Manuscript I-B-VP@3 · I-B-VP-v3.tex · theorem · 93c04eb4

Theorem: Friedrichs Wick squares and stress tensors

Exact Friedrichs observables and analytical checks · lines 1149–1220 · proof included or linked

I-B-VP@3#thm:friedrichs, I-B-VP@3#eq:T-real, I-B-VP@3#eq:T-complex

For one neutral real scalar,

⟨ ϕ^{2} ⟩_{ren} = \frac{α ^{- 2} - 1}{48 π ^{2} r ^{2}},

⟨ T^{μ}_{ν} ⟩_{ren} = \frac{α ^{- 4} - 1}{1440 π ^{2} r ^{4}} diag (1, 1, - 3, 1) + (ξ - \frac{1}{6}) \frac{α ^{- 2} - 1}{48 π ^{2} r ^{4}} diag (4, - 2, 6, 4) .

For one charged complex scalar,

⟨ Φ^{†} Φ ⟩_{ren} = \frac{P _{β}}{48 π ^{2} r ^{2}},

⟨ T^{μ}_{ν} ⟩_{ren} = \frac{C _{β}}{720 π ^{2} r ^{4}} diag (1, 1, - 3, 1) + 2 (ξ - \frac{1}{6}) \frac{P _{β}}{48 π ^{2} r ^{4}} diag (4, - 2, 6, 4) .

Their traces are respectively

(ξ - \frac{1}{6}) \frac{α ^{- 2} - 1}{4 π ^{2} r ^{4}}, (ξ - \frac{1}{6}) \frac{P _{β}}{2 π ^{2} r ^{4}} .

Proof. The Wick squares follow from Lemma lem:expansion. For stress first apply the real differential expression to $Re H_{F, E}$ . Reflections, angular evenness of the real part, translations, boosts, and scaling imply

T^{μ}_{ν} = r^{- 4} diag (a, b, c, a) .

Radial conservation gives $c = - 3 b$ .

For $F = P_{β} / (48 π^{2} r^{2})$ ,

[\partial_{ρ} \partial^{ρ^{'}} H_{F, E}] = \frac{1}{2} Δ_{E} F = \frac{P _{β}}{24 π ^{2} r ^{4}} .

The time-split identities give

T^{τ}_{τ} = - H_{F, E}^{''} (0) + (2 ξ - \frac{1}{2}) \frac{P _{β}}{24 π ^{2} r ^{4}} = \frac{C _{β}}{1440 π ^{2} r ^{4}} + (ξ - \frac{1}{6}) \frac{4 P _{β}}{48 π ^{2} r ^{4}} .

At conformal coupling the trace vanishes, hence $a = b$ . For other couplings use

(g □ - \nabla\nabla)^{μ}_{ν} r^{- 2} = r^{- 4} diag (4, - 2, 6, 4) .

Take $β = 0$ for one real field; multiply by two for one complex stress tensor. The trace formulas follow directly.

Displayed formulas in this passage

I-B-VP@3#eq:T-real · line 1156
$⟨ T^{μ}_{ν} ⟩_{ren} = \frac{α ^{- 4} - 1}{1440 π ^{2} r ^{4}} diag (1, 1, - 3, 1) + (ξ - \frac{1}{6}) \frac{α ^{- 2} - 1}{48 π ^{2} r ^{4}} diag (4, - 2, 6, 4) .$
I-B-VP@3#eq:T-complex · line 1169
$⟨ T^{μ}_{ν} ⟩_{ren} = \frac{C _{β}}{720 π ^{2} r ^{4}} diag (1, 1, - 3, 1) + 2 (ξ - \frac{1}{6}) \frac{P _{β}}{48 π ^{2} r ^{4}} diag (4, - 2, 6, 4) .$
unlabeled · line 1152
$⟨ ϕ^{2} ⟩_{ren} = \frac{α ^{- 2} - 1}{48 π ^{2} r ^{2}},$
unlabeled · line 1165
$⟨ Φ^{†} Φ ⟩_{ren} = \frac{P _{β}}{48 π ^{2} r ^{2}},$
unlabeled · line 1178
$(ξ - \frac{1}{6}) \frac{α ^{- 2} - 1}{4 π ^{2} r ^{4}}, (ξ - \frac{1}{6}) \frac{P _{β}}{2 π ^{2} r ^{4}} .$
unlabeled · line 1193
$T^{μ}_{ν} = r^{- 4} diag (a, b, c, a) .$
unlabeled · line 1199
$[\partial_{ρ} \partial^{ρ^{'}} H_{F, E}] = \frac{1}{2} Δ_{E} F = \frac{P _{β}}{24 π ^{2} r ^{4}} .$
unlabeled · line 1204
$T^{τ}_{τ} = - H_{F, E}^{''} (0) + (2 ξ - \frac{1}{2}) \frac{P _{β}}{24 π ^{2} r ^{4}} = \frac{C _{β}}{1440 π ^{2} r ^{4}} + (ξ - \frac{1}{6}) \frac{4 P _{β}}{48 π ^{2} r ^{4}} .$
unlabeled · line 1214
$(g □ - \nabla\nabla)^{μ}_{ν} r^{- 2} = r^{- 4} diag (4, - 2, 6, 4) .$

Theorem: Friedrichs Wick squares and stress tensors

Displayed formulas in this passage

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