Manuscript III@2 · III.tex · section · 82dad3e5

A scalar symplectic consequence with metric fixed

The coupled first-order jet and retarded harmonic section › A scalar symplectic consequence with metric fixed · lines 1397–1423

Fix the reduced metric $k$ ; do not vary it. Put

c_{k}^{μν} = \frac{1}{2} (tr_{η} k) η^{μν} - k^{μν}, a_{k}^{μν} = η^{μν} + κ c_{k}^{μν}, b_{k} = 1 + \frac{κ}{2} tr_{η} k,

and

P_{k} A = \partial_{μ} (a_{k}^{μν} \partial_{ν} A) - m^{2} b_{k} A .

Expansion gives

P_{k} = P + κ (L_{k} + \frac{1}{2} (tr_{η} k) P) .

Thus $A = A_{0} + κ A_{1}$ solves $P_{k} A = 0$ exactly when

P A_{0} = 0, P A_{1} = - L_{k} A_{0} .

Displayed formulas in this passage

unlabeled · line 1401
$c_{k}^{μν} = \frac{1}{2} (tr_{η} k) η^{μν} - k^{μν}, a_{k}^{μν} = η^{μν} + κ c_{k}^{μν}, b_{k} = 1 + \frac{κ}{2} tr_{η} k,$
unlabeled · line 1410
$P_{k} A = \partial_{μ} (a_{k}^{μν} \partial_{ν} A) - m^{2} b_{k} A .$
unlabeled · line 1415
$P_{k} = P + κ (L_{k} + \frac{1}{2} (tr_{η} k) P) .$
unlabeled · line 1420
$P A_{0} = 0, P A_{1} = - L_{k} A_{0} .$