Weyl-Frame Two-Point Functions

• Weyl-frame two-point functions are defined by employing Weyl rescaling to reinterpret correlators in conformally related settings like flat Robertson–Walker, AdS, and de Sitter spacetimes.
• In Maxwell theory, a tailored gauge condition transforms the Robertson–Walker formulation to the Minkowski Lorenz gauge, enabling the transport of the standard Gupta–Bleuler two-point function.
• For AdS scalars and linearized gravitons, the method either employs a spectral decomposition with Bessel functions or recasts observables in terms of the Weyl tensor to isolate gauge-invariant content.

Searching arXiv for the cited papers to ground the article in current bibliographic records. “Weyl-frame two-point functions” are two-point correlation functions formulated by exploiting a Weyl-related conformal frame in which the field equations, gauge conditions, or spectral decomposition become simpler. In the materials considered here, the notion appears in three distinct but structurally related settings: Maxwell theory in flat Robertson–Walker spacetime, scalar quantum fields in anti-de Sitter space written in Poincaré coordinates, and gauge-invariant graviton observables characterized by the linearized Weyl-tensor two-point function in Minkowski space and in the Poincaré patch of de Sitter space. Across these cases, the common principle is that the underlying dynamical content is preserved or clarified by passing to a conformally related description, while gauge-fixing conditions, potential-level correlators, and the choice of basic observables remain frame-dependent or representation-dependent (Huguet et al., 2013).

1. Definition and conceptual scope

In the present literature, the expression “Weyl frame” does not denote a single standardized formalism. Rather, it refers to using a metric representation related by a Weyl rescaling,

$g_{\mu\nu}=\Omega^2 \tilde g_{\mu\nu},$

in order to construct or reinterpret a two-point function.

The clearest example is the flat Robertson–Walker geometry

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$

which is Weyl-related to Minkowski space by

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$

In that setting, the Maxwell equations in $d=4$ are conformally invariant, while the gauge conditions are not. The two-point function is therefore most efficiently obtained by choosing a Robertson–Walker gauge whose conformal image is the ordinary Lorenz gauge in Minkowski space, quantizing there, and transporting the result back (Huguet et al., 2013).

A related but not identical use of the idea occurs in anti-de Sitter space. In Poincaré coordinates the metric takes the form

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$

so AdS is explicitly presented as a Weyl rescaling of flat half-space. In that frame, the bulk scalar two-point function diagonalizes into a Källén–Lehmann-type superposition of $(d-1)$-dimensional Minkowski two-point functions with Bessel radial profiles. This is the construction most directly adapted to a Weyl-frame reading, even though the paper does not explicitly use that terminology (Moschella, 21 Dec 2025).

In linearized gravity, the phrase acquires a different emphasis. The relevant object is not a potential-level correlator transported between conformal frames, but the linearized Weyl-tensor two-point function itself. In Minkowski space and in the Poincaré patch of de Sitter space, the gauge-invariant graviton two-point function defined by compactly supported transverse smearings is shown to be equivalent to the Weyl-tensor two-point function. This identifies the Weyl tensor as the natural gauge-invariant carrier of the physical content of the graviton correlator in those settings (Higuchi, 2012).

2. Maxwell theory in a Weyl-related Robertson–Walker frame

For flat Robertson–Walker spacetime written in conformal time,

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$

the conformal factor is encoded through

$W_\mu:=\partial_\mu \ln a^2.$

When $a=a(\tau)$ only,

$W_\tau=2\mathcal H,\qquad \mathcal H=\frac{a'}{a},$

and the spatial components vanish. The vector potential $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$0, treated as a one-form, is assigned conformal weight zero, so under the Weyl map it is carried to the same coordinate components. The Maxwell equations are conformally invariant in $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$1, but the gauge conditions are not (Huguet et al., 2013).

This distinction is decisive. If one imposes the ordinary Robertson–Walker Lorenz gauge,

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$2

the gauge-fixed field equation becomes

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$3

which in the conformal Minkowski chart takes the form

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$4

The authors explicitly state that they were not able to find a useful mode basis for these equations.

The simplification comes from reversing the logic of the conformal map. Instead of quantizing directly in the Robertson–Walker Lorenz gauge, one chooses the gauge

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$5

For $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$6,

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$7

This “$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$8-gauge” is engineered so that, under the Weyl transformation to the Minkowski frame, it becomes simply

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$9

The gauge-fixed equations then reduce in the conformal frame to

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$0

which is the standard flat-space Gupta–Bleuler problem.

The corresponding gauge-fixing term added to the Lagrangian is

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$1

and the Euler–Lagrange equations become

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$2

with

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$3

When the gauge condition holds, these reduce to the Maxwell equations.

The significance of this construction is that it isolates the part of the theory that is conformally natural—the Maxwell dynamics—from the part that is frame-dependent—the gauge-fixing sector. The resulting two-point function is therefore not a direct solution of the difficult Robertson–Walker Lorenz-gauge mode problem, but a conformal transport of the ordinary Minkowski Gupta–Bleuler two-point function.

3. Weyl transport of the Maxwell two-point function

Because the conformal-frame equations are

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$4

the mode functions are the standard plane waves,

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$5

Positive frequency is defined with respect to the Minkowski Killing field $g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$6, and the induced state on the Robertson–Walker side is the conformal vacuum. The physical-state condition is the Gupta–Bleuler constraint

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$7

With a polarization basis satisfying

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$8

the Wightman function in the Minkowski frame is

$g_{\mu\nu}=a^2(\tau)\eta_{\mu\nu}.$9

where $d=4$0 is the two-point function of the conformal scalar in Minkowski space. The scalar two-point function transforms as

$d=4$1

Substituting this relation yields the Robertson–Walker expression

$d=4$2

The paper further rewrites the result intrinsically in Robertson–Walker variables as its Eq. (13),

$d=4$3

The scanned text is described as degraded, but the practical content is unambiguous: the vector two-point function is obtained by conformal dressing of the flat-space expression with the scalar conformal factors and the Robertson–Walker metric data (Huguet et al., 2013).

A central structural point is that the transport applies not only to the classical field equation but to the quantization data as well. The paper argues that the solution spaces of the gauge-fixed equations in Robertson–Walker and Minkowski frames are identical as function spaces, and that the sesquilinear products coincide because

$d=4$4

hence

$d=4$5

This is the precise mechanism by which the Minkowski quantization is transported to the Robertson–Walker spacetime.

4. AdS Poincaré coordinates and spectral Weyl-frame representations

In anti-de Sitter space, the key Weyl-frame structure is supplied by the Poincaré patch. The coordinates $d=4$6, $d=4$7, lead to the induced metric

$d=4$8

This is explicitly a Weyl rescaling of the flat half-space metric. The bulk invariant two-point function, however, is first constructed globally and covariantly in embedding-space terms from holomorphic plane waves

$d=4$9

on the universal cover and associated tuboid/chiral-cone domains (Moschella, 21 Dec 2025).

The scalar two-point function is required to satisfy covariance, locality, positive definiteness, and the strong spectral condition formulated as normal analyticity in $ds^2=\frac{1}{u^2}(ds_M^2-du^2),$0. Under these conditions, the reduced function extends analytically to the covering of

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$1

The coordinate-free integral representation is

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$2

with a relative homology cycle

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$3

What makes the Poincaré patch specifically relevant to Weyl-frame two-point functions is the exact diagonalization of this AdS correlator into Minkowski building blocks. In Poincaré coordinates, the paper obtains

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$4

Here $ds^2=\frac{1}{u^2}(ds_M^2-du^2),$5 is the $ds^2=\frac{1}{u^2}(ds_M^2-du^2),$6-dimensional Minkowski Wightman function and the radial dependence is carried by Bessel functions. The same structural decomposition is given for the Euclidean Schwinger function and for the Lorentzian Feynman propagator by replacing the Minkowski leaf correlator with its Euclidean or Feynman counterpart.

This shows that the AdS bulk two-point function can be read as a superposition of flat-space correlators on the Weyl-related leaves. The external factor

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$7

is the natural conformal prefactor associated with the metric $ds^2=\frac{1}{u^2}(ds_M^2-du^2),$8, while the Bessel functions encode the radial mode content. The paper does not formulate this using an explicit Weyl-rescaled field language, but it states the radial equation

$ds^2=\frac{1}{u^2}(ds_M^2-du^2),$9

from which that interpretation follows directly.

A plausible implication is that, in this setting, a Weyl-frame two-point function is less a simple pullback formula than a spectral representation adapted to the conformally flat foliation. The frame reveals how full AdS covariance can coexist with a decomposition into lower-dimensional flat correlators.

5. Weyl-tensor correlators and gauge-invariant graviton two-point functions

For linearized gravity, the physically relevant two-point function is not the unsmeared Wightman function

$(d-1)$0

by itself, because $(d-1)$1 is gauge dependent. Instead, the gauge-invariant graviton two-point function is defined by smearing against smooth, compactly supported, symmetric, transverse test tensors,

$(d-1)$2

with

$(d-1)$3

The central result is that in Minkowski space and in the Poincaré patch of de Sitter space this gauge-invariant smeared graviton two-point function is completely determined by the two-point function of the linearized Weyl tensor (Higuchi, 2012).

In Minkowski space, the linearized Weyl tensor is

$(d-1)$4

with $(d-1)$5 a constant. The proof proceeds by showing that any compactly supported symmetric smearing tensor can be represented in the form

$(d-1)$6

where $(d-1)$7 has the algebraic symmetries appropriate for coupling to the Weyl tensor. After integration by parts, the smeared graviton two-point function is thereby rewritten as a doubly smeared Weyl–Weyl correlator.

In the Poincaré patch of de Sitter space, with metric

$(d-1)$8

the corresponding exactness statement is

$(d-1)$9

For any solution $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$0 of the linearized Einstein equation,

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$1

one then has

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$2

and hence

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$3

This is the exact equivalence formula.

The role of the Weyl tensor here is conceptually distinct from the Weyl rescalings used in the Maxwell and AdS settings. “Weyl-frame” in the conformal sense is not the operative concept; instead, the Weyl tensor supplies the gauge-invariant observable that captures the full physical content of the graviton two-point function under the stated smearing conditions. The commonality is that both uses isolate the invariant content from the representation-dependent one.

6. Hadamard behavior, positivity, and interpretive limits

The Maxwell construction in flat Robertson–Walker space explicitly yields a two-point function with Hadamard short-distance behavior. The argument is that the singular structure is controlled by the conformal scalar factor $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$4, and because

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$5

while near coincidence

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$6

the singularity is inherited from the Minkowski scalar two-point function. The standard Hadamard leading singularity

$ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$7

is therefore reproduced up to smooth tensor factors. The paper does not give a full DeWitt–Schwinger coefficient expansion; its claim is restricted to the short geometric argument establishing the correct short-distance structure (Huguet et al., 2013).

In AdS, the corresponding analytic control is expressed not in Hadamard language in the supplied material, but through covariance, locality, positive definiteness, and maximal analyticity on the covering of the cut plane. Positivity is not automatic for both invariant Klein–Gordon solutions associated with $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$8 and $ds^2=a^2(\tau)\bigl(d\tau^2-d\mathbf{x}^2\bigr),$9; analyticity and locality allow both, but positive definiteness selects the physically admissible quantizations. The paper also emphasizes covering-space subtleties: for generic non-integer $W_\mu:=\partial_\mu \ln a^2.$0, the plane waves are globally meaningful only on the universal cover $W_\mu:=\partial_\mu \ln a^2.$1, whereas for integer $W_\mu:=\partial_\mu \ln a^2.$2 they descend to $W_\mu:=\partial_\mu \ln a^2.$3 (Moschella, 21 Dec 2025).

In the graviton case, the main interpretive issue is the difference between gauge-dependent propagators and gauge-invariant observables. The paper stresses that common graviton propagators in de Sitter gauges may diverge, tend to a constant, or show logarithmic growth at large separation, but that such behavior is not obviously physical because it pertains to $W_\mu:=\partial_\mu \ln a^2.$4, which is gauge dependent. By contrast, the Weyl-tensor two-point function is gauge invariant, and the cited result is that in the Poincaré patch of de Sitter it decays as

$W_\mu:=\partial_\mu \ln a^2.$5

so the gauge-invariant graviton two-point function defined through compactly supported transverse smearings decays in the same way (Higuchi, 2012).

A common misconception is that conformal simplification automatically renders all structures frame-independent. The papers collectively show the opposite. For Maxwell theory, the field equations are conformally invariant but the gauge condition is not. For AdS scalars, the Poincaré patch makes the Weyl-related foliation explicit, but full covariance still depends on the global analytic construction on the covering space. For gravitons, the invariant object is not the potential correlator in a preferred conformal frame but the Weyl-tensor correlator obtained after imposing the correct gauge-invariant smearing procedure.

7. Synthesis across the three settings

The three cases exhibit a shared methodology with different emphases.

Setting	Weyl-related structure	Two-point-function consequence
Flat Robertson–Walker Maxwell field	$W_\mu:=\partial_\mu \ln a^2.$6	Minkowski Gupta–Bleuler quantization is transported to Robertson–Walker spacetime
AdS scalar field in Poincaré patch	$W_\mu:=\partial_\mu \ln a^2.$7	Bulk correlator becomes a Källén–Lehmann-type superposition of Minkowski correlators
Minkowski / de Sitter graviton	conformally flat backgrounds with vanishing background Weyl tensor	Gauge-invariant graviton correlator is equivalent to the Weyl-tensor correlator

In the Maxwell case, the decisive object is the frame-adapted gauge condition

$W_\mu:=\partial_\mu \ln a^2.$8

whose conformal image is the ordinary Minkowski Lorenz gauge. In the AdS case, the decisive object is the Poincaré-coordinate spectral decomposition

$W_\mu:=\partial_\mu \ln a^2.$9

which rewrites the fully covariant bulk correlator in flat-space terms. In the graviton case, the decisive step is the exact representation of admissible smearing tensors in Weyl-type form, leading to the equivalence with the linearized Weyl–Weyl Wightman function.

Taken together, these results support a technically precise understanding of Weyl-frame two-point functions: they are correlators whose construction, simplification, or physical interpretation is controlled by passage to a Weyl-related description. Sometimes this means literal conformal transport of a flat-space correlator, as for the Maxwell field in flat Robertson–Walker spacetime. Sometimes it means a spectral decomposition adapted to a conformally flat foliation, as in AdS Poincaré coordinates. Sometimes it means that the physically meaningful correlator is best expressed in terms of the Weyl tensor rather than the gauge-dependent potential, as for linearized gravity in Minkowski and de Sitter space. The unifying lesson is that conformal structure can simplify the representation of two-point functions, but it does not erase the distinction between invariant dynamics, frame-dependent gauge choices, and genuinely gauge-invariant observables.