Manuscript X-I-ST@1 · X-I-ST.tex · theorem · 9745fd40

Theorem: Exact new residual of the chosen lift

The finite lift and its explicit next residual · lines 1234–1278 · proof included or linked

X-I-ST@1#thm:residual, X-I-ST@1#eq:Rdifference, X-I-ST@1#eq:R4, X-I-ST@1#eq:R5, X-I-ST@1#eq:R6, X-I-ST@1#eq:consistency

As ordinary finite polynomial variational functionals,

R_{λ} - R_{ref} = λ κ^{4} B_{a} .

In particular,

[κ^{4}] R_{λ} [κ^{5}] R_{λ} [κ^{6}] R_{λ} = (S_{1}, S_{3}) + \frac{1}{2} (S_{2}, S_{2}) + λ B_{a}, = (S_{2}, S_{3}), = 0.

Coefficients through order three vanish. The exact lift obeys

(S_{λ}, R_{λ}) = 0.

Proof. All $I_{j}$ and $V_{a}$ are antifield independent. Only $H_{s}$ can bracket nontrivially with $V_{a}$ . Its scalar field rule gives

(H_{s}, V_{a}) = s V_{a} = κ B_{a} .

There is no $s_{2}$ -action on $ϕ$ . This proves (eq:Rdifference) exactly. Expanding the finite polynomial proves (eq:R4)–(eq:R5). The third coefficient $S_{3}$ and $V_{a}$ have no antifields, proving (eq:R6). The lower coefficients vanish by Theorem thm:deformation.

Equation (eq:consistency) is the odd Jacobi identity applied three times to the even Hamiltonian.

Displayed formulas in this passage

X-I-ST@1#eq:Rdifference · line 1237
$R_{λ} - R_{ref} = λ κ^{4} B_{a} .$
X-I-ST@1#eq:R4 · line 1242
$[κ^{4}] R_{λ} [κ^{5}] R_{λ} [κ^{6}] R_{λ} = (S_{1}, S_{3}) + \frac{1}{2} (S_{2}, S_{2}) + λ B_{a}, = (S_{2}, S_{3}), = 0.$
X-I-ST@1#eq:R5 · line 1242
$[κ^{4}] R_{λ} [κ^{5}] R_{λ} [κ^{6}] R_{λ} = (S_{1}, S_{3}) + \frac{1}{2} (S_{2}, S_{2}) + λ B_{a}, = (S_{2}, S_{3}), = 0.$
X-I-ST@1#eq:R6 · line 1242
$[κ^{4}] R_{λ} [κ^{5}] R_{λ} [κ^{6}] R_{λ} = (S_{1}, S_{3}) + \frac{1}{2} (S_{2}, S_{2}) + λ B_{a}, = (S_{2}, S_{3}), = 0.$
X-I-ST@1#eq:consistency · line 1253
$(S_{λ}, R_{λ}) = 0.$
unlabeled · line 1263
$(H_{s}, V_{a}) = s V_{a} = κ B_{a} .$

Theorem: Exact new residual of the chosen lift

Displayed formulas in this passage

Nearby in this section