Complex powers of the wave operator
on asymptotically Minkowski spaces
joint work with Nguyen Viet Dang (Sorbonne Université)

Himeji, March 2023

Michał Wrochna
Cergy Paris Université

Introduction
Consider a Lorentzian manifold (M, g).
The metric has signature (+, −, . . . , −).
2
For instance Minkowski space: R1+d , g0 = dt2 − dy12 − · · · − dyn−1.

The Lorentzian Laplace–Beltrami operator or wave operator:
g =

n−1
X

1

1

|g(x)|− 2 ∂xi |g(x)| 2 g ij (x)∂xji,j=0

On Minkowski space, g = ∂t2 − (∂y21 + · · · + ∂y2n−1 ).
g (+ non-linearity) has rich theory of solving Cauchy problem, asymptotic analysis of solutions, propagation of singularities, etc.

Relatively recently: global theory of g (Fredholm property, Hilbert
space invertibility) Vasy ’13 et al.. Techniques of microlocal andasymptotic analysis in relation with classical dynamics and geometry.
2 / 19

Introduction
Consider a Lorentzian manifold (M, g).
The metric has signature (+, −, . . . , −).
2
For instance Minkowski space: R1+d , g0 = dt2 − dy12 − · · · − dyn−1.

The Lorentzian Laplace–Beltrami operator or wave operator:
g =

n−1
X

1

1

|g(x)|− 2 ∂xi |g(x)| 2 g ij (x)∂xji,j=0

On Minkowski space, g = ∂t2 − (∂y21 + · · · + ∂y2n−1 ).
g (+ non-linearity) has rich theory of solving Cauchy problem, asymptotic analysis of solutions, propagation of singularities, etc.

As opposed to 4g on Riemannian manifold, g is non-elliptic.
How is global g related to geometry of (M, g)?
2 / 19

Spectral zeta function(M, g) compact Riemannian =⇒ 4g has discrete spectrum.
P∞
Recall Riemann zeta ζ(α) = λ=1 λ−α , then spectral zeta:X
C 3 α 7→ ζ4 (α) =λ−α .
λ∈sp(4g )\{0}

Theorem (Minakshisundaram–Pleijel, Seeley)
The function ζ4 (α) = TrL2 4−αis holomorphic on Re α > n2 , withgmeromorphic continuation to α ∈ C and poles at { n2 , n2 − 1, . . . , 1}.

3 / 19

Spectral zeta function(M, g) compact Riemannian =⇒ 4g has discrete spectrum.
P∞
Recall Riemann zeta ζ(α) = λ=1 λ−α , then spectral zeta:X
C 3 α 7→ ζ4 (α) =λ−α .
λ∈sp(4g )\{0}

Theorem (Minakshisundaram–Pleijel, Seeley)
The function ζ4 (α) = TrL2 4−αis holomorphic on Re α > n2 , withgmeromorphic continuation to α ∈ C and poles at { n2 , n2 − 1, . . . , 1}.
+local version with densities:nα 7→ 4−αg (x, x) holomorphic on Re α > 2 , with meromorphic continuationn n
to α ∈ C and poles at { 2 , 2 − 1, . . . , 1}, smooth in x ∈ M .
0−αHere 4−α, sog (x, x ) is the Schwartz kernel of 4
Z

−α
−α
TrL2 4g=
4g (x, x)dx
M
3 / 19

The spectral action principle
The heat kernel expansion (small t expansion of e−t4g (x, x)) relates 4g withinvariants, in particular scalar curvature Rg (x).

Theorem (well-known from elliptic theory)
When dim(M ) = n > 4,R

res TrL2

α= n
2 −1

4−α
g



=

Rg (x)
.
6(4π) Γ n2 − 1
M

n
2

0
Local version for diagonal value x = x0 of Schwartz kernel 4−αg (x, x ):res 4−αg (x, x) =

α= n
2 −1

Rg (x)
.
n6(4π) 2 Γ n2 − 1— This is a “spectral action” for Euclidean gravity: δg Rg = 0 is equivalent
to Einstein equations.
— Poles are geometric ⇒ locality of counterterms in zeta function
regularisation in QFT Hawking ’77
4 / 19Theorem (well-known from elliptic theory / semi-classical analysis)
For any Schwartz function f ,NXf 4g /λ2 (x, x) =λn−2j Cj (f ) aj (x) + O (λn−2N −1 ),j=0where aj (x) are the heat kernel coefficients.
/2 — There is a vectorial version for Dirac operators f D
λ2 .

— Twisting the bundle yields Standard model Lagrangian
Chamseddine–Connes ’97

5 / 19Theorem (well-known from elliptic theory / semi-classical analysis)
For any Schwartz function f ,NXf 4g /λ2 (x, x) =λn−2j Cj (f ) aj (x) + O (λn−2N −1 ),j=0where aj (x) are the heat kernel coefficients.
/2 — There is a vectorial version for Dirac operators f D
λ2 .

— Twisting the bundle yields Standard model Lagrangian
Chamseddine–Connes ’97

But no direct physical meaning unless (M, g) Lorentzian...
Fundamental difficulties: Lorentzian g not elliptic, not bounded frombelow. There is no Lorentzian heat kernel.

5 / 19

So what about g ?
For (M, g) Lorentzian, ∆g becomes g . Two hints:
1. The local geometric quantities (e.g. Rg (x)) still make sense.
— Lorentzian version of local heat kernel coefficients aj (x) bysolving analogous transport equations
2. Recent results show essential self-adjointness of g :— Static spacetimes (e.g. ∂t2 − ∆h with time-independent
coefficients): Dereziński-Siemssen ’18— For perturbations of Minkowski space (and more general
non-trapping Lorentzian scattering spaces):
Vasy ’20 (short-range), Nakamura–Taira ’20 (long-range)(related results: Gérard–Wrochna ’19–’20, Kamiński ’19,
Dereziński-Siemssen ’19, Colin de Verdière–Le Bihan ’20, Taira ’20)
— Asymptotically static spacetimes Nakamura–Taira ’22⇒ f (g ) well-defined!
But is there any relationship between 1. and 2. like in elliptic case?
6 / 19

I. Main results

Main theorem
Assume (M, g) is a (short-range) perturbation of Minkowski space (or moregeneral non-trapping Lorentzian scattering space, see later), of even dimension n.

Theorem(Dang, Wrochna)

For ε > 0, the Schwartz kernel of (g − iε)−α has for Re α > n2 awell-defined on-diagonal restriction (g − iε)−α (x, x), which
extends as a meromorphic function of α ∈ C with poles at { n2 ,nn
2 − 1, 2 − 2, . . . , 1}. Furthermore,limres (g − iε)

ε→0+ α= n2 −1−α(x, x) =Rg (x),ni6(4π) 2 Γ n2 − 1

where Rg (x) is the scalar curvature at x ∈ M .
— Spectral action for gravity. Proof directly in Lorentzian signature.
Perturbations of Minkowski included (no symmetries assumed).
— The ε → 0+ avoids low-frequency problems and responsible forrelationship with Feynman propagator.
7 / 19Main theorem 2
Assume (M, g) is a (short-range) perturbation of Minkowski space (or moregeneral non-trapping Lorentzian scattering space, see later), of even dimension n.

Theorem(Dang, Wrochna)

For any Schwartz f with Fourier transform in ]0, +∞[,NXf (g + iε)/λ2 (x, x) =λn−2j Cj (f ) aj (x) + O (ε, λn−2N −1 ),j=0where aj (x) are Hadamard coefficients.

8 / 19

II. From resolvent to geometric invariants

General plan of proof
1) Let P = g on Lorentzian (M, g). If resolvent exists, (P − iε)−αobtained as contour integral of (P − z)−1 . For α = N + µ > 0:i Im z

γε
ε
2

ε
Re z

(P − iε)−α =

1
2πi

Z

(z − iε)−µ (P − iε)−N (P − z)−1 dz,.

γε2) Construct a Hadamard parametrix HN (z) and show it approximates
the resolvent uniformly in z.
3) Deduce regularity properties, compute poles and get curvature R
from contour integrals of HN (z).
10 / 19

Construction of Hadamard parametrix HN (z):Let Fα (z, |.|g ) be locally given byZ
−α−1 n1Fα (z, x) =eihx,ξi |ξ|2g0 − i0 − zd ξnΓ(α + 1)(2π)(in normal coordinates). Candidate for parametrix of order N :HN (z, .) =NX

uk Fk (z, |.|g ) ∈ D 0 (U ).

k=0solved modulo errors by transport equations thanks to(P − z) (uFα ) = αuFα−1 + (P u)Fα + (hu + 2ρu)Fα−1
2

for all u ∈ C ∞ (M ), where h(x) = bj (x)g0,jk xk and ρ = xk ∂xk .

11 / 19

Hölder–Zygmund and microlocal estimates for Fα (z, |.|η )competition between regularity in x and decay in z
use of Hadamard parametrix wide-spread in QFT; also analytic continuation of
eigenfunctions Zelditch ’18 and Lorentzian local index theory Bär–Strohmaier’20. We followed mostly Hörmander vol. 3 and Sogge ’14
z-dependent Hadamard in compact Riemannian setting used by Sogge ’88,
Dos Santos Ferreira–Kenig–Salo ’14, Bourgain–Shao–Sogge–Yao ’15

Compute poles and get curvature:
Now (P − iε)−α (x, x) expressed by contour integrals of Fβ (z, .).
1
2πi

Z
γε

(z−iε)−α Fk (z, .)dz =(−1)k Γ(−α + 1)
Fk+α−1 (iε, .)Γ(−α − k + 1)Γ(α + k)

scalar curvature in normal coordinates comes from1

P = ∂xk g kj (x)∂xj + g jk (x)(∂xj log |g(x)| 2 )∂xk ,1

transport equation u1 (0) = −P u0 (0) = −P (|g(0)| 4 |g(x)|and gij (x) = g0,ij + 13 Rikjl xk xl + O (|x|3 ).

− 14

)|x=0
12 / 19

Hadamard parametrix HN approximates (P − z)−1 ?
!
NX−1(P − z)uk Fk (z, .)χ = |g| 2 δ∆ + (P uN )FN (z, .)χ + rN (z),k=0where P uN highly regular, and rN singular (but 0 near diagonal).
Applying (P − z)−1 well-defined and yields good errors if (P − z)−1
is shown to have special structure of singularities and mapping
properties uniformly in z.
Think of the distribution (x − i0)−1 on R: it is singular at x = 0, but has good
multiplicative properties like (x − i0)−1 (x − i0)−1 = (x − i0)−2 .

Here, “controlling singularities” means showing existence of
B1 , B2 ∈ Ψ0 (M ), as elliptic as possible s.t.
B1 (P − z)−1 B2∗ : L2 (M ) → C ∞ (M )1with seminorms O(1 + |z|)− 2 . In our case, possible except if B1 forward
connected with B2 (in other words, (P − z)−1 has Feynman wavefront set).
13 / 19

III. Analysis of (P − z)−1

Suppose P = ∂t2 − 4, Im z > 0. Retarded propagator of P − z:√√ei(t−s) −4−z − e−i(t−s) −4−z√
θ(t − s)2i −4 − z
−1

Looks like no chance of k(P − z)−1 k 6 |Im z|. But:

15 / 19

Suppose P = ∂t2 − 4, Im z > 0. Retarded propagator of P − z:√√ei(t−s) −4−z − e−i(t−s) −4−z√
θ(t − s)2i −4 − z
−1

Looks like no chance of k(P − z)−1 k 6 |Im z|. But:“Every particle in Nature has an amplitude to movebackwards in time, and therefore has an anti-particle.”– Richard Feynman√(P − z)−11u (t, .) = −2

Z

e−i|t−s| −4−z√u(s, .)ds.
−4 − z(1)

The boundary value (P − i0)−1 is the Feynman propagator.
But for general P with t-dependent coefficients, nothing like (1) exists...
Start with (1) at infinity, then propagate!
15 / 19

Suppose P = ∂t2 − 4, Im z > 0. Retarded propagator of P − z:√√ei(t−s) −4−z − e−i(t−s) −4−z√
θ(t − s)2i −4 − z
−1

Looks like no chance of k(P − z)−1 k 6 |Im z|. But:“Every particle in Nature has an amplitude to movebackwards in time, and therefore has an anti-particle.”– Richard Feynman√(P − z)−11u (t, .) = −2

Z

e−i|t−s| −4−z√u(s, .)ds.
−4 − z(1)

The boundary value (P − i0)−1 is the Feynman propagator.
Use radial estimates due to Melrose ’94 and Vasy ’13-’19 (or assume g is
a compactly supported perturbation of Minkowski metric g0 ) +propagation estimates Hörmander ’71
15 / 19

Lorentzian scattering spaces
Example: Minkowski metric g0 = dx20 − (dx21 + · · · + dx2n−1 ) on Rn
nextends to radial compactification R defined using boundary-defining1function ρ = (x20 + x21 + · · · + x2n )− 2 . Regularity w.r.t. ρ2 ∂ρ = −∂rDefinition: Lorentzian sc-metrics are C ∞ sections of sc T ∗ M ⊗s sc T ∗ M ,where sc T ∗ M generated by ρ−2 dρ, ρ−1 dy1 , . . . ρ−1 dyn−1 .
Null geodesics lift to null bicharacteristics on sc T ∗ M (rescaled and extended at∂M appropriately)

Definition:(M, g) non-trapping Lorentzian sc-space
if there are sinks/sources L± above ∂M , andnull bicharacteristics flow from and to L− and L+ .
Includes small perturbations of Minkowski
space and asymptotically Minkowski spaces.
16 / 19

From null bicharacteristic flow to global estimates
dynamics of null bicharacteristics in sc T ∗ M⇓
classical quantities increasing along flow⇓
pos. commutator estimates in Ψm,`
sc -calculus⇓
1. Deduce Fredholm property and invertibility of P − z
2. Deduce singularities of (P − z)−1 (x, x0 )

problem of computing WF (P − z)−1 analogous to Dyatlov–Zworski ’16, but
our strategy closer to Vasy–Wrochna ’18 + z-dependent calculus of Shubin’01, parametrix similar to Gérard–Wrochna ’19
work in progress with N.V. Dang and A. Vasy: WF (P − z)−1 directly from
(improved) estimates, also for non-selfadjoint generalisations of the problem
17 / 19

IV. Summary

To sum up...
We have shown relationship of Lorentzian spectral zeta function density
ζg,ε with space-time geometry.
⇒ (Lorentzian!) Gravity can be derived from a spectral action.
I We also get the theorem for ultra-static spacetimes and compactly
supported pertubations. One can conjecture extensions toasymptotically static spacetimes (and beyond, especially if weakening
essential self-adjointness).
I Relationships with QFT on curved spacetimes : (g − iε)−α useful inzeta renormalization.
Remarks: (g − z)−1 is not a retarded or advanced propagator, but a
Feynman propagator: turns out to have better properties in non-linear
problems Gell-Redman–Haber–Vasy ’16.
I Is there a version (even of the Hadamard parametrix) for anti-de Sitter
spacetimes?
Thank you for your attention!
19 / 19

IV. Appendix

Positive commutator estimates
Toy model: P = P ∗ bounded, and ∃ bounded A and D s.t.:[P, iA] > (1 + D2 )s .

(*)

Undo the commutator:
1hAP u, ui − hP Au, uih[P, iA]u, ui =22ihP u, Aui − hAu, P ui=
6 |hP u, Aui|,
2i
By Cauchy–Schwarz,|hP u, Aui| 6 Ck(1 + D2 )−s/2 P ukk(1 + D2 )s/2 uk =: CkP uk−s kuks .
In combination with (*):
kuk2s 6 CkP uk−s kuks ,hence invertibility statement kuks 6 CkP uk−s .

Positive commutator estimates
The existence of suitable A s.t.
[P, iA] > (1 + D2 )s .
is extremely rare. But we can expect to prove it “somewhere in phasespace”.
I If P ∈ Ψs (M ) and A ∈ Ψ` (M ) then [P, iA] ∈ Ψs+`−1 (M ) andσpr ([P, iA]) = {p, a} mod S s+`−2 (M ).
The flow of {p, ·} in {p = 0} is the classical Hamilton flow, or
bicharacteristic flow (note that in {p 6= 0} elliptic theory applies).
I non-compact settings require weighted Sobolev spaces: extra weight(1 + |x|2 )` (Ψm,`sc (M ) calculus)
I non-selfadjointness can be serious trouble (if we know nothing of
P − P ∗ ), or valuable help (for instance P − iε with ε > 0)

Dirac operators2/ satisfies D/ = g + l.o.t. in vector bundle
The Lorentzian Dirac operator D
sense. It is formally self-adjoint w.r.t. the canonical indefinite inner product,but (in general) not for an honest scalar product. However, on Lorentzian2/ satisfies
scattering spaces, P := DP ∗ − P ∈ Ψ1,−1−δ(M )scfor instance for the scalar product h·, γ(n)·iL2 (M ;SM ) used in quantization2/ on non-trapping
work in progress (with N.V. Dang & A. Vasy): P = D
Lorentzian scattering space (M, g) as closed operator.

Conjecture2/ is a closed operator, and:D2/ ) ⊂ R ∪ {some isolated poles in |Im z| 6 R}
sp(DThis uses stronger resolvent estimates using a resolved Ψm,`
sc -calculus
obtained from blowing up the corner of sc T ∗ M .

Dirac operators2/ satisfies D/ = g + l.o.t. in vector bundle
The Lorentzian Dirac operator D
sense. It is formally self-adjoint w.r.t. the canonical indefinite inner product,but (in general) not for an honest scalar product. However, on Lorentzian2/ satisfies
scattering spaces, P := DP ∗ − P ∈ Ψ1,−1−δ(M )scfor instance for the scalar product h·, γ(n)·iL2 (M ;SM ) used in quantization2/ on non-trapping
work in progress (with N.V. Dang & A. Vasy): P = D
Lorentzian scattering space (M, g) as closed operator.

Conjecture2/ is a closed operator, and:D2/ ) ⊂ R ∪ {some isolated poles in |Im z| 6 R}sp(DThe techniques give a fully microlocal implementation of subelliptic
estimate of Taira ’21:
m+ 12 ,`− 12u ∈ Hsc(M ), (P − z)u ∈ Hscm,` (M ) ⇒ u ∈ Hscm,` (M ).

Dirac operators2/ satisfies D/ = g + l.o.t. in vector bundle
The Lorentzian Dirac operator D
sense. It is formally self-adjoint w.r.t. the canonical indefinite inner product,but (in general) not for an honest scalar product. However, on Lorentzian2/ satisfies
scattering spaces, P := DP ∗ − P ∈ Ψ1,−1−δ(M )scfor instance for the scalar product h·, γ(n)·iL2 (M ;SM ) used in quantization2/ on non-trapping
work in progress (with N.V. Dang & A. Vasy): P = D
Lorentzian scattering space (M, g) as closed operator.

Conjecture2/ is a closed operator, and:D2/ ) ⊂ R ∪ {some isolated poles in |Im z| 6 R}sp(DRemark: No role played by indefinite h·, ·iL2 (M ;SM )