Manuscript VII-H@1 · VII-H.tex · section · d20d2a9f

Field, state, source, and readout

Definition sheet for the quantum benchmark › Field, state, source, and readout · lines 493–570

VII-H@1#eq:W, VII-H@1#eq:W-boundary, VII-H@1#eq:AB, VII-H@1#eq:source-expansion, VII-H@1#eq:kubo

Set

M = R^{4}, η = diag (1, - 1, - 1, - 1), z = (t, x), s (z) = t^{2} - ∣ x ∣^{2}, □ = \partial_{t}^{2} - Δ_{x} .

There is no physical reflecting or absorbing boundary.

Let $φ$ be the real massless free scalar in its Minkowski vacuum $ω_{0}$ , with

ω_{0} (φ (x) φ (y)) = ℏ W (x - y), W (t, x) = \int_{R^{3}} \frac{e ^{- i ∣ p ∣ t + i p \cdot x}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p .

Equivalently,

W (t, x) = - \frac{1}{4 π ^{2}} ϵ ↓ 0 lim \frac{1}{( t - i ϵ ) ^{2} - ∣ x ∣ ^{2}} .

The parameter $ℏ > 0$ is retained explicitly.

Define the calibrated vacuum-centred Wick square

Q (x) =: φ (x)^{2} :_{ω_{0}} .

Its field-degree coefficient is one and $ω_{0} (Q (x)) = 0$ . For real compact tests,

A (f) = \int f (x) Q (x) d^{4} x, B (h) = \int h (y) Q (y) d^{4} y .

The external source enters with Hamiltonian density

H_{src} (y) = - λh (y) Q (y) .

The initial state remains $ω_{0}$ . At first source order, the retarded readout is

ω_{0} (A_{λ, h}^{ret} (f)) = λ χ (f, h) + O (λ^{2}),

where, away from coincident source/readout points,

χ (f, h) = \frac{i}{ℏ} \int f (x) h (y) Θ (x^{0} - y^{0}) ω_{0} ([Q (x), Q (y)]) d^{4} x d^{4} y .

The sign follows by expanding source evolution with $H_{src} = - λh Q$ .

The response is connected: its two insertions are joined by vacuum contractions, and the disconnected subtraction $ω_{0} (Q (x)) ω_{0} (Q (y))$ is zero. The selected perturbative order is $N = 1$ in $λ$ . No bulk self-interaction is present.

Displayed formulas in this passage

VII-H@1#eq:W · line 507
$ω_{0} (φ (x) φ (y)) = ℏ W (x - y), W (t, x) = \int_{R^{3}} \frac{e ^{- i ∣ p ∣ t + i p \cdot x}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p .$
VII-H@1#eq:W-boundary · line 518
$W (t, x) = - \frac{1}{4 π ^{2}} ϵ ↓ 0 lim \frac{1}{( t - i ϵ ) ^{2} - ∣ x ∣ ^{2}} .$
VII-H@1#eq:AB · line 535
$A (f) = \int f (x) Q (x) d^{4} x, B (h) = \int h (y) Q (y) d^{4} y .$
VII-H@1#eq:source-expansion · line 547
$ω_{0} (A_{λ, h}^{ret} (f)) = λ χ (f, h) + O (λ^{2}),$
VII-H@1#eq:kubo · line 554
$χ (f, h) = \frac{i}{ℏ} \int f (x) h (y) Θ (x^{0} - y^{0}) ω_{0} ([Q (x), Q (y)]) d^{4} x d^{4} y .$
unlabeled · line 496
$M = R^{4}, η = diag (1, - 1, - 1, - 1), z = (t, x), s (z) = t^{2} - ∣ x ∣^{2}, □ = \partial_{t}^{2} - Δ_{x} .$
unlabeled · line 529
$Q (x) =: φ (x)^{2} :_{ω_{0}} .$
unlabeled · line 542
$H_{src} (y) = - λh (y) Q (y) .$