Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Scalar prescription and its equation normalization

Four contraction kernels in the actual relative slots › Scalar prescription and its equation normalization · lines 1623–1669

Em01-F@3#eq:Wm, Em01-F@3#eq:DTm, Em01-F@3#eq:scalar-kernel-identities

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Define distributionally

W_{m} (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} t + i p \cdot x},

D_{T, m} (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} ∣ t ∣ + i p \cdot x} .

The second Fourier formula defines the extension across $t = 0$ . The symbol $T$ denotes chronological time ordering. For $t > 0$ , $D_{T, m} = W_{m}$ , and for $t < 0$ , $D_{T, m} (x) = W_{m} (- x)$ . Here $W_{m}$ is the vacuum positive-frequency distribution, not a symmetric Hadamard kernel or a parametrix.

Put

Δ_{m} = E_{m}^{R} - E_{m}^{A} .

The mode identities are

P_{m} W_{m} = 0, P_{m} D_{T, m} = - i δ_{D}^{(4)}, W_{m} (x) - W_{m} (- x) = - i Δ_{m} (x) .

Indeed, the first derivative of $e^{- iω ∣ t ∣} / (2 ω)$ jumps by $- i$ at $t = 0$ , while the antisymmetric positive-frequency mode is $- i sin (ω t) / ω$ .

Thus

(- P_{m}) (ℏ D_{T, m}) = i ℏ δ_{D}^{(4)},

and the scalar antisymmetric contraction is $i ℏ$ times the scalar action-Hessian causal inverse.

Displayed formulas in this passage

Em01-F@3#eq:Wm · line 1626
$W_{m} (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} t + i p \cdot x},$
Em01-F@3#eq:DTm · line 1633
$D_{T, m} (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} ∣ t ∣ + i p \cdot x} .$
Em01-F@3#eq:scalar-kernel-identities · line 1652
$P_{m} W_{m} = 0, P_{m} D_{T, m} = - i δ_{D}^{(4)}, W_{m} (x) - W_{m} (- x) = - i Δ_{m} (x) .$
unlabeled · line 1648
$Δ_{m} = E_{m}^{R} - E_{m}^{A} .$
unlabeled · line 1664
$(- P_{m}) (ℏ D_{T, m}) = i ℏ δ_{D}^{(4)},$