Renormalization of Wick polynomials of locally covariant bosonic vector valued fields [arXiv:1411.1302] w/ Valter Moretti [arXiv:1710.01937] w/ Alberto Melati, Valter Moretti Igor Khavkine Institute of Mathematics Czech Academy of Sciences, Prague 07 Jun 2018 AQFT: Where Operator Algebra Meets Microlocal Analysis workshop at Il Palazzone, Cortona, Italy
06 Apr 2018 The best defense is a good offence! — Anonymous Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 1 / 18
Motivation: Nonlinear Local Observables in QFT ◮ Many interesting observables in Field Theory are local and nonlinear. ◮ Examples: ◮ field powers φ 2 , φ ∇ µ φ , φ 2 A µ ◮ charge current ¯ ψγ µ ψ ◮ stress energy tensor T µν = 1 2 ∇ µ φ ∇ ν φ − 1 4 ( ∇ φ ) 2 g µν ◮ In QFT on Minkowski space, these are usually defined using Wick ordering , aka normal ordering , aka vacuum subtraction : � 1 � : φ ( x ) φ ( y ): = φ ( x ) φ ( y ) − � � � φ ( x ) φ ( y ) � , then x → y . ◮ However, it’s not always so simple, especially on curved spacetime: ◮ Do conservation laws remain conserved? ( Anomalies. ) ◮ Is gauge invariance preserved? (Another kind of anomaly .) ◮ How much does the definition depend on the vacuum state? ◮ Is the definition local? ◮ Is the definition covariant? Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 2 / 18
Ambiguity in Definition ◮ If O is any classical local observable, then any quantization prescription O �→ : O : suffers from ambiguities. Why not use : O : ′ = : O : + O ( � ) ? ◮ This is a manifestation of the well-known operator ordering ambiguity in quantum mechanics. ◮ The mapping from classical to quantum observables is a priori non-unique, unless further physical principles are involved. ◮ Can anomalies be cancelled by exploiting these ambiguities? A precise classification of the ambiguities is necessary to answer the question. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 3 / 18
Locally Covariant Fields ◮ We will work in the framework of Locally Covariant QFT on Curved Spacetimes (Hollands-Wald, Brunetti-Fredenhagen-Verch, . . . ). ◮ A QFT is an assignment of an algebra of observables to a spacetime, ( M , g ) → A ( M , g ) . It is locally covariant if ◮ a causal isometric embedding ( M , g ) → ( M ′ , g ′ ) induces an injective homomorphism A ( M , g ) → A ( M ′ , g ′ ) ; ◮ these homomorphisms respect spacelike commutativity, time slice property. ◮ A local field ( M , g ) �→ Φ ( M , g ) is a distribution on M valued in A ( M , g ) . It is locally covariant when Φ ( M , g ) ( f ) ∈ A ( M , g ) respects the inclusions and isomorphisms induced by isometries. ◮ In categorical language, A is a covariant functor from spacetimes to algebras and Φ is a natural transformation from the functor of test functions to the algebra functor A . Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 4 / 18
Result of Hollands and Wald (2001) [arXiv:gr-qc/0103074] ◮ Consider a massive, curvature coupled scalar field L = − 1 2 ( ∇ φ ) 2 − 1 2 m 2 φ 2 − ξ R φ 2 . ◮ To any polynomial P ( φ ) , we can associate a locally covariant local field : P ( φ ): that essentially reduces to the corresponding Wick polynomial on Minkowski space. ◮ The assignment of the field is not unique. Under technical conditions, the ambiguity is precisely characterized as follows: Given two prescriptions : · · · : and : · · · : ′ , there exists a sequence of coefficients C k such that for each n : n − 1 � n � : φ n : ′ − : φ n : = � C n − k : φ k : (setting � = 1) , k k = 0 with each C k = C k [ g , m 2 , ξ ] a scalar diff-op. that depends polynomially on the local Riemann tensor R and its derivatives, depends polynomially on m 2 and depends analytically on ξ . Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 5 / 18
Result of Hollands and Wald (2001) [arXiv:gr-qc/0103074] ◮ Consider a massive, curvature coupled scalar field L = − 1 2 ( ∇ φ ) 2 − 1 2 m 2 φ 2 − ξ R φ 2 . ◮ To any polynomial P ( φ ) , we can associate a locally covariant local field : P ( φ ): that essentially reduces to the corresponding Wick polynomial on Minkowski space. ◮ The assignment of the field is not unique. Under technical conditions, the ambiguity is precisely characterized as follows: Given two prescriptions : · · · : and : · · · : ′ , there exists a sequence of coefficients C k such that for each n : n − 1 � n � : φ n : ′ − : φ n : = � C n − k : φ k : (setting � = 1) , k k = 0 with each C k = C k [ g , m 2 , ξ ] a scalar diff-op. that depends polynomially on the local Riemann tensor R and its derivatives, depends polynomially on m 2 and depends analytically on ξ . Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 5 / 18
Problems with Hollands & Wald ◮ The result of H-W is intuitive and appealing, reducing to the folklore result on Minkowski spacetime. ◮ But: no vectors B µ or spinors ψ , no derivatives ∂ µ φ , no time ordered products T (: φ 2 ( x ): : ¯ ψγ µ ∇ µ ψ ( y ):) , no covariance for background gauge field transformations ( M , g , A ) �→ ( M , g , A + ∂ u ) . ◮ H-W do claim a reasonable result that covers some of these cases, but for a proof they only say that it should be analogous to the scalar case. ◮ The technical conditions involve analyticity in an essential and technically cumbersome way. It is unnatural in smooth differential geometry. ◮ Goal: Eventually address all these issues. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 6 / 18
Our Axioms / Renormalization Conditions ◮ We can essentially reproduce the H-W result, with updated axioms: ◮ normalization , : φ : = φ ◮ commutators , [: A ( x ): , φ ( y )] = i : { A ( x ) , φ ( y ) } : ◮ completeness , ∀ x : [ A , φ ( x )] = 0 ⇐ ⇒ A = α 1 ◮ scaling , ( g , φ, t ) �→ ( µ − 2 g , µ d φ φ, µ d t t ) ⇒ : φ k : �→ µ kd φ (: φ k : + O (log µ )) = ◮ locality and covariance ◮ smoothness , ω (: A g , t ( x ):) is jointly smooth in ( x , s ) under smooth compactly supported variations of ( g s , t s ) , for some non-empty class of states ω (e.g., Hadamard ). ◮ The technical analyticity requirement of H-W ( analyticity upon restriction to analytic ( g , m 2 , ξ ) ) has been replaced by our smoothness axiom with respect to ( g , t ) . ◮ Also, φ = ( φ i ) , t = ( t j ) could be any natural multi-component field , so could ξ . We restrict to tensor fields. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 7 / 18
Conditions on the background fields ◮ The components of the dynamical fields may have different scaling degrees, µ d φ φ = ( µ d i φ i ) . We do not need to require any conditions on the weights s i . ◮ The components of the background fields may also have different scaling degrees, µ d t t = ( µ s j t j ) . Each t j is a component of a covariant tensor of rank ℓ j . A background field t is admissible if ℓ j + s j ≥ 0 (for all j ). When the equality ℓ j + s j = 0 holds, the component t j is said to be marginal . We denote by z = ( t j ) marginal the marginal components. ◮ Example: m 2 ( ℓ = 0, s = 2), ξ ( ℓ = 0, s = 0) ◮ In the physics literature, the scaling weights d i and s j are sometimes called the mass dimension . Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 8 / 18
Theorem (K-Melati-Moretti) Let φ be a multicomponent locally covariant tensor field, coupled to admissible background tensor fields t , with marginal components z . Let { : φ n : } n = 1 , 2 ,... and { : φ n : ′ } n = 1 , 2 ,... be two families of Wick powers of φ . Then there exists a family of locally-covariant c-number fields { C k } k = 1 , 2 ,... , such that C 1 = 0 and, for every k = 1 , 2 , . . . , n − 1 � n � (i) : φ i 1 · · · φ i n : ′ = : φ i 1 · · · φ i n : + � : φ ( i 1 · · · φ i k : C n − k i k + 1 ··· i n ) [ g , t ] , k k = 0 (ii) each C k i 1 ··· i k [ g , t ] is homogeneous of appropriate degree, i 1 ··· i k [ g , t ] = � N k (iii) more precisely C k j = 1 c k j [ g , t ]( P k j ) i 1 ··· i k [ g , t ] for equivariant polynomials P k j [ g , t ]= P k j ( g − 1 , ε , R , ∇ R , t , ∇ t , · · · ) , with smooth invariant scalar c k j [ g , t ] = c k j ( z ) coefficients. N.B.: For mixed Bose-Fermi fields φ , it suffices to use fermionic signs , X ( i 1 ··· i n ) = � σ ∈ S n ( − ) σ X σ i 1 ··· σ i n . But spin equivariance needs more attention! Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 9 / 18
Notes on the proof (1 of 4) We closely follow the structure of our previous work on scalars (which followed the original H-W proof, with greater attention to detail). Starting from normalization , use induction on commutators and completeness to get n − 1 � n � : φ i 1 · · · φ i n : ′ = : φ i 1 · · · φ i n : + � : φ ( i 1 · · · φ i k : C n − k i k + 1 ··· i n ) [ g , t ] , k k = 0 with c -number coefficients C n − k i k + 1 ··· i n [ g , t ] . For scalar φ and t = ( m 2 , ξ ) , we get the H-W formula n − 1 � n � : φ n : ′ − : φ n : = � C n − k [ g , m 2 , ξ ] : φ k : . k k = 0 Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 10 / 18
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