi^dagger(x)$, $eta = pm 1$ $phi_c$ is the charge conjugated field. b) $(phi_c)_c = phi$. c) $hat{a}_c(vec{p}) = eta hat{b}(vec{p})$, $hat{b}_c(vec{p}) = eta hat{a}(vec{p})$. Where $phi_c(x) = int frac{d^3p}{(2pi)^3 2E_p}left(e^{-ipcdot x}hat{a}_c(vec{p}) + e^{ipcdot x}hat{b}^dagger(vec{p}) ight)$. d) $U_c = expleft(ipi int frac{d^3p}{2E_p}left[hat{a}^dagger(vec{p})hat{b}(vec{p}) + hat{b}^dagger(vec{p})hat{a}(vec{p}) - etaig(hat{a}^dagger(vec{p})hat{a}(vec{p}) + hat{b}^dagger(vec{p})hat{b}(vec{p})ig) ight] ight)$. Where $U_c$ is a unitary operator such that: $hat{a}_c(vec{p}) = U_c,hat{a}(vec{p}),U_c^dagger$, $hat{b}_c(vec{p}) = U_c,hat{b}(vec{p}),U_c^dagger$ and therefore $phi_c(x) = U_c,phi(x),U_c^dagger$. Hint: use the Baker–Campbell–Hausdorff formula: $e^{-hat{A}},hat{B},e^{hat{A}} = hat{B} + [hat{B},hat{A}] + frac{1}{2!}[[hat{B},hat{A}],hat{A}] + frac{1}{3!}[[[hat{B},hat{A}],hat{A}],hat{A}] + dots$ e) $U_c,hat{Q},U_c^dagger = -hat{Q}$

          The classical Lagrangian density for the free complex scalar field is given by:
$mathcal{L} = (partial_{mu}phi^*)(partial^{mu}phi) - m^2phi^*phi$
1. Show that:
a) it is invariant under the discrete transformation:
$phi(x) 	o phi_c(x) = eta phi^dagger(x)$,
$eta = pm 1$
$phi_c$ is the charge conjugated field.
b) $(phi_c)_c = phi$.
c) $hat{a}_c(vec{p}) = eta hat{b}(vec{p})$,
$hat{b}_c(vec{p}) = eta hat{a}(vec{p})$.
Where
$phi_c(x) = int frac{d^3p}{(2pi)^3 2E_p}left(e^{-ipcdot x}hat{a}_c(vec{p}) + e^{ipcdot x}hat{b}^dagger(vec{p})
ight)$.
d) $U_c = expleft(ipi int frac{d^3p}{2E_p}left[hat{a}^dagger(vec{p})hat{b}(vec{p}) + hat{b}^dagger(vec{p})hat{a}(vec{p}) - etaig(hat{a}^dagger(vec{p})hat{a}(vec{p}) + hat{b}^dagger(vec{p})hat{b}(vec{p})ig)
ight]
ight)$.
Where $U_c$ is a unitary operator such that:
$hat{a}_c(vec{p}) = U_c,hat{a}(vec{p}),U_c^dagger$, $hat{b}_c(vec{p}) = U_c,hat{b}(vec{p}),U_c^dagger$ and therefore $phi_c(x) = U_c,phi(x),U_c^dagger$.
Hint: use the Baker–Campbell–Hausdorff formula:
$e^{-hat{A}},hat{B},e^{hat{A}} = hat{B} + [hat{B},hat{A}] + 	frac{1}{2!}[[hat{B},hat{A}],hat{A}] + 	frac{1}{3!}[[[hat{B},hat{A}],hat{A}],hat{A}] + dots$
e) $U_c,hat{Q},U_c^dagger = -hat{Q}$

$The classical Lagrangian density for the free complex scalar field is given by: mathcalL = (partialmuphi^*)(partial^muphi) - m^2phi^*phi 1. Show that: a) it is invariant under the discrete transformation: phi(x) o phic(x) = eta phi^dagger(x), eta = pm 1 phic is the charge conjugated field. b) (phic)c = phi. c) hatac(vecp) = eta hatb(vecp), hatbc(vecp) = eta hata(vecp). Where phic(x) = int fracd^3p(2pi)^3 2Epleft(e^-ipcdot xhatac(vecp) + e^ipcdot xhatb^dagger(vecp) ight). d) Uc = expleft(ipi int fracd^3p2Epleft[hata^dagger(vecp)hatb(vecp) + hatb^dagger(vecp)hata(vecp) - etaig(hata^dagger(vecp)hata(vecp) + hatb^dagger(vecp)hatb(vecp)ig) ight] ight). Where Uc is a unitary operator such that: hatac(vecp) = Uc,hata(vecp),Uc^dagger, hatbc(vecp) = Uc,hatb(vecp),Uc^dagger and therefore phic(x) = Uc,phi(x),Uc^dagger. Hint: use the Baker–Campbell–Hausdorff formula: e^-hatA,hatB,e^hatA = hatB + [hatB,hatA] + frac12![[hatB,hatA],hatA] + frac13![[[hatB,hatA],hatA],hatA] + dots e) Uc,hatQ,Uc^dagger = -hatQ$

Added by Sharon J.

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