Quantum Inequalities from Operator Product Expansions Henning - - PowerPoint PPT Presentation

Quantum Inequalities from Operator Product Expansions

Henning Bostelmann

Università di Roma “Tor Vergata”

June 6, 2008 Work in progress with C. J. Fewster, York

Aim of the talk

◮ We consider quantum field theory (on Minkowski space). ◮ Certain expressions which are positive in classical field theory

[φ(x)2] have counterparts in quantum theory which are not positive [:φ 2: (x)].

◮ But certain bounds hold for smeared fields: Quantum inequalities

◮ For a real scalar free field, :φ 2: (f) ≥ −cf 1. ◮ Usually considered for the energy density. ◮ Concepts and proof methods are based on canonical

commutation relations.

◮ What is the correct generalization for interacting fields?

◮ Even conceptually unclear - what is the square? ◮ Formal perturbation theory does not apply.

◮ The talk presents a solution based on the operator product

expansion.

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

An inequality for the Wick square (1/2)

◮ Let φ be a real scalar free field; φ(t) ≡ φ(t,0). ◮ Let ρ be an energy-bounded state. Consider the kernel

F(t,t′) := ρ(φ(t)φ(t′)). It is positive-definite, i.e. 0 ≤

• dt dt′ F(t,t′)g(t)g(t′) = ρ(φ(g)∗φ(g)).

◮ Same holds if F is multiplied by a kernel K of positive type; e.g.

K(t − t′) = (ıπ(t − t′ −ı0))−1.

◮ Use Wick ordering; new variables s = (t + t′)/2, s′ = t − t′:

F(t,t′) = ρ(:φ 2: (s))+∆+(s′)+ R(s,s′).

◮ Here ∆+(s′) is the vacuum two-point function; R(s,s′) is smooth

in s′, vanishes at s′ = 0.

An inequality for the Wick square (2/2)

◮ We obtain

0 ≤ cg + Rρ,g +

• ds ds′ ρ(:φ 2: (s))

ıπ(s′ −ı0) g(s + s′/2)g(s − s′/2)

where cg is independent of ρ; and Rρ,g becomes small when suppg is small.

◮ In fact, for real-valued g, there are simplifications:

◮ g(s + s′/2)g(s − s′/2) is symmetric in s′; ◮ Rρ,g = 0; ◮ (ıπ(s′ −ı0))−1 can be replaced with its symmetric part, δ(s′).

◮ Since the estimate is now independent of ρ, one has:

:φ 2: (g2) ≥ −cg1.

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

Problems when generalizing quantum inequalities

◮ The inequalities have been proved in linear QFT, on flat and

curved spacetime, and in low-dimensional conformal QFT.

◮ Several issues with generalizations to theories with

self-interactions:

◮ Heuristical: “Simple” quantum inequalities cannot hold in

interacting theories (Olum/Graham)

◮ Conceptual: There’s no such thing as a Wick square. ◮ Technical: Proof methods are essentially based on CCR.

◮ Interacting quantum field theories are mostly handled in formal

perturbation theory.

◮ All quantities (fields, S-matrix elements, expectation values. . . )

are formal power series in a “coupling constant”.

◮ No control about convergence (radius of convergence: 0). ◮ For establishing inequalities, we’d have to define when a formal

power series is positive.

When is a formal power series positive? (1/2)

◮ Take a formal power series with real coefficients.

P[g] =

∞

∑

k=0

ckgk, ck ∈ R.

◮ Let us explore some possible definitions for “P[g] ≥ 0”.

When is a formal power series positive? (1/2)

◮ Take a formal power series with real coefficients.

P[g] =

∞

∑

k=0

ckgk, ck ∈ R.

◮ Let us explore some possible definitions for “P[g] ≥ 0”. ◮ Definition 1: P[g] ≥ 0 iff P[g] = Q[g]2 with some formal power

series Q[g].

When is a formal power series positive? (1/2)

◮ Take a formal power series with real coefficients.

P[g] =

∞

∑

k=0

ckgk, ck ∈ R.

◮ Let us explore some possible definitions for “P[g] ≥ 0”. ◮ Definition 1: P[g] ≥ 0 iff P[g] = Q[g]2 with some formal power

series Q[g].

◮ Somewhat abstract, but seems natural (positivity in ∗-algebras). ◮ May be useful.

When is a formal power series positive? (1/2)

◮ Take a formal power series with real coefficients.

P[g] =

∞

∑

k=0

ckgk, ck ∈ R.

◮ Let us explore some possible definitions for “P[g] ≥ 0”. ◮ Definition 1: P[g] ≥ 0 iff P[g] = Q[g]2 with some formal power

series Q[g].

◮ Somewhat abstract, but seems natural (positivity in ∗-algebras). ◮ May be useful.

◮ Definition 2: P[g] ≥ 0 iff P[g] = g2n ∑∞

k=0 dkgk, n ∈ N0, d0 > 0.

When is a formal power series positive? (1/2)

◮ Take a formal power series with real coefficients.

P[g] =

∞

∑

k=0

ckgk, ck ∈ R.

◮ Let us explore some possible definitions for “P[g] ≥ 0”. ◮ Definition 1: P[g] ≥ 0 iff P[g] = Q[g]2 with some formal power

series Q[g].

◮ Somewhat abstract, but seems natural (positivity in ∗-algebras). ◮ May be useful.

◮ Definition 2: P[g] ≥ 0 iff P[g] = g2n ∑∞

k=0 dkgk, n ∈ N0, d0 > 0.

◮ Roughly: P positive iff lowest-order coefficient is positive. ◮ But order 0 corresponds to free field theory. ◮ Not useful; does not capture the effects of interaction.

When is a formal power series positive? (1/2)

◮ Take a formal power series with real coefficients.

P[g] =

∞

∑

k=0

ckgk, ck ∈ R.

◮ Let us explore some possible definitions for “P[g] ≥ 0”. ◮ Definition 1: P[g] ≥ 0 iff P[g] = Q[g]2 with some formal power

series Q[g].

◮ Somewhat abstract, but seems natural (positivity in ∗-algebras). ◮ May be useful.

◮ Definition 2: P[g] ≥ 0 iff P[g] = g2n ∑∞

k=0 dkgk, n ∈ N0, d0 > 0.

◮ Roughly: P positive iff lowest-order coefficient is positive. ◮ But order 0 corresponds to free field theory. ◮ Not useful; does not capture the effects of interaction.

◮ In fact, 1 ⇔ 2 for P = 0.

When is a formal power series positive? (2/2)

Should we consider the following formal power series positive? P[g] =

∞

∑

k=0

(−1)k (2k)! g2k

◮ As a convergent series: Yes, but only for small g. ◮ As a formal series without convergence information: Can only

refer to infinitesimal g.

◮ Order-0 coefficient decides. ◮ Can’t compare the different orders. ◮ Effects of interaction (finite g) are not captured.

◮ Crucial question for physics: Does the physical value of g fall into

the radius of convergence of

√

P?

◮ This information is not accessible in formal perturbation theory.

Nonperturbative approach

◮ Formal perturbation theory does not give the right answer. ◮ Thus, to treat quantum inequalities, we start in a nonperturbative

setting.

◮ Wightman; Haag/Kastler

◮ Pick a field product, say φ(g)∗φ(g) ≥ 0. ◮ Use the operator product expansion to obtain inequalities.

◮ This also tells us what the normal product (“Wick ordering”) is:

composite fields appearing in the OPE.

Idea for obtaining quantum inequalities

◮ There’s one positivity property which survives in the general

case: 0 ≤ φ(g)∗φ(g). 0 ≤

• dt dt′K(t − t′)φ ∗(t)φ(t′)g(t)g(t′).

◮ Expand here φ ∗(t)φ(t′) ≈ ∑N

k=1 ck(t − t′)φk((t + t′)/2)

◮ Insert above, use s = (t + t′)/2 and s′ = (t − t′) to obtain

0 ≤

N

∑

k=1

• ds

fk(s)

• ds′K(s′)ck(s′)g(s + s′/2)g(s − s′/2) φk(s)

=

N

∑

k=1

φk(fk)

◮ So we obtain an inequality between several fields, not just two.

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

Conceptual setting

◮ While we want to describe point fields, it is more natural to start in

the algebraic setting (Haag/Kastler).

◮ local algebras of bounded operators A(O) for spacetime regions

O; operators are “nonlinear in the fields”

◮ Describe how Wightman fields are associated with the local

algebras.

◮ Involves phase space property (assumption!), rather than

computations in a concrete model.

◮ Derive a rigorous version of the operator product expansion.

◮ Obtain detailed estimates on the short-distance expansion.

◮ Use this to obtain inequalities, following the heuristic idea.

Axioms of algebraic QFT (in the vacuum sector)

There is

◮ a Hilbert space H, ◮ to every open region O, a W ∗ algebra A(O) ⊂ B(H), ◮ a strongly continuous unitary representation (x,Λ) → U(x,Λ) of

the Poincaré group on H such that:

◮ A(O1) ⊂ A(O2) whenever O1 ⊂ O2 (isotony); ◮ [A1,A2] = 0 whenever O1 O2, and Ai ∈ A(Oi) (locality); ◮ U(x,Λ)A(O)U(x,Λ)∗ = A(ΛO+ x) (covariance); ◮ The spectrum of the generators of the translation group U(x,1)

falls into the closed forward light cone (positivity of energy);

◮ There is a unique invariant vector Ω for all U(x,1) (the vacuum).

These are very general assumptions; one often adds more specific

• nes (and we will, too!)

Short distance behaviour

◮ We want to consider inequalities between pointlike fields φ(x)

(not elements of the C* algebras).

◮ Must analyze the structure of A(O) for small O. ◮ However, ∩O∋xA(O) = C1. ◮ No bounded operators exist at a point! Unbounded objects must

turn up.

◮ We will use an analogue of the Taylor expansion.

◮ If f is a smooth function on R, and pγ its γ-order Taylor polynomial

at 0:

|f(x)− pγ(x)| ≤ f (γ+1)∞O(|x|γ+1)

as |x| → 0.

◮ Note that ψγ : C∞(R) → L1([−1,1]), f → pγ[f] is linear,

continuous (in the Schwartz topology), and of finite rank.

Taylor expansions in QFT

◮ Set out from a net of C∗ algebras, O → A(O) ⊂ B(H), subject to

the axioms above.

◮ Let Σ := B(H)∗ be the Banach space of normal functionals, with

the norm topology.

◮ Let H ≥ 0 be the generator of time translations (the Hamiltonian),

and R := (1+ H)−1.

◮ For σ ∈ Σ, set σ(ℓ) = σ(R−ℓ · R−ℓ) (where it exists). ◮ Set C∞(Σ) := {σ ∈ Σ | σ(ℓ) < ∞ for all ℓ > 0}, equipped with

the Fréchet topology induced by all those norms.

◮ Let Ξ be the inclusion map C∞(Σ) → Σ. ◮ Let Or be the standard double cone of radius r around 0.

Definition

A noncommutative Taylor expansion is a family {ψγ}γ≥0 of continuous linear finite-rank maps ψγ : C∞(Σ) → Σ with the following property. For every γ ≥ 0, there exists ℓ > 0 such that

(Ξ−ψγ)⌈A(Or)(ℓ) = o(rγ).

Example: Free field

For a real scalar free field φ in physical space-time, such expansion exists and reads as follows. (h is a specific test function in the time-0 plane.)

Ξ = (Ω| · |Ω) · 1

• γ = 0

+ ı(Ω|[ ˙ φ(h), ·]|Ω) · φ(0)

• γ = 1

+ ı

3

∑

k=1

(Ω|[ ˙ φ(xkh), ·]|Ω) · ∂kφ(0) − ı(Ω|[φ(h), ·]|Ω) · ∂0φ(0) − 1

2(Ω|[ ˙

φ(h),[ ˙ φ(h), ·]]|Ω) · :φ 2:(0)                    γ = 2 + ...

Consequences of the short-distance expansion

◮ In the general case, take it as an additional axiom that a

noncommutative Taylor expansion exists (microscopic phase space condition).

◮ Consequences of this condition:

◮ If the rank of ψγ is minimized, then imgψ∗

γ ⊂ C∞(Σ)∗ consists of

pointlike fields. (They fulfill the Wightman axioms after smearing.)

◮ We get all pointlike fields in this way. ◮ Fields can be approximated (with precise estimates) by bounded

• perators.

◮ We can use this to transfer properties of the algebras to the

fields; in particular we obtain products and an operator product expansion.

Precise formulation of the OPE

◮ Let φ be a local field, obtained from imgψ∗ γ as above. ◮ Let K be of positive type, and a boundary value of an analytic

function in the lower half-plane.

◮ Set Π(s′) = K(s′)φ ∗(s′/2)φ(−s′/2). ◮ Fd,m :=

• f ∈ S(R)
• suppf ⊂ [−d,d];∀n ≤ m : f (n)1 ≤ d−n

.

Theorem

There exist ℓ > 0, m ∈ N, and a finite-dimensional projector p in C∞(Σ)∗ onto a space of pointlike fields such that sup

f∈Fd,m

Rℓ Π(f)− pΠ(f)

• Rℓ

d→0

− − → 0.

◮ For a more usual form, write p = ∑k σk(·)φk in a basis:

pΠ(s′) = ∑

k

K(s′)σk(φ ∗(s′/2)φ(−s′/2))φk = ∑

k

Ck(s′)φk.

Quantum inequalities

◮ Now use the positivity relation (in matrix elements):

0 ≤

• dt dt′K(t − t′)φ ∗(t)φ(t′)g(t)g(t′).

◮ Insert the product expansion Π ≈ pΠ. ◮ For large ℓ, m, and g ∈ Fd,m, one obtains:

∑

k

Rℓ φk(fk)Rℓ ≥ −ε(d)1, where fk =

• ds′ Ck(s′)g(s + s′/2)g(s − s′/2),

Ck(s′) = K(s′)σk(φ ∗(s′/2)φ(−s′/2)). and ε(d) → 0 as d → 0.

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

Are these inequalities “trivial” or “nontrivial”?

◮ The quantum inequalities obtained hold between bounded

• perators:

∑

k

Rℓφk(fk)Rℓ ≥ −ε(d)1,

ε(d) → 0 as d → 0.

◮ This kind of inequalities can be “trivial”, e.g.: A ≥ −A. ◮ In our context, an individual term k would be called “trivial” if

sup

g∈Fd,m

Rℓφk(fk)Rℓ → 0 as d → 0.

For in this case, we could leave it out of the sum.

◮ Such trivial terms could always occur!

◮ The OPE is an approximation – we might have picked too many

terms with negligible contribution at small scales.

Answer to the (non)triviality problem

◮ We need to minimize on the number of approximating terms. ◮ Given a product K(t − t′)φ ∗(t)φ(t′), choose the minimal space

Vmin of composite fields needed for approximating it in the OPE.

◮ This is a well defined notion. ◮ We can at least show that this space is never {0}.

◮ Obtain a corresponding inequality,

∑

k

Rℓφk(fk)Rℓ ≥ −ε(d)1,

φk ∈ Vmin.

◮ Quite directly, it follows that those remaining terms are not trivial,

i.e. sup

g Rℓφk(fk)Rℓ → 0 as d → 0.

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

What do these inequalities mean?

◮ Physical relevance of the normal products, or of the φk(fk), will

certainly depend on the model.

◮ What we can do in a model-independent context is to explore

properties of the fk.

◮ Restrict to just one term, drop the index k, so

f(s) =

• ds′ C(s′)g(s + s′/2)g(s − s′/2),

where C(s′) = K(s′)σ(φ ∗(s′/2)φ(−s′/2)) is an OPE coefficient.

◮ Free-field Wick square: f(s) = g(s)2 for real-valued g.

◮ Properties in the general case:

◮ f is smooth; some reminiscence of locality: suppf ⊂ suppg. ◮ Choose σ Hermitean ⇒ C of real type ⇒ f real-valued ◮ One would like f ≥ 0. But this seems to be false in general!

Scaling limit

◮ In the short-distance regime, the QFT should be approximated by

its scaling limit theory.

◮ Limit theory is supposed to be much simpler than the original one

(think of QCD, asymptotic freedom).

◮ So try to analyze our structures in the scaling limit.

◮ The scaling limit can be formulated rigorously, in the algebraic

framework (Buchholz/Verch)

◮ Was recently shown to be compatible with the short-distance

expansion (B.–D’Antoni–Morsella, arXiv:0711.4237, to appear in CMP)

◮ The scaling limit theory is dilation covariant.

◮ although structures may be complicated – representation of the

dilation group on a nonseparable Hilbert space

◮ In simple situations, one expects Nλ C(λ ·) → C(0)(·)

(Fredenhagen/Haag).

◮ In that case, one deals with a one-dimensional representation, i.e.

C(0) is homogeneous.

Homogeneous coefficients

◮ So let us (for simplicity) assume that C is homogeneous:

C(λs′) = λ −β C(s′) with some β ∈ R.

◮ These homogeneous distributions are fully classified; in our case:

C(s′) = c(ı(s′ −ı0))−β , c ∈ R.

◮ Now when is f pointwise positive?

f(s) =

• ds′ C(s′)g(s + s′/2)g(s − s′/2).

◮ Suppose c ≥ 0. Sufficient conditions for f ≥ 0:

◮ β = 1, and g is real-valued. (free-field situation) ◮ −1 < β < 1, and g ≥ 0. ◮ 1 < β < 3, suppg connected, and g logarithmically concave.

◮ The condition c ≥ 0 can always be achieved by absorbing -1 into

the corresponding composite field [φk(fk)]

◮ Note that we have determined the ± sign of the field in this way!

◮ It may be possible in models to exploit the choice of K in order to

arrive at the case β = 1.

Outline

Review of the free field situation Generalizations Technical background Nontriviality Interpretation Summary and Outlook

Summary

◮ We have found a generalized form of quantum inequalities.

◮ Derived from “absolute squares” of fields φ (classically positive) ◮ Generalizes the free field results; applicable in general axiomatic

context

◮ Main technical tool: the operator product expansion

◮ In a rigorous version in the axiomatic context ◮ Derived from a “Taylor expansion” of the local algebras

◮ Form of the inequalities:

∑

k

(1+ H)−ℓφk(fk)(1+ H)−ℓ ≥ −ε(d)1

where φk are composite fields from the OPE of K(t − t′)φ ∗(t)φ(t′).

◮ Interpretation is simplified in the scaling limit.

◮ ± sign of φk can be determined.

Open problems

◮ Can the framework be generalized to curved spacetimes?

◮ Some promising (combinatorial) results for linear fields - Taylor

expansion generalizes.

◮ However, topological details are unclear. ◮ No global Hamiltonian – need to resort to microlocal analysis. ◮ Perturbative versions of the OPE are available (Hollands/Wald);

nonperturbative version to be formulated

◮ What about the energy density?

◮ Heuristically, one would expect that the energy density is a sum of

“absolute squares” as presented here.

◮ However, in a precise sence, only partial results are available for

Noether’s theorem.