Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

An explicit continuous-momentum representation

Retarded and advanced Green distributions › An explicit continuous-momentum representation · lines 969–1008

XX-B@1#eq:mode, XX-B@1#eq:mode-ret, XX-B@1#eq:mode-adv

Choose the global coframe $d t, a (t) d x^{i}$ . Spatial homogeneity writes either normalized operator as

P = - \partial_{t}^{2} + a (t)^{- 2} Δ + B_{0} (t) \partial_{t} + B_{i} (t) \partial_{i} + C (t), P = L, Q .

The matrices are obtained directly from (eq:H), (eq:Fadj), (eq:Bbulk), (eq:Amap), and (eq:Q); this is an exact coordinate representation, not a new operator.

For each $q \in R^{3}$ , let $S_{P} (t, t^{'}; q)$ be the matrix solution of

[- \partial_{t}^{2} + B_{0} (t) \partial_{t} - a (t)^{- 2} \abs q^{2} + i q_{i} B_{i} (t) + C (t)] S_{P} = 0, S_{P} (t^{'}, t^{'}; q) = 0, \partial_{t} S_{P} (t^{'}, t^{'}; q) = - id .

Then

G_{P}^{+} (t, x; t^{'}, y) G_{P}^{-} (t, x; t^{'}, y) = \frac{θ ( t - t ^{'} )}{( 2 π ) ^{3} a ( t ^{'} ) ^{3}} \int_{R^{3}} e^{i q \cdot (x - y)} S_{P} (t, t^{'}; q) d^{3} q, = - \frac{θ ( t ^{'} - t )}{( 2 π ) ^{3} a ( t ^{'} ) ^{3}} \int_{R^{3}} e^{i q \cdot (x - y)} S_{P} (t, t^{'}; q) d^{3} q .

The integrals are distributional Fourier integrals.

Displayed formulas in this passage

XX-B@1#eq:mode · line 985
$[- \partial_{t}^{2} + B_{0} (t) \partial_{t} - a (t)^{- 2} \abs q^{2} + i q_{i} B_{i} (t) + C (t)] S_{P} = 0, S_{P} (t^{'}, t^{'}; q) = 0, \partial_{t} S_{P} (t^{'}, t^{'}; q) = - id .$
XX-B@1#eq:mode-ret · line 995
$G_{P}^{+} (t, x; t^{'}, y) G_{P}^{-} (t, x; t^{'}, y) = \frac{θ ( t - t ^{'} )}{( 2 π ) ^{3} a ( t ^{'} ) ^{3}} \int_{R^{3}} e^{i q \cdot (x - y)} S_{P} (t, t^{'}; q) d^{3} q, = - \frac{θ ( t ^{'} - t )}{( 2 π ) ^{3} a ( t ^{'} ) ^{3}} \int_{R^{3}} e^{i q \cdot (x - y)} S_{P} (t, t^{'}; q) d^{3} q .$
XX-B@1#eq:mode-adv · line 995
$G_{P}^{+} (t, x; t^{'}, y) G_{P}^{-} (t, x; t^{'}, y) = \frac{θ ( t - t ^{'} )}{( 2 π ) ^{3} a ( t ^{'} ) ^{3}} \int_{R^{3}} e^{i q \cdot (x - y)} S_{P} (t, t^{'}; q) d^{3} q, = - \frac{θ ( t ^{'} - t )}{( 2 π ) ^{3} a ( t ^{'} ) ^{3}} \int_{R^{3}} e^{i q \cdot (x - y)} S_{P} (t, t^{'}; q) d^{3} q .$
unlabeled · line 973
$P = - \partial_{t}^{2} + a (t)^{- 2} Δ + B_{0} (t) \partial_{t} + B_{i} (t) \partial_{i} + C (t), P = L, Q .$

An explicit continuous-momentum representation

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