Manuscript XVIII-B@2 · XVIII-B-v2.tex · section · 27ed4f4b

Field and state normalization

The exact two-particle spectral calculation › Field and state normalization · lines 414–442

Let $ω_{k} = ∣ k ∣^{2} + m^{2}$ and

d μ_{m} (k) = \frac{d ^{3} k}{( 2 π ) ^{3} 2 ω _{k}}, k = (ω_{k}, k) .

Use

ϕ (x) = ℏ \int d μ_{m} (k) (a (k) e^{- i k \cdot x} + a^{†} (k) e^{i k \cdot x}),

[a (k), a^{†} (q)] = (2 π)^{3} 2 ω_{k} δ^{3} (k - q), a (k) Ω = 0.

The unordered two-particle completeness relation has factor $1/2!$ . This factor is retained below.

Define

W_{ab c d}^{>} (x) = ⟨ T_{ab} (x) T_{c d} (0) ⟩_{Ω}, W_{ab c d}^{<} (x) = W_{c d ab}^{>} (- x) .

Displayed formulas in this passage

unlabeled · line 417
$d μ_{m} (k) = \frac{d ^{3} k}{( 2 π ) ^{3} 2 ω _{k}}, k = (ω_{k}, k) .$
unlabeled · line 422
$ϕ (x) = ℏ \int d μ_{m} (k) (a (k) e^{- i k \cdot x} + a^{†} (k) e^{i k \cdot x}),$
unlabeled · line 426
$[a (k), a^{†} (q)] = (2 π)^{3} 2 ω_{k} δ^{3} (k - q), a (k) Ω = 0.$
unlabeled · line 436
$W_{ab c d}^{>} (x) = ⟨ T_{ab} (x) T_{c d} (0) ⟩_{Ω}, W_{ab c d}^{<} (x) = W_{c d ab}^{>} (- x) .$