Manuscript IV@1 · IV.tex · section · 351cb2d7

Shared conventions and elementary measurement facts

Shared conventions and elementary measurement facts · lines 415–448

We use $ℏ = 1$ , signature $(- 1, 1, 1, 1)$ , and

□ = - \partial_{t}^{2} + Δ, P = □ - m^{2}, Ω_{m} = - Δ + m^{2}, m > 0.

A fixed reference time is denoted by $t_{0}$ . The spatial Fourier transform is

u (p) = (2 π)^{- 3/2} \int e^{- i p \cdot x} u (x) d x .

Inner products are antilinear in their first entry. Tensor contraction is included in inner products of tensor-valued functions.

The scalar Fock convention is

[a (u), a^{†} (v)] = ⟨ u, v ⟩ I, Φ (u) = a (u) + a^{†} (u),

so $Φ$ is real linear and

[Φ (u), Φ (v)] = 2 i Im ⟨ u, v ⟩ I .

The normalized coherent vector is

Coh (z) = e^{- ∥ z ∥^{2} /2} n = 0 ⨁^{\infty} \frac{z ^{\otimes n}}{n !} .

Its number distribution is Poisson with mean $∥ z ∥^{2}$ .

Displayed formulas in this passage

unlabeled · line 419
$□ = - \partial_{t}^{2} + Δ, P = □ - m^{2}, Ω_{m} = - Δ + m^{2}, m > 0.$
unlabeled · line 426
$u (p) = (2 π)^{- 3/2} \int e^{- i p \cdot x} u (x) d x .$
unlabeled · line 434
$[a (u), a^{†} (v)] = ⟨ u, v ⟩ I, Φ (u) = a (u) + a^{†} (u),$
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$[Φ (u), Φ (v)] = 2 i Im ⟨ u, v ⟩ I .$
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$Coh (z) = e^{- ∥ z ∥^{2} /2} n = 0 ⨁^{\infty} \frac{z ^{\otimes n}}{n !} .$