Manuscript I-B-VP@3 · I-B-VP-v3.tex · proposition · 93c04eb4

Proposition: Matrix observables and radial current

Rotation-breaking nonnegative relations · lines 2383–2433 · proof included or linked

I-B-VP@3#prop:matrix

For $d = 2$ , let $C = C_{W}$ and $K_{j} = K_{ν_{j}} (p r)$ . Then

δ F = \frac{1}{4 π ^{2} α} \int_{0}^{\infty} p d p j k \sum C_{j k} K_{j} K_{k} e^{i (q_{j} - q_{k}) φ},

δ j_{φ} = - \frac{e}{4 π ^{2} α} \int_{0}^{\infty} p d p j k \sum (q_{j} + q_{k}) C_{j k} K_{j} K_{k} e^{i (q_{j} - q_{k}) φ} .

Writing the two labels as $0, - 1$ ,

δ j_{r} = - \frac{e}{2 π ^{2} α} \int_{0}^{\infty} p^{2} d p Im (C_{0, - 1} e^{i φ}) [K_{ν_{0}}^{'} (p r) K_{ν_{- 1}} (p r) - K_{ν_{0}} (p r) K_{ν_{- 1}}^{'} (p r)] .

Also $j_{t} = j_{z} = 0$ . Stress is obtained by applying (eq:pointsplit) to $2 Re δ G_{W}$ . All these operations converge locally and satisfy

\nabla_{μ} T^{μ}_{ν} = 0, T^{μ}_{μ} = (6 ξ - 1) □ F,

\frac{1}{r} \partial_{r} (r j^{r}) + \partial_{φ} j^{φ} = 0.

Generically, unequal deficient orders and off-diagonal boundary data give a nonzero radial current and a nonzero shear component.

At $β = 1/2$ , equal orders force the displayed radial current to vanish for every such relation. Other components may still have angular dependence.

Proof. Apply the derivative difference to (eq:dG). An angular derivative gives $i (q_{j} + q_{k})$ . For a radial derivative combine the $j, k$ and $k, j$ terms using Hermiticity of $C$ ; this gives the imaginary part shown. At equal orders the radial Wronskian factor is zero. All interchanges are justified by Theorem thm:nonnegative-state. The Ward identities are Lemma lem:ward.

Displayed formulas in this passage

unlabeled · line 2387
$δ F = \frac{1}{4 π ^{2} α} \int_{0}^{\infty} p d p j k \sum C_{j k} K_{j} K_{k} e^{i (q_{j} - q_{k}) φ},$
unlabeled · line 2392
$δ j_{φ} = - \frac{e}{4 π ^{2} α} \int_{0}^{\infty} p d p j k \sum (q_{j} + q_{k}) C_{j k} K_{j} K_{k} e^{i (q_{j} - q_{k}) φ} .$
unlabeled · line 2399
$δ j_{r} = - \frac{e}{2 π ^{2} α} \int_{0}^{\infty} p^{2} d p Im (C_{0, - 1} e^{i φ}) [K_{ν_{0}}^{'} (p r) K_{ν_{- 1}} (p r) - K_{ν_{0}} (p r) K_{ν_{- 1}}^{'} (p r)] .$
unlabeled · line 2409
$\nabla_{μ} T^{μ}_{ν} = 0, T^{μ}_{μ} = (6 ξ - 1) □ F,$
unlabeled · line 2413
$\frac{1}{r} \partial_{r} (r j^{r}) + \partial_{φ} j^{φ} = 0.$

Proposition: Matrix observables and radial current

Displayed formulas in this passage

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