The effective theory of strings

Transcription

1 The effective theory of strings How to quantize Lorentzian submanifolds Jochen Zahn based on arxiv: joint work with Dorothea Bahns & Katarzyna Rejzner New trends in AQFT, Frascati, September 2012

2 Motivation We consider d-dimensional Lorentzian submanifolds Σ (world-sheets) of the Minkowski target space M = (R n, η). Let X : Σ M, Σ = ran X. The induced metric is given by g µν = d X a µη ab d X b ν. The Nambu Goto action is defined as S NG ( Σ) = Original motivation (d = 2): Reproduces dual resonance models Σ gdx. Effective theory of vortex lines connecting two quarks Our motivation: Toy model for quantum gravity, diffeomorphism invariance

3 Conventional string theory: The Polyakov action In order to be able to use path integral methods, one usually switches to the Polyakov action, i.e., one introduces an auxiliary metric g on Σ and defines S Pol [g, X ] = gg µν µ X a η ab ν X b dx. Extremal configurations for S NG are also extremal for S Pol. By the equation of motion, g µν (x) = e φ(x) g µν (x), hence there is a new symmetry, local conformal transformations. The critical dimension n = 26 shows up when one requires local conformal invariance also in the quantized theory. It is not clear whether the classical correspondence between S NG and S Pol is still valid in the quantized theory, as there also off-shell configurations are important. The new symmetry might introduce an anomaly. Vacuum corresponds to an event, not a two-dimensional submanifold.

4 Other string quantizations The critical dimension n = 26 also shows up in covariant quantization and quantization in the light cone gauge of the Nambu Goto action. The critical dimension is absent in the Pohlmeyer string and the related loop string. There are also effective string theories in the literature, the Polchinski Strominger and the Lüscher Weisz string. In the former, n = 26 is special, but not the only possible choice. In the latter, no target space dimension seems to be singled out.

5 Our approach We quantize perturbatively. This means that we quantize deviations from a classical string (membrane) configuration. Due to the square root, this theory can only be effective. An important difference to the standard framework (shared by the Lüscher Weisz string) is that we are expanding around a string (membrane). We use methods from perturbative algebraic quantum field theory, quantum field theory on curved spacetimes, and the Batalin Vilkovisky formalism. We show that the Nambu Goto string and its higher dimensional generalizations can be quantized, in the sense of an effective theory, independently of the dimension of the target space.

6 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

7 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

8 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

9 Configuration space The main conceptual step is to realize what perturbation theory means for submanifolds. We choose a background, i.e., a d-dimensional globally hyperbolic submanifold Σ M, which is on-shell (satisfies the Euler Lagrange eqns for S NG ). The dynamical submanifold Σ is then described by the image of ϕ E(Σ). = Γ (Σ, TM) under the exponential map of M, X (x) = exp X (x) (λϕ(x)). ϕ For a flat target space, X a (x) = X a (x) + λϕ a (x). Σ Σ Here X : Σ M is the embedding and λ is a formal parameter of dimension length d/2. The elements of E(Σ) are called configurations. This leads to a field theory on Σ. Physical interpretation: Spontaneous symmetry breaking.

10 Configuration space The main conceptual step is to realize what perturbation theory means for submanifolds. We choose a background, i.e., a d-dimensional globally hyperbolic submanifold Σ M, which is on-shell (satisfies the Euler Lagrange eqns for S NG ). The dynamical submanifold Σ is then described by the image of ϕ E(Σ). = Γ (Σ, TM) under the exponential map of M, X (x) = exp X (x) (λϕ(x)). ϕ For a flat target space, X a (x) = X a (x) + λϕ a (x). Σ Σ Here X : Σ M is the embedding and λ is a formal parameter of dimension length d/2. The elements of E(Σ) are called configurations. This leads to a field theory on Σ. Physical interpretation: Spontaneous symmetry breaking.

11 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. i(σ) ϕ Σ

12 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). i(σ) ϕ Σ

13 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). i(σ) ϕ Σ

14 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). i(σ) ϕ Σ

15 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). E c : Covariant functor Sub Vec i, E c (Σ). = Γ c (Σ, TM). i(σ) ϕ Σ

16 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). E c : Covariant functor Sub Vec i, E c (Σ). = Γ c (Σ, TM). i(σ) ϕ Σ

17 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). E c : Covariant functor Sub Vec i, E c (Σ). = Γ c (Σ, TM). i(σ) ϕ Σ

18 Some categories and functors We do not want to work on a fixed background Σ, but on all possible ones simultaneously, in the spirit of the generally covariant locality principle [BFV03]. This is encoded in the language of category theory. Sub: The objects are d-dimensional, globally hyperbolic, oriented, on-shell submanifolds of M. If for an orientation-preserving isometry i of M we have i(σ) Σ, causally compatible, with compatible orientation, then there is a morphism χ : Σ Σ. Vec (i) : The objects are locally convex vector spaces. The morphisms are continuous linear (injective) maps. Alg: The objects are topological algebras. The morphisms are continuous injective algebra homomorphisms. E: Contravariant functor Sub Vec, E(Σ). = Γ (Σ, TM). E c : Covariant functor Sub Vec i, E c (Σ). = Γ c (Σ, TM). D: Covariant functor Sub Vec i, D(Σ). = C c (Σ). i(σ) ϕ Σ

19 Functionals I A functional is a map F : E(Σ) R. Differentiability and smoothness are defined in terms of the functional derivatives F (1).= d (ϕ), ψ dt F (ϕ + tψ). t=0 In the following, we discuss polynomial functionals, i.e., F (ϕ) = f k (x 1,..., x k )ϕ(x 1 )... ϕ(x k )µ(x 1 )... µ(x k ), k Σ k F = f k (x 1,..., x k )Φ(x 1 )... Φ(x k )µ(x 1 )... µ(x k ). k Σ k Here µ(x) = gdx, f k Γ (Σ k, TM k ), and Φ(x)(ϕ). = ϕ(x). Polynomial functionals can be seen as the Taylor expansion of a functional at ϕ = 0. In the following, we define certain properties of functionals by reference to the f k. There are equivalent definitions that do not rely on the formal expansion.

20 Functionals II The support of a functional is defined as x supp F (x, x 2,..., x k ) supp f k for some x 2,..., x k. A functional is local, if all f k are supported on the diagonal k and WF(f k ) T k. The set of smooth local functionals is denoted by F loc (Σ). F loc is a covariant functor from Sub to Vec i. One defines a commutative product, (F G)(ϕ). = F (ϕ)g(ϕ), and defines F(Σ) as the algebraic completion of F loc (Σ). Its elements are called multilocal functionals. F is a covariant functor from Sub to Alg. The algebra F µc (Σ) of microcausal functionals is the sequential completion of F(Σ) in a topology that enforces the wave front set condition WF(f l ) {(x 1,..., x l ; k 1,..., k l ) k i V + x i i or k i V x i i} =.

21 Vector fields Consider alternating multivector fields on E(Σ), i.e., the space Λ m V(Σ) of smooth maps E(Σ) Λ m E c (Σ). A monomial multivector field (X ϕ)(y 1,..., y m ) = f X (x 1,..., x k, y 1,..., y m )ϕ(x 1 )... ϕ(x k )µ(x 1 )... µ(x k ), with f X compactly supported can be written in terms of the antifield Φ as X = f X (x 1,..., y m )Φ(x 1 )... Φ(x k )Φ (y 1 ) Φ (y m )µ(x 1 )... µ(y m ). By reference to f X, one defines supp X and Λ m V loc (Σ), Λ m V(Σ), Λ m V µc (Σ).

22 Vector fields Consider alternating multivector fields on E(Σ), i.e., the space Λ m V(Σ) of smooth maps E(Σ) Λ m E c (Σ). A monomial multivector field (X ϕ)(y 1,..., y m ) = f X (x 1,..., x k, y 1,..., y m )ϕ(x 1 )... ϕ(x k )µ(x 1 )... µ(x k ), with f X compactly supported can be written in terms of the antifield Φ as X = f X (x 1,..., y m )Φ(x 1 )... Φ(x k )Φ (y 1 ) Φ (y m )µ(x 1 )... µ(y m ). By reference to f X, one defines supp X and Λ m V loc (Σ), Λ m V(Σ), Λ m V µc (Σ). The action of V(Σ) = Λ 1 V(Σ) on F(Σ) is defined as ( X F )(ϕ) =. F (1) (ϕ), X (ϕ) t=0 F (ϕ + tx (ϕ)). = d dt Setting Λ 0 V(Σ). = F(Σ) and defining the product (X Y)(ϕ). = X (ϕ) Y(ϕ), one obtains a graded commutative algebra ΛV(Σ), which naturally carries a Schouten bracket (antibracket) {, } : Λ m V(Σ) Λ n V(Σ) Λ m+n 1 V(Σ).

23 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

24 The action A Lagrangean [FR11a] is defined as a natural transformation between the functors D and F loc, i.e., to each Σ one assigns a linear injective map L Σ : D(Σ) F loc (Σ). Two Lagrangeans L and L are equivalent, L L, if supp(l Σ (f ) L Σ(f )) supp(df ) Σ, f D(Σ). To an equivalence class S of Lagrangeans, one associates the Koszul Tate differential δ : Λ k V(Σ) Λ k 1 V(Σ) defined by δ(f ). = {L(f ), F } for f = 1 on supp F. We denote by F 0 (Σ) the ideal of functionals that vanish on-shell. By definition, ran δ F(Σ) F 0 (Σ), with equality if the eom for S is hyperbolic. Then (ΛV(Σ), δ) is a resolution of F S (Σ). = F(Σ)/F 0 (Σ), the on-shell functionals.

25 The action A Lagrangean [FR11a] is defined as a natural transformation between the functors D and F loc, i.e., to each Σ one assigns a linear injective map L Σ : D(Σ) F loc (Σ). Two Lagrangeans L and L are equivalent, L L, if supp(l Σ (f ) L Σ(f )) supp(df ) Σ, f D(Σ). To an equivalence class S of Lagrangeans, one associates the Koszul Tate differential δ : Λ k V(Σ) Λ k 1 V(Σ) defined by δ(f ). = {L(f ), F } for f = 1 on supp F. We denote by F 0 (Σ) the ideal of functionals that vanish on-shell. By definition, ran δ F(Σ) F 0 (Σ), with equality if the eom for S is hyperbolic. Then (ΛV(Σ), δ) is a resolution of F S (Σ) =. F(Σ)/F 0 (Σ), the on-shell functionals. The Nambu Goto action L(f )(ϕ) =. λ 2 f (x) g[ϕ]dx is reparametrization invariant, so the resulting eom is not hyperbolic. To deal with this, we use the Batalin Vilkovisky formalism.

26 Reparametrization invariance An infinitesimal reparametrization is given by a vector field on Σ, i.e., an element c of g c (Σ). = Γ c (Σ, T Σ). c ϕ Σ Σ For M = R n and in Cartesian coordinates its action on E(Σ) is ρ(c)ϕ a = c µ d X a µ = c µ (dx a µ + λ µ ϕ a ). In the Batalin Vilkovisky formalism, one introduces anticommuting evaluation functionals on g(σ), i.e., C µ (x)(c) = c µ (x), i.e., ghosts, and promotes ρ to the BRST differential γ, γφ a = C µ d X a µ, γc µ = 1 2 λl C C µ = λc λ λ C µ.

27 Reparametrization invariance An infinitesimal reparametrization is given by a vector field on Σ, i.e., an element c of g c (Σ). = Γ c (Σ, T Σ). c ϕ Σ ρ(c)ϕ Σ For M = R n and in Cartesian coordinates its action on E(Σ) is ρ(c)ϕ a = c µ d X a µ = c µ (dx a µ + λ µ ϕ a ). In the Batalin Vilkovisky formalism, one introduces anticommuting evaluation functionals on g(σ), i.e., C µ (x)(c) = c µ (x), i.e., ghosts, and promotes ρ to the BRST differential γ, γφ a = C µ d X a µ, γc µ = 1 2 λl C C µ = λc λ λ C µ.

28 The Batalin Vilkovisky complex Monomials in the Batalin Vilkovisky complex BV(Σ) can be written as F = f F (x 1,..., x l )Φ(x 1 )... Φ(x i )C(x i+1 ) C(x j ) Φ (x j+1 ) Φ (x k )C (x k+1 )... C (x l )µ(x 1 )... µ(x l ). The C s are called antifields of ghosts, they correspond to functional derivatives w.r.t. C. One defines the antibracket ( ) δf {F, G} = δφ α (x) δg δφ α(x) + ( 1)#gh(F ) δf δφ α(x) δg δφ α µ(x), (x) where φ α = Φ, C and φ α = Φ, C. The gradings and the action of the differentials δ and γ is summarized in the following table: #af #pg #gh δ λγ Φ a λ 1 C µ d X µ a C µ C λ λ C µ Φ a h ab µ ( g µν gd X ν b ) µ (C µ Φ a) C µ d X µφ a a λ (C λ C µ) + µ C λ C λ

29 The BRST differential The differential δ was defined via the antibracket with L. We want to do the same for the BRST differential γ. Define θ(f ) =. f γ(φ a )Φ aµ λ f γ(c ν )C ν µ. One may then write the BRST differential γ as γ(f ). = {θ(f ), F } for f = 1 on supp F. It increases #pg by 1, while δ decreases #af by 1. Hence, the BV differential s = γ + δ increases #gh = #pg #af by 1. It can be written as the antibracket with the extended action L ext = L + θ. The nilpotency of s is then equivalent to the classical master equation (CME) {L ext, L ext } 0. The gauge invariant on-shell functionals are given by the 0th cohomology of s.

30 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

31 The free action I The equation of motion derived from the Nambu Goto action is µ ( g µν gd X ) ν a = 0. A natural gauge condition is thus the harmonic gauge µ ( g µν ) g = 0. One introduces Lagrange multipliers B µ and antighosts C µ (and corresponding antifields) and performs a canonical transformation to gauge fix the action. The free part of the gauge fixed action at #af = 0 is then L 0 (f ) = f [ 1 2 (QΦ)a h ab µ µ (QΦ) b B ν dxν a h ab µ µ Φ b] µ i f [ C µ ν ν C µ + C µ R ν µ C ν] µ, where Q projects on the normal bundle N Σ M,. = dxµg a µν dxν c h bc Q = id P. P a b

32 The free action II The free part of the BRST transformation is then The corresponding current is γ 0 Φ a = C µ dx a µ γ 0 C µ = 0 γ 0 C µ = ib µ γ 0 B µ = 0. j µ 0 = µ B ν C ν B ν µ C ν. There is a unique retarded (advanced) propagator ± such that S 0 αβ (x, y) βγ ± (y, z)µ(y) = δαδ(x, γ z), β with φ α = Φ, C, C, B. Set = + and define the free Peierls bracket F, G =. ( 1) (1+ F ) φα δf δg, αβ δφα δφ β. α,β We may relax the condition of global hyperbolicity of Σ by the weaker assumption that, given suitable conditions at timelike boundaries, the wave operator S 0 has unique retarded and advanced propagators.

33 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

34 Observables There is a natural class of observables, namely, given f Cc Ψ(f )(ϕ) = f ( X (x)) g(x)dx. (M) set These measure the intersection of the dynamical submanifold with the support of f. Obviously, they are invariant under reparametrization (and nonperturbatively even under changes of the background). Perturbatively, they give rise to reparametrization invariant local functionals Ψ(f ) Σ (ϕ) = k λ k k! Σ ϕ a1 (x)... ϕ a k (x) a1... ak f (X (x)) g(x)dx, provided that Σ supp f is compact. Defining the covariant functor Tens c(σ). = Γ c (Σ, T M), µ t a1...a k = dx a µt aa1...a k between Sub and Vec i, these can be interpreted as a natural transformation between Tens c and BV loc, i.e., as fields [BFV03].

35 Observables There is a natural class of observables, namely, given f Cc Ψ(f )(ϕ) = f ( X (x)) g(x)dx. (M) set These measure the intersection of the dynamical submanifold with the support of f. Obviously, they are invariant under reparametrization (and nonperturbatively even under changes of the background). Perturbatively, they give rise to reparametrization invariant local functionals Ψ(f ) Σ (ϕ) = k λ k k! Σ ϕ a1 (x)... ϕ a k (x) a1... ak f (X (x)) g(x)dx, provided that Σ supp f is compact. Defining the covariant functor Tens c(σ). = Γ c (Σ, T M), µ t a1...a k = dx a µt aa1...a k between Sub and Vec i, these can be interpreted as a natural transformation between Tens c and BV loc, i.e., as fields [BFV03]. a large class of localized observables, despite diffeomorphism invariance.

36 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

37 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

38 The Dirichlet string As background, we choose the hypersurface given by the coordinates τ, σ as R (0, π) (τ, σ) (τ, σ, 0,..., 0) M. The induced metric in these coordinates is g µν = η µν = diag( 1, 1) and we have dx a µ = 1 a µ. We consider Dirichlet boundary conditions ϕ a (0) = 0 = ϕ a (π), and analogously for the auxiliary fields. The linearized equations of motion are C µ = 0, C µ = 0, B µ = 0, Φ a = 0. The non-vanishing free Peierls brackets are given by C µ (x), C ν (y) = iδ µ ν (x, y), B µ (x), Φ a (y) = dx a µ (x, y), Φ a (x), Φ b (y) = k ab (x, y). Here is the scalar causal propagator corresponding to Dirichlet boundary conditions and k ab = diag(0, 0, 1,..., 1).

39 The two-point function, positivity There is a natural scalar two-point function ω compatible with Dirichlet boundary conditions. Hence, may now define the two-point function ω 2 by ω 2 (C µ (x) C ν (y)) = iδ µ ν ω(x, y), ω 2 (B µ (x)φ a (y)) = dx a µω(x, y), ω 2 (Φ a (x)φ b (y)) = k ab ω(x, y). It is invariant under the free gauge transformation γ 0, i.e., ω 2 γ 0 = 0. In the corresponding Krein space representation the free BRST current gives rise to the charge Q 0 with the following properties: It is nilpotent. The transversal fluctuations, i.e., those corresponding to Φ a for a 2, are in the kernel of Q 0, but not in the image, so they are physical. All other fields are not physical. The representation is positive definite in the sense that Ψ Ψ 0 for Ψ ker Q 0 and Ψ Ψ = 0 for Ψ ker Q 0 iff Ψ ran Q 0.

40 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

41 Quantization We quantize in the framework of perturbative algebraic quantum field theory [BDF09]. This proceeds via deformation quantization, i.e., we deform the classical graded commutative algebra BV(Σ) to (BV(Σ)[[ ]],, ) such that F G = F G + O( ), F G ( 1) F G G F = i F, G + O( 2 ). This is straightforward on the regular functionals, which are polynomials whose integral kernel f F is a test function. On these, one can also define the time-ordered product T, which coincides with for time-ordered arguments. In order to extend to the microcausal functionals, one uses Hadamard two-point functions, implementing a kind of point-splitting. It is possible to preserve the covariance (functoriality) in this construction. To show that two-point functions exist for open strings, one uses a deformation argument, referring to ω 2 of the Dirichlet string. Also the time-ordered product can be extended. To do this in a way that is local and preserves covariance under coordinate changes in Σ (functoriality), one uses techniques from quantum field theory on curved spacetimes [HW02]. Also target space Poincaré covariance can be preserved.

42 Interactions The interacting observables for an interaction term L int are defined via Bogoliubov s formula in the sense of the algebraic adiabatic limit [BF00], i.e., for observables F supported in a causal region O Σ, define R flint (F ) =. ( ) 1 ( ) ilint(f )/ ilint(f )/ e e, T T T F where f D(Σ), f 1 on O. The algebra BV Lint (O) thus obtained does not depend on the choice of f. The inductive limit yields the full algebra BV Lint (Σ), whose elements are formal power series in and λ.

43 Interactions The interacting observables for an interaction term L int are defined via Bogoliubov s formula in the sense of the algebraic adiabatic limit [BF00], i.e., for observables F supported in a causal region O Σ, define R flint (F ) =. ( ) 1 ( ) ilint(f )/ ilint(f )/ e e, T T T F where f D(Σ), f 1 on O. The algebra BV Lint (O) thus obtained does not depend on the choice of f. The inductive limit yields the full algebra BV Lint (Σ), whose elements are formal power series in and λ. There is a time-ordering operator T, implementing T as [FR11b] F T G = T (T 1 F T 1 G). One also defines the time-ordered antibracket as {F, G} T = T {T 1 F, T 1 G}. Defining S 0 = T (L 0 ), S int = T (θ 0 + L 1 + θ 1 ), the CME becomes {S 0 + S int, S 0 + S int } T 0.

44 Fulfillment of the QME In order for the interacting field to be independent of the choice of the gauge fixing, one has to fulfill the quantum master equation (QME) 1 2 {S 0 + S int, S 0 + S int } T i Sint (S 0 + S int ), where Sint is the generalization of the divergence of the standard BV formalism. It is the anomaly term in the Master Ward Identity [BD08]. From the CME we know that violations are at least of O( ). Hence, we replace S int by W, which contains terms of higher order in, At O( ), we thus have to solve One knows that W = j j W j, W 0 = S int. {W 1, S 0 + S int } T i 0 S int (S 0 + S int ). (1) { 0 S int (S 0 + S int ), S 0 + S int } T 0, so (1) can be solved if sf 0, #gh(f ) = 1 implies that F sg.

45 Counterterms and anomalies Hence, cohomological analysis of the BV differential s can be used to determine the admissible counterterms and potential anomalies [BBH00].

46 Counterterms and anomalies Hence, cohomological analysis of the BV differential s can be used to determine the admissible counterterms and potential anomalies [BBH00]. For the free part s 0 of the BV differential s, there are injections π :H g (s) H g (s 0 ), π :H g (s d) H g (s 0 d). Thus, to show triviality of H g (s) or H g (s d), it suffices to show triviality for H g (s 0 ) or H g (s 0 d). We note that s 0 Φ a = C µ dx a µ, and dx has maximal rank. It follows that the ghosts and all their derivatives are part of trivial pairs and can be removed from the cohomology, so that the cohomologies of s 0 at positive ghost number are trivial. Hence, there are no anomalies.

47 Counterterms and anomalies Hence, cohomological analysis of the BV differential s can be used to determine the admissible counterterms and potential anomalies [BBH00]. For the free part s 0 of the BV differential s, there are injections π :H g (s) H g (s 0 ), π :H g (s d) H g (s 0 d). Thus, to show triviality of H g (s) or H g (s d), it suffices to show triviality for H g (s 0 ) or H g (s 0 d). We note that s 0 Φ a = C µ dx a µ, and dx has maximal rank. It follows that the ghosts and all their derivatives are part of trivial pairs and can be removed from the cohomology, so that the cohomologies of s 0 at positive ghost number are trivial. Hence, there are no anomalies. The admissible counterterms may also contain extrinsic curvature terms, such as Π a µν g µλ g νρ η ab Π b λρ. Hence, we are dealing with a modified gravity theory in which also the embedding into an ambient space may be relevant.

48 Outline 1 Setup Configurations, functionals, antifields The action and its symmetries Gauge fixing, the free action Observables 2 Quantization A concrete example: The Dirichlet string The general procedure 3 Conclusion

49 Summary & Outlook Perturbative quantization of the Nambu Goto string as an effective theory. Tools from perturbative algebraic quantum field theory, quantum field theory on curved spacetimes, and the Batalin Vilkovisky formalism. No anomalies for any dimension of the target space. Difference to standard approach: Expansion around a string, not an event.

50 Summary & Outlook Perturbative quantization of the Nambu Goto string as an effective theory. Tools from perturbative algebraic quantum field theory, quantum field theory on curved spacetimes, and the Batalin Vilkovisky formalism. No anomalies for any dimension of the target space. Difference to standard approach: Expansion around a string, not an event. Many open questions: Existence of Hadamard states for the closed string? Relation to the Lüscher Weisz string? Curved target spaces? Alternative gravity theory? Fermions on the world-sheet?

51 Thank you for your attention!

52 References [BBH00] [BD08] [BDF09] [BF00] [BFV03] Glenn Barnich, Friedemann Brandt, and Marc Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000), Ferdinand Brennecke and Michael Dütsch, Removal of violations of the Master Ward Identity in perturbative QFT, Rev. Math. Phys. 20 (2008), Romeo Brunetti, Michael Dütsch, and Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Phys. 13 (2009), no. 5, Romeo Brunetti and Klaus Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208 (2000), Romeo Brunetti, Klaus Fredenhagen, and Rainer Verch, The Generally covariant locality principle: A New paradigm for local quantum field theory, Commun. Math. Phys. 237 (2003), [GGRT73] P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, Quantum dynamics of a massless relativistic string, Nucl. Phys. B 56 (1973), 109. [GPS95] [FR11a] [FR11b] [HW02] Joaquim Gomis, Jordi Paris, and Stuart Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept. 259 (1995), Klaus Fredenhagen and Katarzyna Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory, arxiv: Klaus Fredenhagen and Katarzyna Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, arxiv: Stefan Hollands and Robert M. Wald, Existence of local covariant time ordered products of quantum fields in curved space-time, Commun. Math. Phys. 231 (2002),

53 Canonical quantization, covariant approach For the open string, M = R n, orthonormal gauge, and fixed parameter time τ, one has [GGRT73] with X a (σ) = q a + n=1 1 n cos nσ (αa n + αn a ), P a (σ) = p a + i cos nσ n=1 ( αa n + αn a ), [q a, p b ] = ih ab, [α a n, α b m ] = nδmn h ab. The α ( ) s are represented as annihilation (creation) operators, q a and p b in Schrödinger representation on L 2 (R n ).

54 Canonical quantization, covariant approach For the open string, M = R n, orthonormal gauge, and fixed parameter time τ, one has [GGRT73] with X a (σ) = q a + n=1 1 n cos nσ (αa n + αn a ), P a (σ) = p a + i cos nσ n=1 ( αa n + αn a ), [q a, p b ] = ih ab, [α a n, α b m ] = nδmn h ab. The α ( ) s are represented as annihilation (creation) operators, q a and p b in Schrödinger representation on L 2 (R n ). 1 For state ψ with well-defined excitation numbers, ψ X a (σ) ψ = ψ q a ψ, so describes a particle, not a string.

55 Canonical quantization, covariant approach For the open string, M = R n, orthonormal gauge, and fixed parameter time τ, one has [GGRT73] with X a (σ) = q a + n=1 1 n cos nσ (αa n + αn a ), P a (σ) = p a + i cos nσ n=1 ( αa n + αn a ), [q a, p b ] = ih ab, [α a n, α b m ] = nδmn h ab. The α ( ) s are represented as annihilation (creation) operators, q a and p b in Schrödinger representation on L 2 (R n ). 1 For state ψ with well-defined excitation numbers, ψ X a (σ) ψ = ψ q a ψ, so describes a particle, not a string. 2 The physicality condition ( 1 2 p2 + ) n α nα n a 0 ψ = 0 enforces ψ to be a superposition of eigenvectors of p 2 for a discrete set of eigenvalues. Thus, there are no physical states.

56 Canonical quantization, light cone gauge By singling out a lightlike direction in target space, the parametrization can be fixed completely [GGRT73], by setting X + (σ) = 0, P + (σ) = p +. Then the independent operators are α i n, α i n, q i, p i, p +, q, with i transversal and [q i, p j ] = iδ ij, [α i n, α j m ] = nδmn δ ij, [q, p + ] = i. This singles out a lightlike direction, as states can be arbitrarily well localized in q + and p, which is not possible for the other canonical pairs {q i, p i } and {q, p + }. Hence, target space Lorentz invariance is explicitly broken, by the choice of a parametrization. Note that this is not in contradiction to the statement that the Lorentz algebra M ab = dσ (P a X b P b X a) closes for n = 26, as the action on the q s and p s is non-linear, δ Λ q a Λ a b qb.

57 Fields Fields should be defined simultaneously on all backgrounds [BFV03], i.e., they are natural transformations between the functors Tens c and BV loc, where Tens c (Σ). = Γ c (Σ, T M T Σ). Physical fields are given by the 0th cohomology of s, where [FR11a] ( sψ) Σ (t) = sψ Σ (t) + λ( 1) Ψ Ψ Σ (ρ Σ ( )t). Here ρ eats a vector field on Σ, i.e., it is a ghost. An example is, for t Γ c (Σ, T M), Ψ Σ (t)(ϕ) = t a X a gdx. The relation to the physical excitations as defined by the cohomology of s 0 on the observables is subtle. One has to restrict to fields that are in a certain sense compatible with the perturbative expansion to ensure positivity.