Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Coefficient operators and explicit switch derivatives

The specified switched scalar action › Coefficient operators and explicit switch derivatives · lines 593–657

VII-A@1#eq:Q, VII-A@1#eq:Tquad, VII-A@1#eq:Dgexp, VII-A@1#eq:Vg, VII-A@1#eq:Wg, VII-A@1#eq:Vgexpanded, VII-A@1#eq:Wgexpanded

Define

Q_{h}^{(m)} ψ T_{h}^{(m)} ψ = \partial_{μ} (A^{μν} (h) \partial_{ν} ψ) - m^{2} a (h) ψ, = \partial_{μ} (B^{μν} (h) \partial_{ν} ψ) - m^{2} b (h) ψ .

Then

D_{g} [g] = P_{m} + κ V_{g} [h] + κ^{2} W_{g} [h, k] + O (κ^{3}),

where

V_{g} [h] ψ W_{g} [h, k] ψ = \partial_{μ} (g A^{μν} (h) \partial_{ν} ψ) - m^{2} g a (h) ψ = Q_{gh}^{(m)} ψ, = \partial_{μ} (g (A^{μν} (k) + B^{μν} (h)) \partial_{ν} ψ) - m^{2} g (a (k) + b (h)) ψ .

In expanded form,

V_{g} [h] ψ W_{g} [h, k] ψ = g Q_{h}^{(m)} ψ + (\partial_{μ} g) A^{μν} (h) \partial_{ν} ψ, = g (Q_{k}^{(m)} + T_{h}^{(m)}) ψ + (\partial_{μ} g) (A^{μν} (k) + B^{μν} (h)) \partial_{ν} ψ .

These identities display all derivatives of the switch in the coefficient operators. In particular, $V_{g} [h] \neq = g Q_{h}^{(m)}$ in general.

Displayed formulas in this passage

VII-A@1#eq:Q · line 596
$Q_{h}^{(m)} ψ T_{h}^{(m)} ψ = \partial_{μ} (A^{μν} (h) \partial_{ν} ψ) - m^{2} a (h) ψ, = \partial_{μ} (B^{μν} (h) \partial_{ν} ψ) - m^{2} b (h) ψ .$
VII-A@1#eq:Tquad · line 596
$Q_{h}^{(m)} ψ T_{h}^{(m)} ψ = \partial_{μ} (A^{μν} (h) \partial_{ν} ψ) - m^{2} a (h) ψ, = \partial_{μ} (B^{μν} (h) \partial_{ν} ψ) - m^{2} b (h) ψ .$
VII-A@1#eq:Dgexp · line 610
$D_{g} [g] = P_{m} + κ V_{g} [h] + κ^{2} W_{g} [h, k] + O (κ^{3}),$
VII-A@1#eq:Vg · line 618
$V_{g} [h] ψ W_{g} [h, k] ψ = \partial_{μ} (g A^{μν} (h) \partial_{ν} ψ) - m^{2} g a (h) ψ = Q_{gh}^{(m)} ψ, = \partial_{μ} (g (A^{μν} (k) + B^{μν} (h)) \partial_{ν} ψ) - m^{2} g (a (k) + b (h)) ψ .$
VII-A@1#eq:Wg · line 618
$V_{g} [h] ψ W_{g} [h, k] ψ = \partial_{μ} (g A^{μν} (h) \partial_{ν} ψ) - m^{2} g a (h) ψ = Q_{gh}^{(m)} ψ, = \partial_{μ} (g (A^{μν} (k) + B^{μν} (h)) \partial_{ν} ψ) - m^{2} g (a (k) + b (h)) ψ .$
VII-A@1#eq:Vgexpanded · line 637
$V_{g} [h] ψ W_{g} [h, k] ψ = g Q_{h}^{(m)} ψ + (\partial_{μ} g) A^{μν} (h) \partial_{ν} ψ, = g (Q_{k}^{(m)} + T_{h}^{(m)}) ψ + (\partial_{μ} g) (A^{μν} (k) + B^{μν} (h)) \partial_{ν} ψ .$
VII-A@1#eq:Wgexpanded · line 637
$V_{g} [h] ψ W_{g} [h, k] ψ = g Q_{h}^{(m)} ψ + (\partial_{μ} g) A^{μν} (h) \partial_{ν} ψ, = g (Q_{k}^{(m)} + T_{h}^{(m)}) ψ + (\partial_{μ} g) (A^{μν} (k) + B^{μν} (h)) \partial_{ν} ψ .$