a^{*}

a^{†}

[a_{p}, a_{q}^{†}] = (2 π)^{3} δ^{3} (p - q), [a_{p}, a_{q}] = [a_{p}^{†}, a_{q}^{†}] = 0.

Together with the normalization in part a, they give

[ϕ (t, x), π (t, y)] = i δ^{3} (x - y)

and vanishing equal-time commutators between two fields or two conjugate momenta.

(2 π)^{3} δ^{3} (p \pm q)

. The

aa

and

a^{†} a^{†}

coefficients cancel because

ω_{p}^{2} = p^{2} + m^{2}

. The remaining terms give

H = \frac{1}{2} \int \frac{d ^{3} p}{( 2 π ) ^{3}} ω_{p} (a_{p}^{†} a_{p} + a_{p} a_{p}^{†}) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} ω_{p} a_{p}^{†} a_{p} + \frac{( 2 π ) ^{3} δ ^{3} ( 0 )}{2} \int \frac{d ^{3} p}{( 2 π ) ^{3}} ω_{p} .

This is a continuum of quantum harmonic oscillators. In a finite periodic volume

V

(2 π)^{3} δ^{3} (0) = V

. The Fock vacuum satisfies

a_{p} ∣0 ⟩ = 0

E = \frac{⟨ 0∣ H ∣0 ⟩}{V} = \frac{1}{2} \int \frac{d ^{3} p}{( 2 π ) ^{3}} p^{2} + m^{2} .

Normal ordering would subtract this constant; here it must be retained because it is the quantity being calculated.^[ref]

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