Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

The quadratic normalization

The signed action and its Euler derivatives › The quadratic normalization · lines 691–793

XI-A-ST@1#eq:stressvariation, XI-A-ST@1#eq:bs, XI-A-ST@1#eq:as, XI-A-ST@1#eq:vertex, XI-A-ST@1#eq:secondvertex, XI-A-ST@1#eq:Hdefinition, XI-A-ST@1#eq:J1

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Proof. These are XI-A :TT and :Newton CheungXIA. Raising both spatial indices in (eq:TT) contributes two minus signs, so the time-derivative coefficient remains positive. In (eq:Newtonmetric), $H ˉ_{00} = 4 Φ_{N}$ , $H ˉ_{ij} = 0$ , and $D (H) = 0$ . Equation (eq:Glin) gives $G_{lin}^{00} (H) = 2Δ Φ_{N}$ . The retained unswitched equation is $G^{μν} = κ^{2} T^{μν} /4$ , giving (eq:Newton). Neither calculation inserts $G$ in place of $q$ .

The Newtonian source is an auxiliary static comparator. It is not a solution of the compact-time switched system. Neither sign diagnostic computes vacuum energy.

metric coupling and the actual scalar switch

Let $p_{μ} = \partial_{μ} ϕ$ , and put

ℓ_{q} = \frac{1}{2} (q^{α β} p_{α} p_{β} - m^{2} ϕ^{2}), T_{q}^{μν} = q^{μα} q^{ν β} p_{α} p_{β} - q^{μν} ℓ_{q} .

Write $T_{0} = T_{η}$ .

The covariant metric variation is

δ_{q} L_{q} = - \frac{1}{2} ρ_{q} T_{q}^{μν} δ q_{μν} .

Set

ρ_{q} = 1 + κ b_{1} + κ^{2} b_{2} + O (κ^{3}), ρ_{q} q^{μν} = η^{μν} + κ a_{1}^{μν} + κ^{2} a_{2}^{μν} + O (κ^{3}),

where

b_{1} a_{1}^{μν} = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, b_{2} a_{2}^{μν} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{α β} γ^{α β}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} η^{μν} .

The matter vertices are therefore

V_{1, g} V_{2, g} = \frac{1}{2} \int g (a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2}) d^{4} x = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} d^{4} x, = \frac{1}{2} \int g (a_{2}^{μν} p_{μ} p_{ν} - m^{2} b_{2} ϕ^{2}) d^{4} x .

There is one factor $g$ , not $g^{2}$ , in the second coefficient.

Define $H_{n}^{μν}$ by

ρ_{q} G^{μν} (q) = κ H_{1}^{μν} + κ^{2} H_{2}^{μν} + κ^{3} H_{3}^{μν} + O (κ^{4}) .

Thus $H_{1} = G_{lin}$ . Appendix app:coefficients gives a finite coefficient prescription.

Also,

ρ_{q} T_{q}^{μν} = T_{0}^{μν} + κ J_{1}^{μν} + O (κ^{2}),

with

J_{1}^{μν} = b_{1} T_{0}^{μν} - γ^{μα} p_{α} p^{ν} - γ^{ν α} p_{α} p^{μ} + γ^{μν} L_{η} + \frac{1}{2} η^{μν} γ^{α β} p_{α} p_{β} .

Displayed formulas in this passage

XI-A-ST@1#eq:stressvariation · line 725
$δ_{q} L_{q} = - \frac{1}{2} ρ_{q} T_{q}^{μν} δ q_{μν} .$
XI-A-ST@1#eq:bs · line 738
$b_{1} a_{1}^{μν} = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, b_{2} a_{2}^{μν} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{α β} γ^{α β}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} η^{μν} .$
XI-A-ST@1#eq:as · line 738
$b_{1} a_{1}^{μν} = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, b_{2} a_{2}^{μν} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{α β} γ^{α β}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} η^{μν} .$
XI-A-ST@1#eq:vertex · line 751
$V_{1, g} V_{2, g} = \frac{1}{2} \int g (a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2}) d^{4} x = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} d^{4} x, = \frac{1}{2} \int g (a_{2}^{μν} p_{μ} p_{ν} - m^{2} b_{2} ϕ^{2}) d^{4} x .$
XI-A-ST@1#eq:secondvertex · line 751
$V_{1, g} V_{2, g} = \frac{1}{2} \int g (a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2}) d^{4} x = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} d^{4} x, = \frac{1}{2} \int g (a_{2}^{μν} p_{μ} p_{ν} - m^{2} b_{2} ϕ^{2}) d^{4} x .$
XI-A-ST@1#eq:Hdefinition · line 766
$ρ_{q} G^{μν} (q) = κ H_{1}^{μν} + κ^{2} H_{2}^{μν} + κ^{3} H_{3}^{μν} + O (κ^{4}) .$
XI-A-ST@1#eq:J1 · line 783
$J_{1}^{μν} = b_{1} T_{0}^{μν} - γ^{μα} p_{α} p^{ν} - γ^{ν α} p_{α} p^{μ} + γ^{μν} L_{η} + \frac{1}{2} η^{μν} γ^{α β} p_{α} p_{β} .$
unlabeled · line 715
$ℓ_{q} = \frac{1}{2} (q^{α β} p_{α} p_{β} - m^{2} ϕ^{2}), T_{q}^{μν} = q^{μα} q^{ν β} p_{α} p_{β} - q^{μν} ℓ_{q} .$
unlabeled · line 730
$ρ_{q} = 1 + κ b_{1} + κ^{2} b_{2} + O (κ^{3}), ρ_{q} q^{μν} = η^{μν} + κ a_{1}^{μν} + κ^{2} a_{2}^{μν} + O (κ^{3}),$
unlabeled · line 778
$ρ_{q} T_{q}^{μν} = T_{0}^{μν} + κ J_{1}^{μν} + O (κ^{2}),$

The quadratic normalization

Displayed formulas in this passage

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