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 ) ≥ − c f 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. � dt dt ′ F ( t , t ′ ) g ( t ) g ( t ′ ) = ρ ( φ ( g ) ∗ φ ( g )) . 0 ≤ ◮ Same holds if F is multiplied by a kernel K of positive type; e.g. K ( t − t ′ ) = ( ı π ( t − t ′ − ı 0 )) − 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 � ds ds ′ ρ (: φ 2 : ( s )) ı π ( s ′ − ı 0 ) g ( s + s ′ / 2 ) g ( s − s ′ / 2 ) 0 ≤ c g + R ρ , g + where c g is independent of ρ ; and R ρ , g becomes small when supp g is small. ◮ In fact, for real-valued g , there are simplifications: ◮ g ( s + s ′ / 2 ) g ( s − s ′ / 2 ) is symmetric in s ′ ; ◮ R ρ , g = 0; ◮ ( ı π ( s ′ − ı 0 )) − 1 can be replaced with its symmetric part, δ ( s ′ ) . ◮ Since the estimate is now independent of ρ , one has: : φ 2 : ( g 2 ) ≥ − c g 1 .

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. ∞ ∑ c k g k , P [ g ] = c k ∈ R . k = 0 ◮ 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. ∞ ∑ c k g k , P [ g ] = c k ∈ R . k = 0 ◮ 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. ∞ ∑ c k g k , P [ g ] = c k ∈ R . k = 0 ◮ 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. ∞ ∑ c k g k , P [ g ] = c k ∈ R . k = 0 ◮ 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 ] = g 2 n ∑ ∞ k = 0 d k g k , n ∈ N 0 , d 0 > 0.

When is a formal power series positive? (1/2) ◮ Take a formal power series with real coefficients. ∞ ∑ c k g k , P [ g ] = c k ∈ R . k = 0 ◮ 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 ] = g 2 n ∑ ∞ k = 0 d k g k , n ∈ N 0 , d 0 > 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. ∞ ∑ c k g k , P [ g ] = c k ∈ R . k = 0 ◮ 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 ] = g 2 n ∑ ∞ k = 0 d k g k , n ∈ N 0 , d 0 > 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? ∞ ( − 1 ) k ∑ ( 2 k )! g 2 k P [ g ] = k = 0 ◮ 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 ) . � dt dt ′ K ( t − t ′ ) φ ∗ ( t ) φ ( t ′ ) g ( t ) g ( t ′ ) . 0 ≤ ◮ Expand here φ ∗ ( t ) φ ( t ′ ) ≈ ∑ N k = 1 c k ( t − t ′ ) φ k (( t + t ′ ) / 2 ) ◮ Insert above, use s = ( t + t ′ ) / 2 and s ′ = ( t − t ′ ) to obtain f k ( s ) � �� � N � � ds ′ K ( s ′ ) c k ( s ′ ) g ( s + s ′ / 2 ) g ( s − s ′ / 2 ) φ k ( s ) ∑ 0 ≤ ds k = 1 N ∑ = φ k ( f k ) k = 1 ◮ 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 ( O 1 ) ⊂ A ( O 2 ) whenever O 1 ⊂ O 2 (isotony); ◮ [ A 1 , A 2 ] = 0 whenever O 1 �� O 2 , and A i ∈ A ( O i ) (locality); ◮ U ( x , Λ) A ( O ) U ( x , Λ) ∗ = A (Λ O + 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 ones (and we will, too!)