ε ε ε x1 ε ε 4 ε Perturbative Quantum Field Theory and Vertex Algebras Stefan Hollands School of Mathematics Cardiff University = y Magnify ε2 x 2 x 3 Different scalings Oi O O O O O O O O O O O 1 i 2 i 3 i i1 4 i5 i6 i2 i3 i4 i5 i6 y x ε2 x 5 x 6 y ε 0 Factorization successive mergers y Ci j 1i 2 i 3 Ci l 4 kj Ci k 5 i 6 Ci l 1 i 2 i 3 i 4 i5 i 6 27/03/2008
Outline Introduction Operator Product Expansions Deformations Vertex algebras and perturbation theory Conclusions
Different approaches to QFT: Path-integral: Z[j] = dφexp( is/ + j,φ ). Intuitive, easy to remember, relation to statistical mechanics (t iτ), classical mathematics tools. But: difficult to make rigorous ( perturbation theory). S-matrix: Clear-cut relation to scattering experiments, perturbative formulation, graphical representation. But: Not first principle, not appropriate in curved space, bound states? Wightman s or other axioms: Mathematically rigorous, conceptually clean. But: Not constructive, no interesting examples in 4 dimensions. This talk: New formulation in terms of OPE/consistency conditions. Easy to remember, mathematically rigorous, constructive, conceptually clean, works on manifolds. But: Only short distance physics.
Different approaches to QFT: Path-integral: Z[j] = dφexp( is/ + j,φ ). Intuitive, easy to remember, relation to statistical mechanics (t iτ), classical mathematics tools. But: difficult to make rigorous ( perturbation theory). S-matrix: Clear-cut relation to scattering experiments, perturbative formulation, graphical representation. But: Not first principle, not appropriate in curved space, bound states? Wightman s or other axioms: Mathematically rigorous, conceptually clean. But: Not constructive, no interesting examples in 4 dimensions. This talk: New formulation in terms of OPE/consistency conditions. Easy to remember, mathematically rigorous, constructive, conceptually clean, works on manifolds. But: Only short distance physics.
Different approaches to QFT: Path-integral: Z[j] = dφexp( is/ + j,φ ). Intuitive, easy to remember, relation to statistical mechanics (t iτ), classical mathematics tools. But: difficult to make rigorous ( perturbation theory). S-matrix: Clear-cut relation to scattering experiments, perturbative formulation, graphical representation. But: Not first principle, not appropriate in curved space, bound states? Wightman s or other axioms: Mathematically rigorous, conceptually clean. But: Not constructive, no interesting examples in 4 dimensions. This talk: New formulation in terms of OPE/consistency conditions. Easy to remember, mathematically rigorous, constructive, conceptually clean, works on manifolds. But: Only short distance physics.
Different approaches to QFT: Path-integral: Z[j] = dφexp( is/ + j,φ ). Intuitive, easy to remember, relation to statistical mechanics (t iτ), classical mathematics tools. But: difficult to make rigorous ( perturbation theory). S-matrix: Clear-cut relation to scattering experiments, perturbative formulation, graphical representation. But: Not first principle, not appropriate in curved space, bound states? Wightman s or other axioms: Mathematically rigorous, conceptually clean. But: Not constructive, no interesting examples in 4 dimensions. This talk: New formulation in terms of OPE/consistency conditions. Easy to remember, mathematically rigorous, constructive, conceptually clean, works on manifolds. But: Only short distance physics.
Main tool in my approach: OPE General formula: [Wilson, Zimmermann 1969,..., S.H. 2006] φ a1 (x 1 ) φ an (x n ) Ψ Ca b 1...a n (x 1,...,x n ) φ b (x n ) Ψ }{{} φ b OPE coefficients structure constants Physical idea: Separate the short distance regime of theory (large energies ) from the energy scale of the state (small) E 4 ρ Ψ. Application: OPE-coefficients may be calculated within perturbation theory (Yang-Mills-type theories) applications deep inelastic scattering in QCD.
Axiomatization of QFT I propose to axiomatize quantum field theory as a collection of fields (vectors in an abstract vector space V ) and operator product coefficients C(x 1,...,x n ) : V V V, each of with is an analytic function on (R D ) n \ {diagonals}, subject to Covariance Local (anti-) commutativity Analyticity (Euclidean framework) Consistency (Associativity) Hermitian adjoint Consequences: New intrinsic formulation of perturbation theory Constructive tool
Considering the product of quantum fields at three different spacetime points, associativity of the field operators, φ a (x 1 )(φ b (x 2 )φ c (x 3 )) = (φ a (x 1 )φ b (x 2 )) φ c (x 3 ), yields the consistency condition Cac e (x 1,x 3 )Cbd c (x 2,x 3 ) = Cab c (x 1,x 2 )Ccd e (x 2,x 3 ) c c on domain D 3 = {r 12 < r 23 < r 13 }. Idea: Elevate the OPE to an axiom of QFT, i.e. define a QFT by a set of coefficients Cab c (x,y) satisfying the consistency condition (among other axioms)
Mathematical formulation of the consistency condition: Postulate that ( ) C(x 2,x 3 ) C(x 1,x 2 ) id ( ) = C(x 1,x 3 ) id C(x 2,x 3 ), Here, we view C(x 1,x 2 ) abstractly as a mapping V V V ( index-free notation ), where V is the space of all composite fields of the given theory. The above equation is valid in the sense of analytic functions on domain D 3 = {r 12 < r 23 < r 13 }. Key Idea: The mappings C(x 1,x 2,... ) define (and hence determine) the quantum field theory! Coherence theorem: All higher order C s and consistency conditions follow from this one. (Analogy (AB)C = A(BC) implies higher associativity conditions such as (AB)(CD) = (A(BC))D etc. in ordinary algebra).
Perturbation theory Suppose we have a family of QFT s depending on parameter: Coupling parameter: λ. t Hooft limit: ǫ = 1/N. Classical limit: -expansion. Expand OPE-coefficients: C i (x 1,x 2 ) := di dλ ic(x 1,x 2 ;λ). λ=0 Then C i should satisfy order by order version of consistency condition. Lowest order condition determines higher order ones. = Conditions have formulation in terms of Hochschild cohomology.
Idea: Express perturbative consistency condition in term of differential. Let f n (x 1,...,x n ) : V V V, (x 1,...,x n ) D n. We next introduce a boundary operator b on such objects by the formula (bf n )(x 1,...,x n+1 ) := C 0 (x 1,x n+1 )(id f n (x 2,...,x n )) n + ( 1) i f n (x 1,..., x i,...,x n+1 )(id i 1 C 0 (x i,x i+1 ) id n i ) i=1 +( 1) n C 0 (x n,x n+1 )(f n (x 1,...,x n ) id). A calculation reveals b 2 = 0.
1 The first order consistency condition states that C 1 must satisfy bc 1 = 0. 2 If C 1 arises from a field redefinition (a map z : V V ), then this means that C 1 = bz 1. = {1st order perturbations C 1 } = H 2 (b) = ker b/ran b 3 At i-th order, we get a condition of the form bc i = w i, where bw i = 0, which we want to solve for C i (with w i defined by lower order perturbations). = ith order obstruction w i H 3 (b) = ker b/ran b
Gauge theories For gauge theories (e.g. Yang-Mills) need a further modification: BRST symmetry (e.g. Yang-Mills: sa = du iλ[a,u],su = λ/2i[u,u],...) BRST-transformation defines map s(λ) : V V. Must satisfy compatibility condition sc(x 1,x 2 ) = C(x 1,x 2 ) ( s id + γ s ). Expand: s i := di dλ is(λ). λ=0 Then s i,c i should satisfy order by order version of compatibility condition.
= Conditions can be reformulated in terms of modified Hochschild cohomology: Define new differential B by (Bf n )(x 1,...,x n ) := sf n (x 1,...,x n ) Then one can prove n f n (x 1,...,x n )(γ i 1 s id n i ). i=1 B 2 = 0 = {b,b}, so δ = b + B defines new differential. We can then discuss associativity and BRST condition simultaneously for C i,s i in terms of δ.
Connection to Vertex algebras arises as follows: We view this set of coefficients as matrix elements of operators Y (x,φ a ) on the space V spanned by the fields φ a : C c ab (x) = φ c Y (φ a,x) φ b This is very useful to construct the OPE in non-trivial perturbative QFT s! (rest of this talk). From now: φ a a.
OPE vertex algebras Axioms imply that Y satisfy axioms of a vertex algebra : An OPE vertex algebra is a 4-tuple (V, Y, µ, 0 ), where V is a vector space, µ End(V ) a derivation, µ = 1,..., D, 0 V, and Y : V End(V ) O(R D \ {diagonals}), linear in V, satisfying: Vacuum: Y (x, 0 ) = 1 V, µ 0 = 0, Y (x, a) 0 = a + O(x) Compatible derivations: Y (x, µ a) = µ Y (x, a) Euclidean invariance Consistency condition: Y (x, a)y (y, b) = Y (y, Y (x y, a)b) for x > y > x y Quasisymmetry: Y (x, a)b = exp(x )Y ( x, b)a Scaling degree: sd x=0 Y (x, a) dim(a)
How to construct OPE vertex algebras? How to characterize a QFT? E.g. by a classical field equation: ϕ = λϕ 3 This yields some restrictions on the OPE coefficients and thus on the vertex operators: Y (x,ϕ) = λy (x,ϕ 3 ) We want to exploit these relations and develop an iterative construction scheme
Construction of OPE vertex algebras Perturbative construction of the QFT associated to the scalar field satisfying the field equation ϕ = λϕ 3 : Construct (formal) power series of vertex operators Y (x,a) = λ i Y i (x,a) satisfying (1) i=0 a) the field equation, Y (x,ϕ) = λy (x,ϕ 3 ) Y i (x,ϕ) = Y i 1 (x,ϕ 3 ) b) the consistency condition, i i Y k (x,a)y i k (x,b) = Y k (y,y i k (x y,a)b) k=0 k=0
Computing higher order vertex operators Start with the vertex operators of the free field (0-th order vertex operators) Invert the field equation to get to the next order: Y 1 (x,ϕ) = 1 Y 0 (x,ϕ 3 ) Use the consistency condition in the limit x y to find the first order vertex operators with non-linear vector arguments: Y 1 (x, ϕ 2 ) = lim y x {Y 1 (y, ϕ)y 0 (x, ϕ) } a a Y 1(y x, ϕ) ϕ Y 0 (x, a) + (0 1) or, more generally, Y i (x, a b) = lim y x j=0 i Y j (y, a)y i j (x, b) counterterms
The Euclidean free field Consider the Euclidean free field ϕ(x) in D 3 dimensions (Schwinger two point-function G(x,y) = x y 2 D ). We define the corresponding OPE vertex algebra: V =unital, free commutative ring generated by 1, ϕ and its symmetric trace free derivatives, l,m ϕ = c l S l,m ( )ϕ, where S l,m (ˆx) are the spherical harmonics in D dimensions and c l is some normalisation constant. We introduce creation and annihilition operators on V, ] b + l,m 1 = l,m ϕ, b l,m 1 = 0, [b l,m,b +l,m = 1
Y can be read off the OPE of the free field normal ordered products Y (x, ϕ) = const. r (D 2)/2 l=0 N(l,D) m=1 1 ω(d, l) [r l+(d 2)/2 S l,m (ˆx)b +l,m + r l (D 2)/2 S l,m (ˆx)b l,m ] r = x, ω(l, D) = 2l + D 2 N(l, D) = number of linearly independent spherical harmonics S l,m (ˆx) of degree l in D dimensions
1 Calculate the first order vertex operator Y 1 (x,ϕ): Y 1 (x,ϕ) = 1 Y 0 (x,ϕ 3 ) 2 Use the consistency condition to find Y 1 (x,ϕ 2 ) and Y 1 (x,ϕ 3 ) 3 Go to 2nd order by Y 2 (x,ϕ) = 1 Y 1 (x,ϕ 3 ), and so on 4 vertex operators Y j (x,ϕ p ), p > 3 can also be calculated by using the consistency condition
Formula for the iteration step : Y i+1 (x, ϕ) = 1 Y i (x, ϕ 3 ) [ i ] = 1 lim Y j (y, ϕ)y i j (x, ϕ 2 ) counterterms y x j=0 [ i [ i j = 1 lim Y j (y 1, ϕ) lim Y k (y 1, ϕ)y i j k (x, ϕ) y 1 x y 2 x j=0 k=0 ] ] more counterterms counterterms Dropping the counterterms and limits for the moment, this reads i i j Y i+1 (x, ϕ) = 1 Y j (x, ϕ)y k (x, ϕ)y i j k (x, ϕ) j=0 k=0
This suggests a graphical representation by trees: Y i+1 (x, ϕ) Y j (x, ϕ) Y k (x, ϕ) Y i j k (x, ϕ)
Diagrammatic rules for writing down an integral expression for Y n (x,ϕ): Draw all trees with n 4-valent vertices (labeled by 1,..., n) With the vertex i, associate a number δ i C \ Z and a unit vector ˆx i With the line between vertices i and j, associate a momentum ν ij C \ Z Label the leaves by numbers 1,...,n L. With the leaf j, associate numbers l j,m j N (m j N(l j,d))
Now to each tree, we apply the following graphical rules: ν ij π sin πν ij P( ˆx i ˆx j, ν ij, D) ν ik1 δ i r 2+δi δ(2 + ν ik1 + ν ik2 + ν ik3 ν ik4 + δ i ) ν ik2 ν ik3 ν ik4 K D ω(l j ) 1/2 S lj,m j (ˆx)r lj b + l j,m j l j, m j l j, m j K D ω(l j ) 1/2 S lj,m j (ˆx)r lj D+2 b lj,m j
Write down all these factors and integrate over all ˆx i S D 1 dˆx i integrate over all ν ij C dν ij integrate over all δ i 1 2πi dδi δ i take the sum over all l j,m j (Here, the expression becomes ill-defined consideration of counterterms/renormalization necessary)
Example ˆx ( 1 2πi ) 3 K 4 D dδ3 S D 1 dˆx 3 S D 1 dˆx 2 π sin π(l 3 l 2 D+4+δ 1) dδ2 dδ1 δ 3 δ 2 δ 1 S D 1 dˆx 1 ˆx 2 l 1, m 1 ˆx 3 ˆx 1 l 4, m 4 P( ˆx 1 ˆx 2, l 3 l 2 D + 4 + δ 1, D) π sin π(l 1+l 3 l 2 D+6+δ 1+δ 2) P( ˆx 2 ˆx 3, l 1 + l 3 l 2 D + 6 + δ 1 + δ 2, D) π sin π(l 1+l 3 l 2 l 4 2D+10+ δ P( ˆx ˆx i) 3, l 2, m 2 l 3, m 3, l 1 + l 3 l 2 l 4 2D + 10 + δ i, D) S l1,m 1 (ˆx)S l2,m 2 (ˆx)S l3,m 3 (ˆx)S l4,m 4 (ˆx) b + l 1,m 1 b l2,m 2 b + l 3,m 3 b l4,m 4 r l1+l3 l2 l4 2D+10+
Conclusions 1 The OPE can be used to give a general definition of QFT independent of Lagrangians or special states (such as vacuum). 2 One can impose powerful consistency conditions on the OPE. These incorporate algebraic content of QFT. 3 Perturbations can be characterized intrinsically 4 Consistency conditions together with field equations give rise to new and efficient scheme for pert. calculations. 5 Renormalization not needed.