Yang{Mills for probabilists - University of Utahfiras/FPD18/Talks/Chatterjee.pdfSourav Chatterjee...

Yang–Mills for probabilists

Sourav Chatterjee

Sourav Chatterjee Yang–Mills for probabilists

Fundamental forces

I The four fundamental forces:

I The electromagnetic force. Interaction of light and matter.I The weak force. Interactions between sub-atomic particles.I The strong force. Force that holds together quarks that form

protons and neutrons.I Gravity.

I Physical theories:I Electromagnetic force: Quantum electrodynamics (QED).

Dirac, Feynman, Tomonaga, Schwinger, Dyson, ...I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ...I Strong force: Quantum chromodynamics (QCD). Yang, Mills,

Gross, Politzer, Wilczek, Wilson, ...I Gravity: General relativity (GR). Einstein.

Fundamental forces

I The four fundamental forces:I The electromagnetic force. Interaction of light and matter.

I The weak force. Interactions between sub-atomic particles.I The strong force. Force that holds together quarks that form

Fundamental forces

I The four fundamental forces:I The electromagnetic force. Interaction of light and matter.I The weak force. Interactions between sub-atomic particles.

I The strong force. Force that holds together quarks that formprotons and neutrons.

I Gravity.

Fundamental forces

I The four fundamental forces:I The electromagnetic force. Interaction of light and matter.I The weak force. Interactions between sub-atomic particles.I The strong force. Force that holds together quarks that form

protons and neutrons.

I Gravity.

Fundamental forces

I Physical theories:

I Electromagnetic force: Quantum electrodynamics (QED).Dirac, Feynman, Tomonaga, Schwinger, Dyson, ...

I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ...I Strong force: Quantum chromodynamics (QCD). Yang, Mills,

Fundamental forces

Dirac, Feynman, Tomonaga, Schwinger, Dyson, ...

I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ...I Strong force: Quantum chromodynamics (QCD). Yang, Mills,

Fundamental forces

Dirac, Feynman, Tomonaga, Schwinger, Dyson, ...I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ...

I Strong force: Quantum chromodynamics (QCD). Yang, Mills,Gross, Politzer, Wilczek, Wilson, ...

I Gravity: General relativity (GR). Einstein.

Fundamental forces

Gross, Politzer, Wilczek, Wilson, ...

I Gravity: General relativity (GR). Einstein.

Fundamental forces

Quantum field theories

I The theories explaining the first three forces are known asquantum field theories.

I The various quantum field theories are unified into one bodyknown as the Standard Model.

I The first quantum field theory was quantum electrodynamics.

I Gave astonishingly accurate predictions, matching up to 10decimal places with experimental data!

I The other theories have similar success stories.

I However, there is no rigorous mathematical foundation forthese theories. (Clay millennium problem of Yang–Millsexistence.)

I Even from the point of view of theoretical physicists, there arevery important unsolved theoretical problems — quarkconfinement, mass gap, etc.

Quantum gravity

I There is no quantum theory of gravity. Finding such a theoryis the holy grail of modern physics.

I Quantum effects are important at very small distances, wheregravity is unimportant.

I Gravity is felt at large distances, between very massive bodies,where quantum effects are negligible.

I So, the unification seems like only an academic goal, exceptthat...

I Both gravity and quantum effects manifest themselves inblack holes. Small black holes can potentially form whenparticles collide with each other at very high speeds, as inparticle accelerators.

Quantum gravity

More about quantum gravity

I In classical physics, particles move along deterministictrajectories.

I In quantum physics, the trajectories are probabilistic(although the notion of probability is replaced by complexprobability amplitudes).

I General relativity is a classical theory, in the sense that thestructure of curved spacetime is deterministic.

I A quantum theory of gravity would replace the fixedspacetime by a randomly fluctuating spacetime (randomRiemannian manifold).

I The most promising approach: String theory.

I Roughly speaking, strings moving randomly trace out randomsurfaces. Higher dimensional strings, known as branes, traceout higher dimensional random manifolds.

Gauge-string duality

I There is a theoretical result of Weinberg and Witten (1980)that it is impossible to generate gravity using quantum fieldtheories in the traditional sense (hard to explain without goinginto details).

I However, Weinberg and Witten have the unstated assumptionthat one is looking for both theories in the same dimension.

I In 1997, Maldacena made the remarkable discovery thatcertain quantum field theories are ‘dual’ to certain stringtheories in one dimension higher!

I Duality means that any calculation in one theory correspondsto some calculation in the other theory.

I Maldacena’s discovery is known as AdS-CFT duality orgauge-string duality.

I The principle of going to one dimension higher is known asthe holographic principle.

Plan for the remainder of the talk

I This concludes the presentation of the physics background.

I In the remaining part of the talk, I will present some concretemathematical problem for probabilists.

I The physics connections will not be discussed in any greatdepth due to time constraints. I will only say one or twosentences for each problem, connecting the math problemswith the physics problems mentioned before.

Quantum Yang–Mills theories

I The quantum field theories describing the electromagnetic,weak and strong forces are known as quantum Yang–Millstheories.

I Quantum Yang–Mills theories are defined in Minkowskispacetime. Euclidean Yang–Mills theories are ‘Wick-rotated’quantum Yang–Mills theories that are defined in Euclideanspacetime.

I They are formally probability measures on spaces ofconnections on certain principal bundles.

I They have lattice analogs, known as lattice gauge theories,that are rigorously defined probabilistic models.

I Euclidean Yang–Mills theories are supposed to be scalinglimits of lattice gauge theories.

Constructive field theory

I The problem of rigorously constructing Euclidean Yang–Millstheories, and then using them to construct quantumYang–Mills theories, is the problem of Yang–Mills existence.

I A standard approach is via the program of constructivequantum field theory.

I The plan there is to first define Euclidean Yang–Mills theoriesas probability measures on appropriate spaces of generalizedfunctions; then show that these probability measures satisfycertain axioms (the Osterwalder–Schrader axioms); this wouldthen imply that the theory can be ‘quantized’ to obtain thedesired quantum Yang–Mills theories.

I The constructive field theory program waged a valiant battlefor more than thirty years (1960–1990), making sense ofvarious quantum field theories in two and three dimensions,but never quite reached its ultimate goal of constructing 4Dquantum Yang–Mills theories. May be revival possible?

I The plan there is to first define Euclidean Yang–Mills theoriesas probability measures on appropriate spaces of generalizedfunctions;

then show that these probability measures satisfycertain axioms (the Osterwalder–Schrader axioms); this wouldthen imply that the theory can be ‘quantized’ to obtain thedesired quantum Yang–Mills theories.

I The plan there is to first define Euclidean Yang–Mills theoriesas probability measures on appropriate spaces of generalizedfunctions; then show that these probability measures satisfycertain axioms (the Osterwalder–Schrader axioms);

this wouldthen imply that the theory can be ‘quantized’ to obtain thedesired quantum Yang–Mills theories.

Towards the formal definition of Euclidean YM theories

I Ingredients: dimension n and gauge group G .

I Assume that G is a closed subgroup of U(N) for some N.I Examples (all in n = 4):

I Quantum electrodynamics: G = U(1).I Electroweak theory: G = SU(2).I Quantum chromodynamics: G = SU(3).

I Let g be the Lie algebra of G .

I Then g is a subspace of the space of all N × Nskew-Hermitian matrices.

I Assume that G is a closed subgroup of U(N) for some N.

I Examples (all in n = 4):I Quantum electrodynamics: G = U(1).I Electroweak theory: G = SU(2).I Quantum chromodynamics: G = SU(3).

I Quantum electrodynamics: G = U(1).

I Electroweak theory: G = SU(2).I Quantum chromodynamics: G = SU(3).

I Quantum electrodynamics: G = U(1).I Electroweak theory: G = SU(2).

I Quantum chromodynamics: G = SU(3).

Connections and curvature

I A smooth G connection form on Rn is a smooth map from Rn

into gn.

I If A is a G connection form, its value A(x) at x is an n-tuple(A1(x), . . . ,An(x)) of skew-Hermitian matrices. In thelanguage of differential forms,

A =n∑

Ajdxj .

I The curvature form F of A is the g-valued 2-form

F = dA + A ∧ A .

I This means that at each x , F (x) is an n × n array ofskew-Hermitian matrices of order N, whose (j , k)th entry isthe matrix

Fjk(x) =∂Ak

∂xj−∂Aj

∂xk+ [Aj(x),Ak(x)].

into gn.I If A is a G connection form, its value A(x) at x is an n-tuple

(A1(x), . . . ,An(x)) of skew-Hermitian matrices. In thelanguage of differential forms,

A =n∑

Ajdxj .

F = dA + A ∧ A .

Fjk(x) =∂Ak

∂xj−∂Aj

A =n∑

Ajdxj .

F = dA + A ∧ A .

Fjk(x) =∂Ak

∂xj−∂Aj

A =n∑

Ajdxj .

F = dA + A ∧ A .

Fjk(x) =∂Ak

∂xj−∂Aj

The Yang–Mills action

I Let A be the space of all smooth G connection forms on Rn.The Yang–Mills action on this space is the function

SYM(A) := −∫Rn

Tr(F ∧ ∗F ),

where F is the curvature form of A and ∗ denotes the Hodgestar operator, assuming that this integral is finite.

I Explicitly, this is

SYM(A) = −∫Rn

n∑j ,k=1

Tr(Fjk(x)2)dx .

The Yang–Mills action

I Let A be the space of all smooth G connection forms on Rn.The Yang–Mills action on this space is the function

SYM(A) := −∫Rn

Tr(F ∧ ∗F ),

where F is the curvature form of A and ∗ denotes the Hodgestar operator, assuming that this integral is finite.

I Explicitly, this is

SYM(A) = −∫Rn

n∑j ,k=1

Tr(Fjk(x)2)dx .

Formal definition of Euclidean YM theories

I The Euclidean Yang–Mills theory with gauge group G on Rn

is formally described as the probability measure

dµ(A) =1

(− 1

4g2SYM(A)

where:

I A ∈ A, the space of all smooth G connection forms on Rn,I SYM is the Yang–Mills action,I

dA =n∏

∏x∈Rn

d(Aj(x))

is ‘infinite-dimensional Lebesgue measure’ on A,I g is a parameter called the coupling strength, andI Z is the normalizing constant that makes this a probability

measure.

dµ(A) =1

(− 1

4g2SYM(A)

where:I A ∈ A, the space of all smooth G connection forms on Rn,

I SYM is the Yang–Mills action,I

dA =n∏

∏x∈Rn

d(Aj(x))

measure.

dµ(A) =1

(− 1

4g2SYM(A)

where:I A ∈ A, the space of all smooth G connection forms on Rn,I SYM is the Yang–Mills action,

dA =n∏

∏x∈Rn

d(Aj(x))

measure.

dµ(A) =1

(− 1

4g2SYM(A)

where:I A ∈ A, the space of all smooth G connection forms on Rn,I SYM is the Yang–Mills action,I

dA =n∏

∏x∈Rn

d(Aj(x))

is ‘infinite-dimensional Lebesgue measure’ on A,

I g is a parameter called the coupling strength, andI Z is the normalizing constant that makes this a probability

measure.

dµ(A) =1

(− 1

4g2SYM(A)

dA =n∏

∏x∈Rn

d(Aj(x))

is ‘infinite-dimensional Lebesgue measure’ on A,I g is a parameter called the coupling strength, and

I Z is the normalizing constant that makes this a probabilitymeasure.

dµ(A) =1

(− 1

4g2SYM(A)

dA =n∏

∏x∈Rn

d(Aj(x))

measure.

Problem

I The above description of Euclidean Yang–Mills theory withgauge group G is not directly mathematically meaningfulbecause of the problems associated with the definitionLebesgue measure on A.

I While it has been possible to give rigorous meanings to similardescriptions of Brownian motion and various quantum fieldtheories in dimensions two and three, 4D EuclideanYang–Mills theories have so far remained largely intractable.

Problem

I The above description of Euclidean Yang–Mills theory withgauge group G is not directly mathematically meaningfulbecause of the problems associated with the definitionLebesgue measure on A.

I While it has been possible to give rigorous meanings to similardescriptions of Brownian motion and various quantum fieldtheories in dimensions two and three, 4D EuclideanYang–Mills theories have so far remained largely intractable.

Lattice gauge theories

I Wilson (1974) proposed a discretization of EuclideanYang–Mills theories, now known as lattice gauge theories.

I The lattice gauge theory with gauge group G on a finite setΛ ⊆ Zn is defined as follows.

I Suppose that for any two adjacent vertices x , y ∈ Λ, we havea group element U(x , y) ∈ G , with U(y , x) = U(x , y)−1.

I Let G (Λ) denote the set of all such configurations.I A square bounded by four edges is called a plaquette. Let

P(Λ) denote the set of all plaquettes in Λ.I For a plaquette p ∈ P(Λ) with vertices x1, x2, x3, x4 in

anti-clockwise order, and a configuration U ∈ G (Λ), define

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

I The Wilson action of U is defined as

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

I Let G (Λ) denote the set of all such configurations.

I A square bounded by four edges is called a plaquette. LetP(Λ) denote the set of all plaquettes in Λ.

I For a plaquette p ∈ P(Λ) with vertices x1, x2, x3, x4 inanti-clockwise order, and a configuration U ∈ G (Λ), define

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

P(Λ) denote the set of all plaquettes in Λ.

I For a plaquette p ∈ P(Λ) with vertices x1, x2, x3, x4 inanti-clockwise order, and a configuration U ∈ G (Λ), define

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

SW(U) :=∑

p∈P(Λ)

Re(Tr(I − Up)).

Definition of lattice gauge theory

I Let σΛ be the product Haar measure on G (Λ).

I Given β > 0, let µΛ,β be the probability measure on G (Λ)defined as

dµΛ,β(U) :=1

Ze−βSW(U)dσΛ(U),

where Z is the normalizing constant.

I This probability measure is called the lattice gauge theory onΛ for the gauge group G , with inverse coupling strength β.

I An infinite volume limit of the theory is a weak limit of theabove probability measures as Λ ↑ Zn.

I The infinite volume limit may or may not be unique.

I The uniqueness (or non-uniqueness) is in general unknown forlattice gauge theories in dimension four when β is large.

dµΛ,β(U) :=1

From LGT to Euclidean YM theory: Wilson’s heuristic

I Discretize Rn as εZn for some small ε.

I Take a G connection form A =∑n

j=1 Ajdxj .

I Let e1, . . . , en denote the standard basis vectors of Rn.

I For a directed edge (x , x + εej) of εZn, define

U(x , x + εej) := eεAj (x),

and let U(x + εej , x) := U(x , x + εej)−1.

I This defines a configuration of unitary matrices assigned todirected edges of εZn.

j=1 Ajdxj .

Wilson’s heuristic, continued

I By the Baker–Campbell–Hausdorff formula for products ofmatrix exponentials, one can derive the formal approximation

SW(U) ≈ −ε4−n

4SYM(A).

I The above heuristic was used by Wilson to justify theapproximation of Euclidean Yang–Mills theory by lattice gaugetheory, scaling the inverse coupling strength β like ε4−n as thelattice spacing ε→ 0.

Wilson’s heuristic, continued

I By the Baker–Campbell–Hausdorff formula for products ofmatrix exponentials, one can derive the formal approximation

SW(U) ≈ −ε4−n

4SYM(A).

I The above heuristic was used by Wilson to justify theapproximation of Euclidean Yang–Mills theory by lattice gaugetheory, scaling the inverse coupling strength β like ε4−n as thelattice spacing ε→ 0.

Scaling in dimension four

I The most important dimension is n = 4, because spacetime isfour-dimensional.

I In the above formulation, β does not scale with ε at all whenn = 4.

I Currently, however, the general belief in the physicscommunity is that β should scale like some multiple oflog(1/ε) in dimension four.

Wilson loops

I Suppose that we have a lattice gauge theory on Λ ⊆ Zn withgauge group G .

I Given a loop γ with directed edges e1, . . . , em, the Wilsonloop variable Wγ is defined as

Wγ := Tr(U(e1)U(e2) · · ·U(em)).

Wilson loops

I Suppose that we have a lattice gauge theory on Λ ⊆ Zn withgauge group G .

I Given a loop γ with directed edges e1, . . . , em, the Wilsonloop variable Wγ is defined as

Wγ := Tr(U(e1)U(e2) · · ·U(em)).

Open problem #1: Yang–Mills existence

I The problem has many parts.

I But the main step is to show that 4D non-Abelian latticegauge theories have nontrivial continuum limits.

I The description of the limit is part of the problem.

I The most important groups are SU(2) and SU(3).

I Large body of work in 2D. Less in 3D. Almost none in 4D,except for a very long series of papers by Ba laban that peoplefind very difficult to understand. May be someone can take offfrom where Ba laban stopped? Or revive the project usingdifferent ideas?

Open problem #2: Yang–Mills mass gap

I Again, the problem has many parts, but the main step is toshow that 4D non-Abelian lattice gauge theories haveexponential decay of correlations at any β.

I There are standard techniques for showing exponential decayof correlations at small β (e.g. by Dobrushin’s condition).Showing exponential decay at large β is conjectured for manymodels in statistical physics, but most of these problems,including the YM mass gap, are open.

I Even physicists do not think they have a proof of mass gap.

I A solution of this problem will explain, roughly speaking, whymass exists in the universe.

Open problem #3: Quark confinement

I Suppose that we are given a 4D non-Abelian lattice gaugetheory.

I Show that for any β, there are constants C (β) and c(β) suchthat for any loop γ,

|〈Wγ〉| ≤ C (β)e−c(β)area(γ),

where 〈Wγ〉 is the expected value of the Wilson loop variableWγ and area(γ) is the minimal surface area enclosed by γ.

I Showing for rectangles is good enough.I There is a proof at small β by Osterwalder & Seiler (1978).I Proof at large β for 3D U(1) theory by Gopfert and Mack

(1982).I Disproof at large β for 4D U(1) theory by Guth (1980) and

Frohlich & Spencer (1982).I Proof of this conjecture will explain why we do not observe

free quarks in nature. This is one of the biggest mysteries ofparticle physics.

I Showing for rectangles is good enough.

I There is a proof at small β by Osterwalder & Seiler (1978).I Proof at large β for 3D U(1) theory by Gopfert and Mack

I Showing for rectangles is good enough.I There is a proof at small β by Osterwalder & Seiler (1978).

I Proof at large β for 3D U(1) theory by Gopfert and Mack(1982).

I Disproof at large β for 4D U(1) theory by Guth (1980) andFrohlich & Spencer (1982).

I Proof of this conjecture will explain why we do not observefree quarks in nature. This is one of the biggest mysteries ofparticle physics.

(1982).

I Disproof at large β for 4D U(1) theory by Guth (1980) andFrohlich & Spencer (1982).

Frohlich & Spencer (1982).

Open problem #4: Gauge-string duality

I Recall that gauge-string duality is an attempt to unifyquantum field theories and gravity.

I Technically speaking, this problem can be discussed only aftersolving the problem of YM existence.

I The main step is to show that Wilson loop expectations in acontinuum Yang–Mills theory can be expressed as integralsover trajectories of strings in a string theory, where thetrajectories are in one dimension higher.

Some results about gauge-string duality

I Consider SO(N) lattice gauge theory on Zn, n arbitrary.

I In recent work (C., 2015 and C. & Jafarov, 2016), we gave aformula for Wilson loop expectations in this theory asasymptotic series expansions in 1/N, where each coefficient inthe series arises as a sum over trajectories in a certain latticestring theory, where the trajectories are in Zn+1.

I This proves a version of gauge-string duality and theholographic principle. Possibly the first rigorous result.

I The expansion was proved only at small β (strong coupling).Will be a very important breakthrough to prove somethingsimilar at large β.

I In 2D, the terms were explicitly evaluated by Basu & Ganguly(2016) using combinatorial techniques. May be the techniquescan extend to higher dimensions?

The master loop equation

The following is a generalization of what are called Makeenko–Migdalequations or master loop equations. They hold at all β, and give the startingpoint for the proof of the 1/N expansion and gauge-string duality.

Theorem (C., 2015)Consider SO(N) LGT on Zn. For a collection of loops s = (`1, . . . , `m), define

φ(s) :=〈W`1W`2 · · ·W`m 〉

Let |s| be the total number of edges in s. Then

(N − 1)|s|φ(s) =∑

s′∈T−(s)

φ(s ′)−∑

s′∈T+(s)

φ(s ′) + N∑

s′∈S−(s)

φ(s ′)

− N∑

s′∈S+(s)

φ(s ′) +1

∑s′∈M−(s)

φ(s ′)− 1

∑s′∈M+(s)

φ(s ′)

+ Nβ∑

s′∈D−(s)

φ(s ′)− Nβ∑

s′∈D+(s)

φ(s ′),

where T±, S±, M± and D± are certain operations that produce new collectionsof loops from old.

The master loop equation

The following is a generalization of what are called Makeenko–Migdalequations or master loop equations. They hold at all β, and give the startingpoint for the proof of the 1/N expansion and gauge-string duality.

Theorem (C., 2015)Consider SO(N) LGT on Zn. For a collection of loops s = (`1, . . . , `m), define

φ(s) :=〈W`1W`2 · · ·W`m 〉

Let |s| be the total number of edges in s. Then

(N − 1)|s|φ(s) =∑

s′∈T−(s)

φ(s ′)−∑

s′∈T+(s)

φ(s ′) + N∑

s′∈S−(s)

φ(s ′)

− N∑

s′∈S+(s)

φ(s ′) +1

∑s′∈M−(s)

φ(s ′)− 1

∑s′∈M+(s)

φ(s ′)

+ Nβ∑

s′∈D−(s)

φ(s ′)− Nβ∑

s′∈D+(s)

φ(s ′),

where T±, S±, M± and D± are certain operations that produce new collectionsof loops from old.

Final remarks

I I have a very recent preprint on arXiv, with the same title asthis talk, that contains most of this talk in greater detail.

I The preprint also has an extensive review of the mathematicalliterature on these topics, which I did not cover in this talk.

I Special thanks to David Brydges, Erhard Seiler and SteveShenker for teaching me most of what I know aboutYang–Mills theories, lattice gauge theories and quantum fieldtheories.

Final remarks

Yang{Mills for probabilists - University of Utahfiras/FPD18/Talks/Chatterjee.pdfSourav Chatterjee...

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