Locality

Mathematical reflections onlocality
Sylvie Paycha
University of Potsdam

joint work with Li Guo and Bin Zhang

Women in Mathematics, RIMS, Kyoto, 8th September2022

Locality

I. The principle of locality revisitedLocality

The principle of locality states that an object is influenced directly
only by its immediate surroundings.
Thus, one can separate events located in different regions ofspace-time and should be able to measure them independently.
Our aim
Propose a mathematical framework which encompasses themain features of the locality principle in QFT;
use this framework to carry out renormalisation (evaluate
meromorphic germs at their poles) in accordance with thelocality principle.

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Causal separation
Light cone, past and future
Pd−1
In the Minkowski space (Rd , g ), where g (x, y ) = −x0 y0 + j=1 xj yj is
the Lorentzian scalar product, there is a notion of "past" and "future":(picture downloaded from Wikipedia)

Two sets S1 and S2 are causally separated (S1 kS2 ) if and only if
Si does not lie in the future of Sj for i 6= j.

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Locality in axiomatic QFT

The Wightman field ϕ : S(Rd ) → O(H) obeys the locality axiom
Supp(f1 )kSupp(f2 ) =⇒ [ϕ(f1 ), ϕ(f2 )] = 0.

(1)

The (relative) scattering matrix Sf satisfies the locality condition
Supp(f1 )kSupp(f2 )=⇒ Sf (f1 + f2 ) = Sf (f1 ) Sf (f2 )=⇒ [Sf (f1 ), Sf (f2 )] = 0.

(2)

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Mathematical interpretation
We introduce two binary relations
on sets:
O1 >0 O2 ⇔ [O1 , O2 ] = 0,(3)

f1 >f2 ⇔ Supp(f1 )kSupp(f2 ).

(4)

on test functions:Interpretation of (1) as a locality map (see later)
f1 >f2 =⇒ ϕ(f1 )>0 ϕ(f2 ).

(5)

Interpretation of (2) as a locality morphism (see later)
f1 >f2 =⇒ Sf (f1 + f2 ) = Sf (f1 ) Sf (f2 ).

(6)

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II. Locality as a symmetric binary relation

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Algebraic locality
Definition of locality
A locality set is a couple (X , >) where X is a set and > ⊆ X × X is asymmetric relation on X , called locality relation (or independencerelation) of the locality set.
x1 >x2 ⇐⇒ (x1 , x2 ) ∈ >, ∀x1 , x2 ∈ X .

First examples of locality
X >Y ⇐⇒ X ∩Y = ∅ on subsets X , Y of a set Z .
X >Y ⇐⇒ X ⊥Y on subsets X , Y of an euclidean vector space V .

(almost-)Separation of supports
Let U ⊂ Rn be an open subset and  ≥ 0. Two functions φ, ψ ∈ D(U)are independent i.e., φ > ψ whenever d (Supp(φ), Supp(ψ)) > .
For  = 0, this amounts to disjointness of supports, otherwise to-separation of supports.

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Further examples
Probability theory: independence of events
Given a probability space P := (Ω, Σ, P) and two events A, B ∈ Σ:A > B ⇐⇒ P(A ∩ B) = P(A) P(B).

Geometry: transversal manifolds
Given two submanifolds L1 and L2 of a manifold M:
L1 > L2 ⇐⇒ L1 t L2 ⇐⇒ Tx L1 + Tx L2 = Tx M ∀x ∈ L1 ∩ L2 .

Number theory: coprime numbers
Given two positive integers m, n in N:
m > n ⇐⇒ m ∧ n = 1.

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Partial products
Locality set: (X , >),Polar set: U > := {x ∈ X , x > u∀u ∈ U} for U ⊆ X ;Graph of the locality relation: > = {(x1 , x2 ) ∈ X 2 ,Partial product: mX : X × X ⊃ > −→ X i.e.

x1 > x2 };mX (>) ⊂ X .

(X , mX , >) locality semi-groupsemi-group condition: ∀U ⊆ X ,mX (U > × U > ) ∩ > ⊆ U >or equivalently(x1 >u1 and x2 >u2∀u1 , u2 ∈ U) =⇒ (mX (x1 , x2 )>w∀w ∈ U) .

Counterexample
Equip R with the locality relation x > y ⇐⇒ x + y 6∈Z.
(R, >, +) is NOT a locality semi-group: for U = {1/3} we have (1/3, 1/3) ∈ (U > × U > ) ∩ > but
1/3+1/3 = 2/3∈U/ >Locality

Locality category
Locality structures
set X

locality set (X , >);semi-group (X , mX )locality semi-group (X , mX , >, );vector space (V , +, ·)locality vector space (V , +, ·, >)(U ⊂ V =⇒ U > vector space);
algebra (A, +, ·, mA )locality algebra (A, +, ·, mA , >).

Locality morphisms: f : (X , >X ) → (Y , >Y )locality map:
(f × f )(>X ) ⊂ >Yor equivalently x1 >X x2 =⇒ f (x1 )>Y f (x2 );locality semi-group morphism f : (X , mX , >X ) → (Y , mY , >Y ):f is a locality map such that
x1 >X x2 =⇒ f (mX (x1 , x2 )) = mY (f (x1 ), f (x2 )).
······

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III. Evaluating meromorphic germs at poles in QFT

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Functions of several variables in QFT
Speer’s analytic renormalisation [JMP 1967] revisited
Eugene Speer considers Feynman amplitudes given by the coefficients ofthe perturbation-series expansion of the S matrix in a Lagrangian field
theory (with non zero mass).

Excerpt of Speer’s article
In this paper we apply a method of defining divergent quantities which
was originated by Riesz and has been used in various contexts by manyauthors. [....] We find it necessary to consider functions of several
complex variables z1 , · · · , zk , one associated with each line of theFeynman graph. The main difficulty is the extension of the above[Riesz’s] treatment of poles to the more complicated singularities which
occur in several complex variables...

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Brain teaser(We assume the poles are at zero)
Speer shows [Theorem 1] that the divergent expressions lie in the filteredFeyn(Ck ) consisting of Feynman
algebra MFeyn (C∞ ) := ∪∞k=1 Mkfunctions f : C → C,Xh(z1 , · · · , zk )f =,L=zj , Ji ⊂ {1, · · · , k}, h holom. at zeroiss
L1 1 · · · Lm mj∈Ji

Questions:
1How to evaluate f consistently at the poles z1 = · · · = zk = 0?

2What freedom of choice do we have for the evaluator?

Evaluating a fraction with a linear pole at zero
1?
z1 − z20?
f (z1 , z2 ) ==z1 + z2 |z1 =0,z2 =0 10000 ?

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Speer’s generalised evaluators
They consist of a family E = {Ek , ∈ N} of linear forms
Ek : MFeyn (Ck ) → C, compatible with the filtration, which fulfill thefollowing conditions
1(extend ev0 ) E is the ordinary evaluation ev0 at zero on holom.
germs;

2(partial multiplicativity) E(f1 · f2 ) = E(f1 ) · E(f2 ) if f1 and f2
depend on different sets (later called independent) of variables zi ;3

E is invariant under permutations of the variables Ek ◦ σ ∗ = Ek forany σ ∈ Σk , with σ ∗ f (z1 , · · · , zk ) := f (zσ(1) , · · · , zσ(k) );4(continuity) If fn (~zk ) · Ls11 · · · Lsmm

uniformly−→ g (~zk ) as holomorphicn→∞germs, then Ek (fn ) −→ Ek ( lim fn ).
n→∞

n→∞

Drawback: Speer’s approach depends on the choice of coordinates
z1 , · · · , zk , · · · .

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IV. Locality on meromorphic germs comes to the rescue

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Back to the locality principle in QFT
kWe consider M := M(C∞ ) := ∪∞k=1 M(C ) consisting of meromorphic
functions/germs f : Ck → C with linear poles at zero,f =h(z1 , · · · , zk ),L1 s 1 · · · Lm s m

Li linear in z1 , · · · , zk , h holom. at zero

Aim: evaluate meromorphic germs at poles according to the principle oflocality: "two events separated in space can be measured independently"
Principle of locality: factorisation on independent events=⇒Meas|a and{z b} independent factorisation∈A(a ∨ b) = Meas(a) · Meas(b).
| {z }

concatenation

We shall later equip M with a locality relation >;Principle of locality revisited: locality evaluatorsf > g =⇒ E(f · g ) = E(f ) E(g ) for two meromorphic germs f and g in an
appropriate subalgebra M• of M.

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Locality on/independence of meromorphic
germs
Meromorphic germs with linear poles,`m )M(Ck ) 3 f = h(`Ls11,···, h holomorphic germ, si ∈ Z≥0 ,···Lsn
1knk`i : C → C, Lj : C → C linear forms with real coefficients (lie in L(Ck )).

Locality on meromorphic germs: orthogonality
Dependence set Dep(f ) := h`1 , · · · , `m , L1 , · · · , Ln i.
Q inner product on Rk induces one on L(Ck )f1 ⊥Q f2 ⇐⇒ Dep(f1 )⊥Q Dep(f2 ).
kQpolar germs: M•Q− (C ) 3 f ⇐= h ⊥ Li for all i = 1, · · · , n.

Theorem: (L. Guo, S.-P., B. Zhang/ N. Berline, M. Vergne 2015)
QkM• (Ck ) = M+ (Ck ) ⊕ M•Q− (C )

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Where we standDataM• , ⊥Q an (locality) algebra of meromorphic germs at zerowith a prescribed type of poles (e.g. Chen ⊂ Speer ⊂Feynman);M+ ⊂ M• the algebra of holomorphic germs at zero;
the evaluation at zero: ev0 : M+ → C;
the Galois group GalQ (M• /M+ ) of (locality)
isomorphisms of M• , ⊥Q ;hQM•Q− is generated by polar germs f = g with h ⊥ g .

Orthogonal projection⊥Q induces a splittingM• = M+ ⊕Q M•Q−and π+ Q : M• −→ M+

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Theorem [Guo, S.P., Zhang 2022]
Definition
A locality evaluator at zero E : M• −→ C is a linear form which i)
extends the ordinary evaluation ev0 at zero and ii) factorises on
independent germs (or is a locality character):

f1 ⊥Q f2 =⇒ E(f1 )⊥Q E(f2 ).
Example: Minimal subtraction scheme:
E MS :π+ Q

ev0M• −→ M+ −→C is a locality evaluator.

Theorem
Given an inner product Q, a locality evaluator at zero E : M• −→ C is of
the form: E = ev0 ◦ π+ Q ◦TE.
|{z}
| {z }E MS

GalQ (M• /M+ )Locality

THANK YOU FOR YOUR ATTENTION!

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P. Clavier, L. Guo, B. Zhang and S. P., An algebraic formulationof the locality principle in renormalisation, European Journal ofMathematics, Volume 5 (2019) 356-394
P. Clavier, L. Guo, B. Zhang and S. P., Renormalisation via
locality morphisms, Revista Colombiana de Matemáticas, Volume 53(2019) 113-141
P. Clavier, L. Guo, B. Zhang and S. P., Renormalisation andlocality: branched zeta values, in "Algebraic Combinatorics,Resurgence, Moulds and Applications (Carma)" Vol. 2 ,Eds. F.
Chapoton, F. Fauvet, C. Malvenuto, J.-Y. Thibon, Irma Lectures inMathematics and Theoretical Physics 32, European Math. Soc.
(2020) 85–132
P. Clavier, L. Guo, B. Zhang and S. P., Locality andrenormalisation: universal properties and integrals on trees, Journalof Mathematical Physics 61, 022301 (2020)

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L. Guo, B. Zhang and S. P., Renormalisation and theEuler-Maclaurin formula on cones, Duke Math J., 166 (3) (2017)537–571.
L. Guo, B. Zhang and S. P., A conical approach to Laurent
expansions for multivariate meromorphic germs with linear poles,
Pacific Journal of Mathematics 307 (2020) 159–196.

L. Guo, B. Zhang and S. P., Galois groups of meromorphic
germs and multiparameter renormalisation (2022) (Preprint)
L. Guo, B. Zhang and S. P., Mathematical reflections onlocality (2022) (Preprint)
R. Dahmen, A. Schmeding and S. P., A topological splittingof the space of meromorphic germs in several variables andcontinuous evaluators, arXiv:2206.13993 (2022)