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Finite heavy-scalar Gaussian matching

Physical scope

Exact deduction for a finite positive quadratic heavy-scalar regulator and spectrally band-limited sources; no interacting continuum theory or empirical result is asserted.

Statement

Assume the Axiom of Choice. Let $d \geq 1$ , equip $R^{d}$ with inner product $u^{T} v$ and norm $∥ u ∥ = (u^{T} u)^{1 / 2}$ , let $A$ be a real symmetric positive-semidefinite $d \times d$ matrix, let $M > 0$ , and put $K = M^{2} I + A$ , where $I$ is the identity on $R^{d}$ . Then $K$ is positive definite. Use natural units with $χ$ in mass-inverse units, $A$ and $K$ in mass-squared units, and a source $J \in R^{d}$ in mass units. Let $E \geq 0$ . Define $C^{d} = R^{d} + i R^{d}$ with Hermitian inner product $⟨ z, w ⟩ = \sum_{k = 1}^{d} z_{k} \overline{w_{k}}$ and the complex-linear map $A_{C} (x + i y) = A x + i A y$ . Real symmetry and nonnegativity make $A_{C}$ self-adjoint and nonnegative; use its normalized spectral projection-valued measure $E_{A_{C}}$ , whose Borel-set values are orthogonal projections, whose total value is the identity, and which is strongly countably additive, as in the positive heavy-mode Gaussian matching-data definition. Set $P_{E} : = E_{A_{C}} ([0, E^{2}])$ , restricted to $R^{d}$ , and assume $J = P_{E} J$ . Thus $J$ lies in the sum of the eigenspaces of $A$ with eigenvalues in $[0, E^{2}]$ . Let $γ_{K^{- 1}} = N_{d} (0, K^{- 1})$ be the centered normalized Gaussian law whose every linear projection $u^{T} χ$ is normal with mean zero and variance $u^{T} K^{- 1} u$ , and define

$Z_{H} (J) : = \int_{R^{d}} e^{J^{T} χ} d γ_{K^{- 1}} (χ)$

$Δ S_{W} (J) : = - \log Z_{H} (J)$

For each integer $N \geq 0$ , put

S_{N} J : = M^{- 2} \sum_{r = 0}^{N} (- A / M^{2})^{r} J .

Then the finite Gaussian expectation $Z_{H} (J)$ is finite and positive, and

$Z_{H} (J) = \exp ⁣ (\frac{1}{2} J^{T} K^{- 1} J)$

$Δ S_{W} (J) = - \frac{1}{2} J^{T} K^{- 1} J$

The inverse-kernel and action remainders satisfy

∥ K^{- 1} J - S_{N} J ∥ \leq M^{- 2} (E^{2} / M^{2})^{N + 1} ∥ J ∥,

$∣ Δ S_{W} (J) + \frac{1}{2} J^{T} S_{N} J ∣ \leq \frac{∥ J ∥^{2}}{2 M^{2}} (E^{2} / M^{2})^{N + 1}$

These are exact conditional statements for a finite positive quadratic heavy sector. The remainder is small in the stated band when $E < M$ ; no claim is made for unrestricted sources, arbitrary interactions, a continuum measure, or a differentiated Wilsonian flow.

Statement references

Finite-regulator EFT model and scale data

Finite-regulator Wilsonian action

Finite-regulator derivative expansion and truncation

Positive heavy-mode Gaussian matching data

Finite-dimensional heavy Gaussian source shift

Low-momentum matching has a controlled remainder

Facts & Assumptions

Given: AC; finite $d \geq 1$ ; real symmetric $A \geq 0$ ; $M > 0$ ; $E \geq 0$ ; and $J = P_{E} J$ .

[A1]

The Axiom of Choice: AC is the assumption used by the finite Gaussian law and density suppliers.

[A2]

AC supplies countable selections and prescribed serial paths: AC supplies countable choice, as required by the positive spectral-calculus supplier.

[F1]

Finite-regulator EFT model and scale data: the regulator has a finite nonempty vertex set $X$ , retained coordinates $ℓ \in R^{X}$ , integrated coordinates $χ \in R^{X}$ , and a real Borel ultraviolet action with a specified Borel probability measure for $χ$ . In its positive Gaussian submodel, $A$ is real symmetric positive semidefinite and has range at most one on the regulator graph, $\sqrt{∥ A ∥} \leq Λ$ , $K = M^{2} I + A$ , and a source $J (ℓ)$ has mass units and is supported in the spectral band $[0, E^{2}]$ . The positive scales $μ_{W}, M, Λ$ and the bound $E \geq 0$ are separately named and have no imposed numerical relations.

[F2]

Finite-regulator Wilsonian action: at fixed retained coordinates, the Wilsonian action is the negative logarithm of the finite high-coordinate integral under the specified normalized measure.

[F3]

Finite-regulator derivative expansion and truncation: $S_{N} J = M^{- 2} \sum_{r = 0}^{N} (- A / M^{2})^{r} J$ .

[F4]

Positive heavy-mode Gaussian matching data: for a nonnegative self-adjoint $A$ and $M > 0$ , $K = M^{2} I + A$ , $K^{- 1} = (M^{2} + A)^{- 1}$ , and $Δ S_{e f f} (J) = - \frac{1}{2} ⟨ J, K^{- 1} J ⟩$ .

[F5]

Finite-dimensional heavy Gaussian source shift: for finite $d$ , symmetric positive-definite $K$ , and $j \in R^{d}$ , the normalized finite Gaussian source integral is $\exp (\frac{1}{2} j^{T} K^{- 1} j)$ .

[F6]

Low-momentum matching has a controlled remainder: for $J = P_{E} J$ , the truncation above obeys the stated vector and action remainder bounds; the useful small-parameter regime is $E < M$ .

[A3]

For the Wilsonian interpretation, take $X = {1, \dots, d}$ with the complete graph, choose $Λ > 0$ with $\sqrt{∥ A ∥} \leq Λ$ and any $μ_{W} > 0$ , and fix the retained configuration $ℓ_{*} = 0 \in R^{X}$ . Use the constant Borel source map $J (ℓ^{'}) \equiv J$ for all $ℓ^{'} \in R^{X}$ , identify the integrated coordinate $h$ of the Wilsonian-action definition with $χ$ , and set the normalized Borel Gaussian probability $ν_{H} = γ_{K^{- 1}}$ and $S_{U V} (ℓ^{'}, χ) = - J (ℓ^{'})^{T} χ$ . The complete graph makes the given matrix range at most one, and the constant-source action is real-valued and Borel; since $J = P_{E} J$ , this source map obeys the stated band restriction. In particular, $J = J (ℓ_{*})$ .

Proof

technique · finite Gaussian source shift followed by the exact positive-spectral remainder theorem

Given: The hypotheses in the Statement.

1.1givenF1F4algebra

For every nonzero $x \in R^{d}$ , $x^{T} K x = M^{2} ∥ x ∥^{2} + x^{T} A x > 0$ , so $K$ is symmetric positive definite and the finite Gaussian supplier applies. On the complexification, real symmetry makes $A$ self-adjoint; for $x, y \in R^{d}$ , the cross terms in $⟨ x + i y, A (x + i y) ⟩$ cancel because $x^{T} A y = y^{T} A x$ , leaving $x^{T} A x + y^{T} A y \geq 0$ . Thus the positive-operator hypotheses of [F4] also hold.

2.1A1A3F2F5givenstep 1.1algebra

In the model specialization [A3], $e^{- S_{U V} (ℓ_{*}, χ)} = e^{J (ℓ_{*})^{T} χ} = e^{J^{T} χ}$ and $d ν_{H} (χ) = d γ_{K^{- 1}} (χ)$ . Therefore the Wilsonian integral in [F2] is exactly

Z_{H} (ℓ_{*}) = \int_{R^{d}} e^{- S_{U V} (ℓ_{*}, χ)} d ν_{H} (χ) = \int_{R^{d}} e^{J^{T} χ} d γ_{K^{- 1}} (χ) = Z_{H} (J) .

By [F5] and step 1.1 this value is finite and strictly positive, so $ℓ_{*}$ lies in the domain of the Wilsonian-action definition. Hence $S_{W} (ℓ_{*}; μ_{W}) = - \log Z_{H} (ℓ_{*}) = - \log Z_{H} (J) = Δ S_{W} (J)$ . At zero source the normalized Gaussian integral is $1$ , so this is also the source-induced action increment with zero baseline; no determinant factor remains.

3.1A1A2F3F4F6givenstep 1.1step 2.1∎

By [F4] and step 2.1, the heavy Gaussian effective-action contribution equals $Δ S_{W} (J)$ . Apply [F6] to the band-limited source $J = P_{E} J$ and the positive operator verified in step 1.1; its vector and action inequalities are the two displayed remainder bounds. When the additional condition $E < M$ holds, the factor $E^{2} / M^{2}$ is less than one and the bound decreases geometrically with $N$ . This conclusion is conditional and remains within the finite Gaussian sector.

Finite heavy-scalar Gaussian matching

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources