lemmaroutine

Divergence-free positive source tensor from holomorphic flow

Holomorphic flow gives a symmetric PSD tensor with divergence-free columns

abstract positive conserved tensor for the wave current

Lemma·lem:current-flow·§2.2 Two auxiliary flows

Statement (paper)

Let

\mathscr P=\sum_\mu C^\mu\partial_{\mathsf X^\mu}

have constant real symmetric coefficients. For a holomorphic solution

\mathscr P f_{\rm sys}=0

and the real displacement , put, wherever defined,

G=\overline{f_{\rm sys}(\mathsf X-\mathrm i q)},\qquad V_{\ell a}=\sum_\mu(C^\mu w_\ell)_a\,\partial_{\mathsf X^\mu},\qquad D_{\ell a}=\partial_{w_{\ell a}}+\mathrm i V_{\ell a},\qquad \mathcal Z_\ell=\sum_a G_aD_{\ell a}.

In the real coordinates

(\mathsf X,w_1,w_2)

, the tensor

B=\Rea\sum_{\ell=1}^2\mathcal Z_\ell\otimes\overline{\mathcal Z_\ell}

is symmetric and positive semidefinite, and each of its columns has zero divergence.

In plain English

For a first-order system

\mathscr P f_{\rm sys}=0

with constant real symmetric coefficients and holomorphic

f_{\rm sys}

, the tensor

B=\Rea\sum_{\ell=1}^2\mathcal Z_\ell\otimes\overline{\mathcal Z_\ell}

in the real coordinates

(\mathsf X,w_1,w_2)

is symmetric and positive semidefinite, and each of its columns is divergence-free.

Why it's here

This is the structural input for the wave current: positivity makes the current a nonnegative quantity, and zero divergence of the columns gives conservation, so the later propositions can treat the current as a conserved flux. The mechanism is that the operators

D_{\ell a}=\partial_{w_{\ell a}}+\mathrm i V_{\ell a}

combine with the holomorphic equation

\mathscr P f_{\rm sys}=0

, so the divergence terms cancel, while

B

is a real part of a sum of rank-one Hermitian terms, which gives symmetry and positivity.

Classified routine:

Direct computation: symmetry and positivity hold because B is the real part of a sum of Hermitian rank-one terms, and the divergence cancels using holomorphy and the constant-coefficient equation, much as divergence-free energy-momentum tensors arise for constant-coefficient symmetric systems. It is an auxiliary structural lemma, not one of the paper's main new ideas.

Uses

Nothing else in the paper

Used by

propWave current is positive and divergence-freeclaimConserved current with Gaussian multiplier symbols

Sources

PaperOpen

Section: §2.2 Two auxiliary flows
Page: —
Label: lem:current-flow
Line: sections/current.tex:104

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