CoarseGaugeCoarseMatrices #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
Provider: coarse graining intertwines with a constant skew shift of the field #
This module closes the obligation named in the module docstring of
Algsuperdiff.Section3.Provider.Diffusivity.ApproximateRecurrence.PrincipalResponseGauge:
the coefficient-field half of ABK26. Where that module proved the
block-algebra identity blockVecDot_blockMatrixOfCoarseMatrices_gaugeShift at
the coarse package carrier, here we show that the coarse package of the shifted
field really is the gauge shift of the coarse package:
coarseMatrices U (a - hbar) = gaugeShiftCoarseMatrices hbar (coarseMatrices U a)
for a constant skew hbar, i.e. sigma and sigmaStar^{-1} are unchanged
and kappa drops by hbar. Combining the two gives the manuscript's opening
equality of Step 3 at the coefficient-field carrier,
blockVecDot_coarseBlockMatrix_sub_const_skew.
How it is proved #
CoarseGaugeResponse.responseJ_sub_const_skew turns the field shift into the
load shear J(U,p,q;a - hbar) = J(U,p,q - hbar p;a). CoarseGraining's
Homogenization.Book.Ch02.responseJ_eq_coarseMatrices_formula_canonical
(unconditional) writes J(U,p,q;a) as (1/2) p.sigma p + (1/2) (q + kappa p).sigmaStarInv (q + kappa p) - p.q. Because hbar is skew, p. hbar p = 0,
so the shear absorbs entirely into the completed square and reproduces the same
normal form with kappa replaced by kappa - hbar. The canonical
CoarseGraining entry formulas -- polarizations of J(U,0,.), of
mixedResponse, and of the corrected p-response -- are then read off one at
a time. The single nondegeneracy input is CoarseGraining's unconditional
Homogenization.Book.Ch02.isUnit_det_sigmaStarInvCoarse, needed only because
kappaCoarse is defined as sigmaStar * (sigmaStarInv kappa).
Every hypothesis is a source premise: hbar skew (it is an average of skew
matrices) and b a coefficient object representing a - hbar. No proof step
is carried as a hypothesis.
Two quadratic-form helpers #
The coarse matrices of the shifted field #
The pure-flux coarse matrix does not see a constant skew shift.
The normal form of J(U,.,.;a - hbar): the same sigma and
sigmaStarInv, with kappa replaced by kappa - hbar.
The coupling matrix drops by the constant skew shift.
The diffusivity matrix does not see a constant skew shift.
The intertwining #
Coarse graining intertwines with a constant skew shift of the field.
For a constant skew hbar and any coefficient object b representing a - hbar, the coarse matrix package of b is the gauge shift of the coarse matrix
package of a: sigma and sigmaStarInv are unchanged and kappa drops by
hbar. This is the coefficient-field half of ABK26.
ABK26 at the coefficient-field carrier.
Shearing the doubled load by bfG_{-hbar} exactly compensates the constant skew
shift a |-> a - hbar of the coefficient field, in the doubled quadratic form
of the coarse block matrix bfA(U; .).
The block-algebra half is
blockVecDot_blockMatrixOfCoarseMatrices_gaugeShift; the coarse-graining half
is coarseMatrices_sub_const_skew.