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Engineering scaling of the Gaussian gradient and quartic actions

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ , meaning every sequence $(X_{n})_{n \in N}$ of nonempty sets has a function $f$ with $f (n) \in X_{n}$ for every $n$ . Let integer $d \geq 1$ , dimensionless scale factor $b > 0$ , and real function $φ^{'} \in C_{c}^{\infty} (R^{d})$ be given, where $C_{c}^{\infty} (R^{d})$ denotes the functions $ψ : R^{d} \to R$ whose partial derivatives of every order exist and are continuous and whose support $supp ψ : = \overline{{x : ψ (x) \neq 0}}$ is compact. Define

$φ_{b} (x) : = b^{- (d - 2) / 2} φ^{'} (x / b), x \in R^{d} .$

For any $ψ \in C_{c}^{\infty} (R^{d})$ , define its Gaussian gradient action and quartic integral by

$S_{2} [ψ] : = \frac{1}{2} \int_{R^{d}} ∣ \nabla ψ (x) ∣^{2} d λ_{d} (x), Q_{4} [ψ] : = \frac{1}{4!} \int_{R^{d}} ψ (x)^{4} d λ_{d} (x),$

where $\nabla$ is the vector of first partial derivatives. Then

$S_{2} [φ_{b}] = S_{2} [φ^{'}], Q_{4} [φ_{b}] = b^{4 - d} Q_{4} [φ^{'}] .$

Consequently a coefficient $g$ multiplying $Q_{4}$ transforms to $g^{'} = b^{4 - d} g$ when the action is re-expressed in the rescaled coordinate $x^{'} = x / b$ . This is engineering power counting on smooth test fields; it neither constructs an interacting probability measure nor proves an interacting critical exponent.

Statement references

The Axiom of Countable Choice ( ${A C}_{ω}$ )

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$

Facts & Assumptions

Given: ${A C}_{ω}$ , integer $d \geq 1$ , $b > 0$ , and a real smooth compactly supported $φ^{'}$ on $R^{d}$ .

[F1]

Under ${A C}_{ω}$ , a $C^{1}$ diffeomorphism changes a nonnegative Lebesgue integral by its absolute Jacobian (The Axiom of Countable Choice ( ${A C}_{ω}$ ), A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

[F2]

The derivative of a composition is given by the chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

Proof

1.1givenF1F2algebra

The map $T (x^{'}) = b x^{'}$ is a $C^{1}$ diffeomorphism of $R^{d}$ with Jacobian determinant $b^{d}$ . By [F2], each partial derivative satisfies $\partial_{x_{j}} φ_{b} (x) = b^{- (d - 2) / 2} b^{- 1} (\partial_{x_{j}^{'}} φ^{'}) (x / b) = b^{- d / 2} (\partial_{x_{j}^{'}} φ^{'}) (x / b) .$ Thus $∣ \nabla φ_{b} (x) ∣^{2} = b^{- d} ∣ \nabla φ^{'} (x / b) ∣^{2}$ . Applying [F1] with $T$ to this nonnegative integrand gives $S_{2} [φ_{b}] = \frac{1}{2} \int_{R^{d}} b^{- d} ∣ \nabla φ^{'} (x^{'}) ∣^{2} b^{d} d λ_{d} (x^{'}) = S_{2} [φ^{'}] .$ The compact support and smoothness assumptions make these integrands measurable and finite.

2.1givenF1algebra∎

The same change of variables gives $Q_{4} [φ_{b}] = \frac{1}{4!} \int_{R^{d}} b^{- 2 (d - 2)} φ^{'} (x / b)^{4} d λ_{d} (x) = b^{d - 2 (d - 2)} Q_{4} [φ^{'}] = b^{4 - d} Q_{4} [φ^{'}] .$ Therefore $g Q_{4} [φ_{b}] = g^{'} Q_{4} [φ^{'}]$ with $g^{'} = b^{4 - d} g$ , as claimed.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Statistical Field Theory, complete Cambridge lecture notes