Manuscript Em03-A@1 · Em03-A-v1.tex · section · 6d54627d

Canonical current normalization

Quantum input, parametrix, and the thermal calculation › Canonical current normalization · lines 657–699

Em03-A@1#eq:currentexpansion, Em03-A@1#eq:currentCCR, Em03-A@1#eq:vacuumcurrent, Em03-A@1#eq:vacuumpositive

For a normalized radiation field $ψ (s)$ , write $j (s) = \partial_{s} ψ (s)$ . The positive-frequency convention is

j (s) = - i \int_{0}^{\infty} \frac{ω}{4 π} (a (ω) e^{- i ω s} - a (ω)^{*} e^{i ω s}) d ω, [a (ω), a (ω^{'})^{*}] = δ (ω - ω^{'}) .

This yields

[j (s), j (s^{'})] = \frac{i}{2} δ^{'} (s - s^{'}) .

The vacuum current covariance is the tempered distribution

C_{0} (s - s^{'}) = - \frac{1}{4 π ( s - s ^{'} - i 0 ) ^{2}} = ϵ ↓ 0 lim \int_{0}^{\infty} \frac{ω}{4 π} e^{- i ω (s - s^{'}) - ϵ ω} d ω .

In particular, for any complex compact test $g$ ,

\int_{R^{2}} g (s) C_{0} (s - s^{'}) g (s^{'}) d s d s^{'} = \int_{0}^{\infty} \frac{ω}{4 π} \int_{R} g (s) e^{i ω s} d s^{2} d ω \geq 0.

This formula also proves finite covariance on compact smooth tests.

We work with currents rather than an unsmeared massless scalar zero mode. The number observables used below have frequency support bounded away from zero.

Displayed formulas in this passage

Em03-A@1#eq:currentexpansion · line 661
$j (s) = - i \int_{0}^{\infty} \frac{ω}{4 π} (a (ω) e^{- i ω s} - a (ω)^{*} e^{i ω s}) d ω, [a (ω), a (ω^{'})^{*}] = δ (ω - ω^{'}) .$
Em03-A@1#eq:currentCCR · line 671
$[j (s), j (s^{'})] = \frac{i}{2} δ^{'} (s - s^{'}) .$
Em03-A@1#eq:vacuumcurrent · line 677
$C_{0} (s - s^{'}) = - \frac{1}{4 π ( s - s ^{'} - i 0 ) ^{2}} = ϵ ↓ 0 lim \int_{0}^{\infty} \frac{ω}{4 π} e^{- i ω (s - s^{'}) - ϵ ω} d ω .$
Em03-A@1#eq:vacuumpositive · line 686
$\int_{R^{2}} g (s) C_{0} (s - s^{'}) g (s^{'}) d s d s^{'} = \int_{0}^{\infty} \frac{ω}{4 π} \int_{R} g (s) e^{i ω s} d s^{2} d ω \geq 0.$