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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
Renormalization of Wick polynomials of locally covariant bosonic - - PowerPoint PPT Presentation
Renormalization of Wick polynomials of locally covariant bosonic - - PowerPoint PPT Presentation
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,
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Motivation: Nonlinear Local Observables in QFT
◮ Many interesting observables in Field Theory are local and
nonlinear.
◮ Examples:
◮ field powers φ2, φ∇µφ, φ2Aµ ◮ charge current ¯
ψγµψ
◮ stress energy tensor Tµν = 1
2∇µφ∇νφ − 1 4(∇φ)2gµν
◮ In QFT on Minkowski space, these are usually defined using Wick
- rdering, 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 07/06/2018 2 / 18
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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
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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
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Result of Hollands and Wald (2001) [arXiv:gr-qc/0103074]
◮ Consider a massive, curvature coupled scalar field
L = −1 2(∇φ)2 − 1 2m2φ2 − ξRφ2.
◮ 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
- f coefficients Ck such that for each n:
:φn:′ − :φn: =
n−1
- k=0
n k
- Cn−k:φk:
(setting = 1), with each Ck = Ck[g, m2, ξ] a scalar diff-op. that depends polynomially on the local Riemann tensor R and its derivatives, depends polynomially on m2 and depends analytically on ξ.
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 5 / 18
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Result of Hollands and Wald (2001) [arXiv:gr-qc/0103074]
◮ Consider a massive, curvature coupled scalar field
L = −1 2(∇φ)2 − 1 2m2φ2 − ξRφ2.
◮ 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
- f coefficients Ck such that for each n:
:φn:′ − :φn: =
n−1
- k=0
n k
- Cn−k:φk:
(setting = 1), with each Ck = Ck[g, m2, ξ] a scalar diff-op. that depends polynomially on the local Riemann tensor R and its derivatives, depends polynomially on m2 and depends analytically on ξ.
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 5 / 18
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Problems with Hollands & Wald
◮ The result of H-W is intuitive and appealing, reducing to the
folklore result on Minkowski spacetime.
◮ But: no vectors Bµ or spinors ψ, no derivatives ∂µφ, no time
- rdered products T(:φ2(x): : ¯
ψγµ∇µψ(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
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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−2g, µdφφ, µdtt)
= ⇒ :φk: → µkdφ(:φk: + O(log µ))
◮ locality and covariance ◮ smoothness, ω(:Ag,t(x):) is jointly smooth in (x, s) under smooth
compactly supported variations of (gs, ts), for some non-empty class of states ω (e.g., Hadamard).
◮ The technical analyticity requirement of H-W (analyticity upon
restriction to analytic (g, m2, ξ)) has been replaced by our smoothness axiom with respect to (g, t).
◮ Also, φ = (φi), t = (tj) 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
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Conditions on the background fields
◮ The components of the dynamical fields may have different
scaling degrees, µdφφ = (µdiφi). We do not need to require any conditions on the weights si.
◮ The components of the background fields may also have different
scaling degrees, µdtt = (µsjtj). Each tj is a component of a covariant tensor of rank ℓj. A background field t is admissible if ℓj + sj ≥ 0 (for all j). When the equality ℓj + sj = 0 holds, the component tj is said to be
- marginal. We denote by z = (tj)marginal the marginal components.
◮ Example: m2 (ℓ = 0, s = 2), ξ (ℓ = 0, s = 0) ◮ In the physics literature, the scaling weights di and sj are
sometimes called the mass dimension.
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 8 / 18
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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 {:φn:}n=1,2,... and {:φn:′}n=1,2,... be two families of Wick powers of φ. Then there exists a family of locally-covariant c-number fields {Ck}k=1,2,..., such that C1 = 0 and, for every k = 1, 2, . . ., (i) :φi1 · · · φin:′ = :φi1 · · · φin: +
n−1
- k=0
n k
- :φ(i1 · · · φik: Cn−k
ik+1···in)[g, t] ,
(ii) each Ck
i1···ik[g, t] is homogeneous of appropriate degree,
(iii) more precisely Ck
i1···ik[g, t] = Nk j=1 ck j [g, t](Pk j )i1···ik[g, t] for
equivariant polynomials Pk
j [g, t]=Pk j (g−1, ε, R, ∇R, t, ∇t, · · · ),
with smooth invariant scalar ck
j [g, t] = ck j (z) coefficients.
N.B.: For mixed Bose-Fermi fields φ, it suffices to use fermionic signs, X(i1···in) =
σ∈Sn(−)σXσi1···σin. But spin equivariance needs more attention!
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 9 / 18
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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 :φi1 · · · φin:′ = :φi1 · · · φin: +
n−1
- k=0
n k
- :φ(i1 · · · φik: Cn−k
ik+1···in)[g, t] ,
with c-number coefficients Cn−k
ik+1···in[g, t].
For scalar φ and t = (m2, ξ), we get the H-W formula :φn:′ − :φn: =
n−1
- k=0
n k
- Cn−k[g, m2, ξ] :φk: .
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 10 / 18
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Notes on the proof (2 of 4)
Using locality and smoothness, we conclude that the coefficients (g, t) → Ck[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:1411.7499].
Key place where the analyticity was previously used by H-W.
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 11 / 18
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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
- f the differential operators Ck can be refined to
u · Ck[g, t] = u · Ck(x, g, ∂g, . . . , t, ∂t, . . .) = Pk
g(R, ∇R, · · · , t, ∇t; u) ,
where Pk
g are homogeneous g-isotropic scalar functions, which is
linear in u auxiliary tensors.
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 12 / 18
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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 p0 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 07/06/2018 13 / 18
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Smooth invariant theory: what we need
There is no way to know whether our smooth invariant Pk
g(R, . . .) has
level sets that can be spearated by invariant polynomials!
Definition
V—vector space, (finite) {pi}—polynomials on V, Z ⊂ V—dense open subset with connected components {Zj} (finite). A smooth scalar σ on V is locally a smooth function of {pi} w.r.t Z if σ|Zj = Σj({pi}|Zj) (smooth) for each j, σ = [Σ]Z({pi}).
Conjecture (Extended Luna-Richardson)
For any fin-dim. rational O(g) or SO(g) representation V, there exists an invariant polynomial p0 such that any smooth g-isotropic scalar σ
- n V is locally a smooth function of invariant polynomials {pi} w.r.t Z,
σ = [Σ]Z({pi}). A careful adaptation of Luna’s original argument should suffice.
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 14 / 18
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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 · Ck[g, t] = Pk
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 07/06/2018 15 / 18
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Example: scalar Klein-Gordon, with derivative
Scalar Scalar Klein-Gordon in n-dimensions: gφ − m2φ + ξR φ = 0 ,
- Φ = (φ, ∇aφ), Φ → µ
n−2 2 Φ
- .
Admissible: m2 (ℓ + s = 0 + 2), ξ (ℓ + s = 0 + 0); marginal: ξ.
:φ2:′ :φ∇aφ:′ :∇(aφ∇b)φ:′ = :φ2: :φ∇aφ: :∇(aφ∇b)φ: + α1m2 + α2R + Aξ,m2 β1∇aR + Bξ,m2 gab
- γ1m4 + γ2m2R + γ3R2
+
- γ4m2 + γ5
- Rab + Cξ,m2
with smooth {α, β, γ}j = {α, β, γ}j(ξ), where also
Aξ,m2 = α3∇aξ∇aξ + α4ξ , Bξ,m2 = β2∇am2 + β3m2∇aξ + β4R∇aξ + β5Rab∇bξ + β6(∇bξ∇bξ)∇aξ + β7ξ∇aξ + β8∇bξ∇(b∇a)ξ + β9∇aξ , Cξ,m2 = γ6∇(aξ∇b)m2 + γ7m2∇(aξ∇b)ξ + γ8R∇aξ∇bξ + γ9Rab(∇ξ)2 + γ10Rc(a∇b)ξ∇cξ + γ11gab∇cξ∇cm2 + γ12gabm2(∇ξ)2 + γ13gabR(∇ξ)2 + γ14gabRbc∇bξ∇cξ + γ15∇(a∇b)m2 + γ16m2∇(a∇b)ξ + γ17ξ∇(a∇b)ξ + γ18R∇(a∇b)ξ + γ19Rabξ + γ20gabm2 + γ21gabm2ξ + γ22gab(ξ)2 + γ23gabRξ + γ24∇(aξ∇b)ξ + γ25∇(a∇b)ξ + γ26gab∇cξ∇cξ + γ27gab2ξ . Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 16 / 18
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Example: Vector Klein-Gordon
Vector Klein-Gordon in n-dimensions: gAa − m2Aa + ξb
aR Ab = 0 ,
- Ab → µ
n−2 4 Ab
- .
Admissible: m2 (ℓ + s = 0 + 2), ξb
a (ℓ + s = 2 − 2); marginal: ξb a.
:AaAb:′ = :AaAb: + (y1m2 + y2R)gab + y3Rab + (y4m2 + y5R)ξab + Bξ , where
Bξ = y6gabξc
c + y7∇(a∇b)ξc c + y8gab∇cξd d ∇cξd d + y9gcd ∇(aξcd ∇b)ξc c
+ y10
- ∇(a∇b)ξcd
- ξcd + y11∇(aξcd ∇b)ξcd + y12gab (ξcd ) ξcd + y13gab∇cξde∇cξde
+ y14ξabξc
c + y15ξab∇cξd d ∇cξd d + y16ξab + y17ξab (ξcd ) ξcd + y18ξab∇cξde∇cξde
+ y19ξcd ∇(aξcd ∇b)ξc
c + y20ξcd ξef ∇(aξef ∇b)ξcd ,
with smooth yj = yj(tr ξ = ξa
a, tr ξ2 = ξb aξa b, tr ξ3, . . . , tr ξn) (locally).
Stable orbit types are separated by the matrix discriminant p0(ξ) = disc(ξ) = det
- tr ξi+j−2n
i,j=1 .
Igor Khavkine (CAS, Prague) Wick powers in vector LCQFT Cortona 07/06/2018 17 / 18
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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 07/06/2018 18 / 18
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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 07/06/2018 18 / 18
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