Renormalization of Wick polynomials of locally covariant bosonic vector valued fields

Transcription

1 Renormalization of Wick polynomials of locally covariant bosonic vector valued fields [arxiv: ] w/ Valter Moretti [arxiv: ] 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

2 06 Apr 2018 The best defense is a good offence! Anonymous Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

3 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 µφ 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: :φ(x)φ(y): = φ(x)φ(y) ( 1 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/ / 18

4 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/ / 18

5 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/ / 18

6 Result of Hollands and Wald (2001) [arxiv:gr-qc/ ] Consider a massive, curvature coupled scalar field L = 1 2 ( φ)2 1 2 m2 φ 2 ξrφ 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 :φ n : :φ n : = k=0 ( n k ) C n k :φ k : (setting = 1), 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/ / 18

7 Result of Hollands and Wald (2001) [arxiv:gr-qc/ ] Consider a massive, curvature coupled scalar field L = 1 2 ( φ)2 1 2 m2 φ 2 ξrφ 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 :φ n : :φ n : = k=0 ( n k ) C n k :φ k : (setting = 1), 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/ / 18

8 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/ / 18

9 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 = α1 scaling, (g, φ, t) (µ 2 g, µ d φ φ, µ dt t) = :φ k : µ kd φ (:φ k : + O(log µ)) locality and covariance smoothness, ω(:ag,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/ / 18

10 Conditions on the background fields The components of the dynamical fields may have different scaling degrees, µ d φφ = (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 tt = (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/ / 18

11 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 (i) :φ i1 φ in : = :φ i1 φ in : + k=0 ( n k ) :φ (i1 φ ik : C n k i k+1 i n) [g, t], (ii) each C k i 1 i k [g, t] is homogeneous of appropriate degree, (iii) more precisely C k i 1 i k [g, t] = N k j=1 ck j [g, t](p k j ) i1 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 (i1 i n) = σ S n ( ) σ X σi1 σi n. But spin equivariance needs more attention! Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

12 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 i1 φ in : = :φ i1 φ in : + :φ k (i1 φ ik : C n k i k+1 i n) [g, t], 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/ / 18

13 Notes on the proof (2 of 4) Using locality and smoothness, we conclude that the coefficients (g, t) C k [g, t] are local and regular. Theorem (Peetre-Slovák) A map C C that is local (compatible with restriction to smaller domains) and regular (maps smooth families to smooth families) must be a smooth differential operator of locally bounded order. Original result for linear maps, Peetre (1959, 1960). Extension to nonlinear maps, Slovák (1988). Great exposition, Navarro-Sancho [arxiv: ]. Key place where the analyticity was previously used by H-W. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

14 Notes on the proof (3 of 4) Theorem (Thomas Replacement) A smooth homogeneous tensor function of g, g,... T, T,... is equivariant under diffeomorphisms iff it is a smooth homogeneous pointwise g-isotropic function of R, R,... T, T,... and ε. Original, T.Y. Thomas (1920s). More modern, Slovák (1992). Concise, self-contained proof (our paper). Using covariance (under diffeomorphisms) and scaling, the structure of the differential operators C k can be refined to u C k [g, t] = u C k (x, g, g,..., t, t,...) = P k g(r, R,, t, t; u), where P k g are homogeneous g-isotropic scalar functions, which is linear in u auxiliary tensors. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

15 Smooth invariant theory: state of the art ATTN: Smooth invariant theory of real groups may be complicated! Note that O(g) and SO(g) are not compact, but are algebraic and linearly reductive with linearly reductive complexification (fin-dim. rational representations decompose into irreps). Theorem (Richardson, 1973) For any fin-dim. rational representation V of O(g) or SO(g), there exists an invariant polynomial p 0 such that each connected component of V \ p 1 0 (0) is filled by a single orbit type. Theorem (Luna, 1976) A smooth g-isotropic scalar, on a fin-dim. rational representation, whose level sets can be separated by invariant polynomials, is a smooth function of g-isotropic polynomials. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

16 Smooth invariant theory: what we need There is no way to know whether our smooth invariant P k g(r,...) has level sets that can be spearated by invariant polynomials! Definition V vector space, (finite) {p i } polynomials on V, Z V dense open subset with connected components {Z j } (finite). A smooth scalar σ on V is locally a smooth function of {p i } w.r.t Z if σ Zj = Σ j ({p i } Zj ) (smooth) for each j, σ = [Σ] Z ({p i }). Conjecture (Extended Luna-Richardson) For any fin-dim. rational O(g) or SO(g) representation V, there exists an invariant polynomial p 0 such that any smooth g-isotropic scalar σ on V is locally a smooth function of invariant polynomials {p i } w.r.t Z, σ = [Σ] Z ({p i }). A careful adaptation of Luna s original argument should suffice. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

17 Notes on the proof (4 of 4) Theorem (FFT of Invariant Theory) Scalar g-isotropic polynomials are generated by (a) outer products, (b) index contractions with g, (c) index contractions with ε. Original, Weyl (1930s). Textbook, Procesi (2007). Concise, complete, self-contained proof (our paper). Theorem (Folklore) A positive weight, homogeneous function that is smooth around zero is a polynomial. Thus, with only admissible background fields t, u C k [g, t] = P k g(r, R,..., t, t,... ; u) is a sum of homogeneous invariant polynomials, whose coefficients are (locally) smooth functions of (finitely many) invariant scalar polynomials in (marginal components) z. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

18 Example: scalar Klein-Gordon, with derivative Scalar Scalar Klein-Gordon in n-dimensions: ) g φ m 2 φ + ξr φ = 0, (Φ = (φ, a φ), Φ µ n 2 2 Φ. Admissible: m 2 (l + s = 0 + 2), ξ (l + s = 0 + 0); marginal: ξ. :φ 2 : :φ 2 : α 1 m 2 + α 2 R + A ξ,m 2 :φ aφ: = :φ aφ: + β 1 ar + B ξ,m 2 : (a φ b) φ: ( : (a φ b) φ: g ab γ1 m 4 + γ 2 m 2 R + γ 3 R 2) + ( γ 4 m 2 + γ 5 ) R ab + C ξ,m 2 with smooth {α, β, γ} j = {α, β, γ} j (ξ), where also C ξ,m 2 = γ 6 (a ξ b) m 2 + γ 7 m 2 (a ξ b) ξ + γ 8 R aξ b ξ + γ 9 R ab ( ξ) 2 A ξ,m 2 = α 3 a ξ aξ + α 4 ξ, B ξ,m 2 = β 2 am 2 + β 3 m 2 aξ + β 4 R aξ + β 5 R ab b ξ + β 6 ( b ξ b ξ) aξ + β 7 ξ aξ + β 8 b ξ (b a) ξ + β 9 a ξ, + γ 10 R c(a b) ξ c ξ + γ 11 g ab c ξ c m 2 + γ 12 g ab m 2 ( ξ) 2 + γ 13 g ab R( ξ) 2 + γ 14 g ab R bc b ξ c ξ + γ 15 (a b) m 2 + γ 16 m 2 (a b) ξ + γ 17 ξ (a b) ξ + γ 18 R (a b) ξ + γ 19 R ab ξ + γ 20 g ab m 2 + γ 21 g ab m 2 ξ + γ 22 g ab ( ξ) 2 + γ 23 g ab R ξ + γ 24 (a ξ b) ξ + γ 25 (a b) ξ + γ 26 g ab c ξ c ξ + γ 27 g ab 2 ξ. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

19 Example: Vector Klein-Gordon Vector Klein-Gordon in n-dimensions: g A a m 2 A a + ξ b ar A b = 0, (A b µ n 2 4 Ab ). Admissible: m 2 (l + s = 0 + 2), ξ b a (l + s = 2 2); marginal: ξ b a. :A a A b : = :A a A b : + (y 1 m 2 + y 2 R)g ab + y 3 R ab + (y 4 m 2 + y 5 R)ξ ab + B ξ, where B ξ = y 6 g ab ξ c c + y 7 (a b) ξ c c + y 8g ab c ξ d d c ξd d + y 9g cd (a ξ cd b) ξ c c ( ) + y 10 (a b) ξ cd ξ cd + y 11 (a ξ cd b) ξ cd + y 12 g ab ( ξ cd ) ξ cd + y 13 g ab c ξ de c ξ de + y 14 ξ ab ξ c c + y 15ξ ab c ξ d d c ξd d + y 16 ξ ab + y 17 ξ ab ( ξ cd ) ξ cd + y 18 ξ ab c ξ de c ξ de + y 19 ξ cd (a ξ cd b) ξ c c + y 20ξ cd ξ ef (a ξ ef b) ξ cd, with smooth y j = y j (tr ξ = ξa, a tr ξ 2 = ξaξ b b a, tr ξ3,..., tr ξ n ) (locally). Stable orbit types are separated by the matrix discriminant p 0 (ξ) = disc(ξ) = det (tr ξ i+j 2) n i,j=1. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

20 Discussion In the earlier K-Moretti work, we reproved the classification finite renormalizations of locally covariant scalar Wick polynomials, by replacing the analyticity axiom of Hollands-Wald by a smoothness axiom (Peetre-Slovák theorem, classical invariant theory,... ). In K-Melati-Moretti we have extended our earlier work to cover locally covariant tensor valued Wick polynomials (smooth invariant theory,... ). Reminder: need to check that the smoothness axiom is verified! New remark: need to improve the state of the art on smooth invariants of reductive groups (extended Luna-Richardson). It remains to generalize the results to tensor and spinor fields, background gauge fields, Wick products with derivatives and time ordered products. Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

21 Discussion In the earlier K-Moretti work, we reproved the classification finite renormalizations of locally covariant scalar Wick polynomials, by replacing the analyticity axiom of Hollands-Wald by a smoothness axiom (Peetre-Slovák theorem, classical invariant theory,... ). In K-Melati-Moretti we have extended our earlier work to cover locally covariant tensor valued Wick polynomials (smooth invariant theory,... ). Reminder: need to check that the smoothness axiom is verified! New remark: need to improve the state of the art on smooth invariants of reductive groups (extended Luna-Richardson). It remains to generalize the results to tensor and spinor fields, background gauge fields, Wick products with derivatives and time ordered products. Thank you for your attention! Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/ / 18

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