aa r X i v : . [ phy s i c s . g e n - ph ] F e b A MATHEMATICAL APPROACH TO QUANTUMFIELD THEORY
ALEXANDER ROI STOYANOVSKY
Abstract.
We develop a mathematical theory of quantizationof multidimensional variational principles, and compare it withtraditional constructions of quantum field theory. We conjecturethat mathematical realization of quantum field theory axioms, ingeneral, does not exist.
Introduction
The purpose of this paper is to present an attempt to construct amathematical version of quantum field theory, which we consider aspart of a mathematical generalization of the theory of linear partialdifferential equations to the case when the bicharacteristics are notcurves but are (multidimensional) surfaces. We think that this theorycan be interesting from mathematical point of view. We also compareit with traditional quantum field theory. The result is a conjecture thatquantum field theory axioms, in general, can be neither mathematicallyimplemented nor disproved, like Continuum Hypothesis.The paper consists of three Sections. In § § renormalized Tomonaga–Schwinger equation . In § S -matrix, the Wightman functions, andthe Green functions for the ϕ d model of scalar field in the spacetime R n +1 for arbitrary d and n . It seems clear that, if these objects are welldefined, i. e. do not depend on regularization of the theory, then theysatisfy the axioms and the perturbative expansions of quantum fieldtheory. However, these objects are, in general, ill defined. We thinkthat this is the problem not of our approach but of traditional quantum field theory. We conjecture that quantum field theory is not a mathe-matical theory, i. e. its axioms cannot be, in general, mathematicallyrealized. 1. Equations of classical field theory
The formalism of classical field theory equations that we shall needhas been developed in the papers [1–4] and in the book [5]. For thesake of completeness, let us briefly recall it here.Consider the action functional of the form(1) J = Z D F ( x , . . . , x n , ϕ , . . . , ϕ m , ϕ x , . . . , ϕ mx n ) dx . . . dx n , where x = ( x , . . . , x n ) is a space-time point; ϕ i = ϕ i ( x ) are real smoothfield functions, i = 1 , . . . , m ; ϕ ix j = ∂ϕ i /∂x j , j = 0 , . . . , n , and inte-gration goes over a domain D in the space-time R n +1 with the smoothboundary C = ∂ D .1.1. Formula for variation of action.
We shall need the well knownformula for variation of action on a domain with moving boundary,which is used, for instance, in derivation of the Noether theorem. Let usrecall this formula. Let s = ( s , . . . , s n ) be parameters on the boundarysurface C : x = x ( s ). Let C ( ε ) : x = x ( s, ε ) be a deformation of theparameterized surface C depending on a small parameter ε , x ( s,
0) = x ( s ), and let ϕ i ( x, ε ) be a deformation of the field function ϕ i ( x ) = ϕ i ( x,
0) depending on ε . Put ϕ i ( s, ε ) = ϕ i ( x ( s, ε ) , ε ), and let δ denotethe variation, i. e. the differential with respect to the parameter ε at ε = 0. Let J = J ( ε ) be the integral (1) with the field functions ϕ i ( x, ε ) over the domain D ( ε ) with the boundary C ( ε ). Then one hasthe following formula for variation of the functional J :(2) δJ = Z D X i F ϕ i − X j ∂∂x j F ϕ ixj ! δϕ i ( x ) dx + Z C (cid:16)X π i ( s ) δϕ i ( s ) − X H j ( s ) δx j ( s ) (cid:17) ds, MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 3 where F ϕ i = ∂F/∂ϕ i , F ϕ ixj = ∂F/∂ϕ ix j ,(3) π i = X l F ϕ ixl ( − l ∂ ( x , . . . , b x l , . . . , x n ) ∂ ( s , . . . , s n ) ,H j = X l = j X i F ϕ ixl ϕ ix j ! ( − l ∂ ( x , . . . , b x l , . . . , x n ) ∂ ( s , . . . , s n )+ X i F ϕ ixj ϕ ix j − F ! ( − j ∂ ( x , . . . , b x j , . . . , x n ) ∂ ( s , . . . , s n ) . Here ∂ ( x ,...,x n ) ∂ ( s ,...,s n ) is the Jacobian, and the hat over a variable means thatthe variable is omitted.If the boundary functions are fixed, δϕ i ( s ) = δx j ( s ) = 0, then thecondition δJ = 0 yields the Euler–Lagrange equations(4) F ϕ i − X j ∂∂x j F ϕ ixj = 0 , i = 1 , . . . , m. This is a system of nonlinear second order partial differential equationsfor the functions ϕ i ( x ).1.2. The generalized Hamilton–Jacobi equation.
Let us assumethat for any functions x j ( s ) , ϕ i ( s ) sufficiently close to certain fixed func-tions, there exist the unique field functions ϕ i ( x ) satisfying the Euler–Lagrange equations (4) with the boundary conditions ϕ i ( x ( s )) = ϕ i ( s ).Denote by S = S ( ϕ i ( s ) , x j ( s )) the integral (1) with these field func-tions ϕ ( x ) over the domain D with the boundary C : x = x ( s ). Let usderive the equations satisfied by the functional S ( ϕ i ( s ) , x j ( s )).To this end, note that by formula (3), the quantities(5) π i ( s ) = δSδϕ i ( s ) , H j ( s ) = − δSδx j ( s )(for the definition of variational derivatives δS/δϕ i ( s ) and δS/δx j ( s ),see Subsect. 2.0 below) depend not only on the functions ϕ i ( s ), x j ( s ),but also on the derivatives ϕ ix j . These m ( n + 1) derivatives are relatedby mn equations(6) X j ϕ ix j x js k = ϕ is k , i = 1 , . . . , m, k = 1 , . . . , n. Therefore, m + n + 1 quantities (5) depend on m ( n + 1) − mn = m free parameters. Hence they are related by n + 1 equations. n of these ALEXANDER ROI STOYANOVSKY equations are easy to find:(7) X i ϕ is k π i − X j x js k H j = 0 , k = 1 , . . . , n. These equations express the fact that the value of the functional S doesnot depend on a parameterization of the surface C .The remaining ( n + 1)-th equation depends on the form of the func-tion F . Denote it by(8) H (cid:0) x j ( s ) , ϕ i ( s ) , x js k ( s ) , ϕ is k ( s ) , π i ( s ) , − H j ( s ) (cid:1) = 0 . From equalities (5) we obtain the following system of equations on thefunctional S , called the generalized Hamilton–Jacobi equation :(9) P i ϕ is k δSδϕ i ( s ) + P j x js k δSδx j ( s ) = 0 , k = 1 , . . . , n, H (cid:16) x j ( s ) , ϕ i ( s ) , x js k ( s ) , ϕ is k ( s ) , δSδϕ i ( s ) , δSδx j ( s ) (cid:17) = 0 . Examples.
1) (Scalar field with self-action) [4, 5] Let m = 1 and(10) F ( x j , ϕ, ϕ x j ) = 12 ϕ x − X j =0 ϕ x j ! − V ( x, ϕ )= 12 ϕ x µ ϕ x µ − V ( x, ϕ ) , where we have introduced Greek indices µ instead of j , and raising andlowering indices goes using the Lorentz metric(11) dx = ( dx ) − X j =0 ( dx j ) . Then the generalized Hamilton–Jacobi equation has the following form:(12) x µs k δSδx µ ( s ) + ϕ s k δSδϕ ( s ) = 0 , k = 1 , . . . , n, vol · δSδ n ( s ) + (cid:16) δSδϕ ( s ) (cid:17) + vol (cid:0) dϕ ( s ) + V ( x ( s ) , ϕ ( s )) (cid:1) = 0 , where vol is the volume element on the surface x = x ( s ),(13) vol = D µ D µ , D µ = ( − µ ∂ ( x , . . . , c x µ , . . . , x n ) ∂ ( s , . . . , s n ) ,δS/δ n ( s ) is the normal derivative,(14) vol · δSδ n ( s ) = D µ δSδx µ ( s ) , MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 5 and dϕ ( s ) is the scalar square of the differential dϕ ( s ) of the function ϕ ( s ) on the surface x = x ( s ),(15) vol dϕ ( s ) =( D µ ϕ x µ ) − ( D µ D µ )( ϕ x ν ϕ x ν ) = − X µ<ν X k,l D µνk D µν,l ϕ s k ϕ s l ,D µνk = ( − k + µ + ν ∂ ( x , . . . , c x µ , . . . , b x ν , . . . , x n ) ∂ ( s , . . . , b s k , . . . , s n ) .
2) (The Plateau problem) [5] Assume that one considers ( n + 1)-dimensional surfaces D with the boundary C : x = x ( s ) in the Eu-clidean space R N with coordinates ( x , . . . , x N ), and the role of inte-gral (1) is played by the area of the surface D . Then the generalizedHamilton–Jacobi equation is(16) P j x js k δSδx j ( s ) = 0 , ≤ k ≤ n, P j (cid:16) δSδx j ( s ) (cid:17) = P j <... From n + 1equations (7, 8) one can, in general, express H j as functions(17) H j = H j ( s ) = H j ( x ( s ) , x s k ( s ) , ϕ i ( s ) , ϕ is k ( s ) , π i ( s )) . Let x j = x j ( s , . . . , s n , t ), j = 0 , . . . , n , be arbitrary functions, and let(18) H = H ( t ) = H ( t, ϕ i ( · ) , π i ( · ))= Z X j H j ( x ( s, t ) , x s k ( s, t ) , ϕ i ( s ) , ϕ is k ( s ) , π i ( s )) x jt ( s, t ) ds, where ϕ i ( · ), π i ( · ) are two functions of s . Then the Euler–Lagrangeequations (4) are equivalent to the following generalized canonical Ham-ilton equations :(19) ( ∂ϕ i ( s,t ) ∂t = δHδπ i ( s ) ( t, ϕ i ( · , t ) , π i ( · , t )) , ∂π i ( s,t ) ∂t = − δHδϕ i ( s ) ( t, ϕ i ( · , t ) , π i ( · , t )) . Another form of these equations is the following one. LetΦ( x j ( · ); ϕ i ( · ) , π i ( · ))be a functional of functions x j ( s ), ϕ i ( s ), π i ( s ). Then the condition(20) d Φ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) /dt = 0 ALEXANDER ROI STOYANOVSKY is equivalent to the equation(21) ∂ Φ( x j ( · , t ); ϕ i ( · ) , π i ( · )) ∂t = { Φ( x j ( · , t ); ϕ i ( · ) , π i ( · )) , H ( t ) } , where(22) { Φ , Φ } = Z X i (cid:18) δ Φ δπ i ( s ) δ Φ δϕ i ( s ) − δ Φ δϕ i ( s ) δ Φ δπ i ( s ) (cid:19) ds is the Poisson bracket of two functionals Φ ( ϕ i ( · ) , π i ( · )), Φ ( ϕ i ( · ) , π i ( · )).Equivalently, the equation is(23) δ Φ δx j ( s ) = { Φ , H j ( s ) } . Let us call by a classical field theory observable a functional Φ( x j ( · ); ϕ i ( · ), π i ( · )) satisfying equations (20), (21), or (23). Observables forma commutative Poisson algebra with respect to multiplication of func-tionals and the Poisson bracket (22). Equations (20), (21), or (23)mean that Φ actually depends not on x j ( s ), ϕ i ( s ), π i ( s ) but only onthe solution ϕ i ( x ) of the Euler–Lagrange equations (4) with the initialconditions ϕ i ( x ( s )) = ϕ i ( s ), π i ( x ( s )) = π i ( s ).The system of equations (23) satisfies the Frobenius integrability(zero curvature) condition,(24) δH j ( s ) δx j ′ ( s ′ ) − δH j ′ ( s ′ ) δx j ( s ) − { H j ( s ) , H j ′ ( s ′ ) } = 0 . This condition means that the solution of the system (23) is well de-fined. Example. For the scalar field with self-action (10, 12), the system ofgeneralized canonical Hamilton equations is equivalent to the followingsystem of equations for a functional Φ( x µ ( · ); ϕ ( · ) , π ( · )):(25) ( x µs k ( s ) δ Φ δx µ ( s ) = { Φ , ϕ s k ( s ) π ( s ) } , k = 1 , . . . , n, vol · δ Φ δ n ( s ) = (cid:8) Φ , π ( s ) + vol (cid:0) dϕ ( s ) + V ( x ( s ) , ϕ ( s )) (cid:1)(cid:9) . The Euler–Lagrange equations and the equivalent generalized canon-ical Hamilton equations are called the characteristics equations for thegeneralized Hamilton–Jacobi equation. Regarding the integration the-ory of the generalized Hamilton–Jacobi equation using integration ofthe characteristics equations, see [13] (for the case m = n = 1) and [2,5] (for the general case). Problem. Conversely, integrate Euler–Lagrange equations withlarge symmetry groups (such as the Einstein or Yang–Mills equations)using integration of the generalized Hamilton–Jacobi equation. MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 7 Symmetries. Let G be a Lie group of symmetries of the ac-tion (1), i. e. transformations ( x, ϕ ) → ( e x, e ϕ ) = P ( x, ϕ ) of variables( x, ϕ ) = ( x j , ϕ i ) preserving the integral (1), and let g be the Lie alge-bra of G . Let α = dP ( ε ) /dε | ε =0 ∈ g , where P ( ε ) is a curve in G with P (0) = 1, and let(26) δ e ϕ iα ( x, ϕ ) = a iα ( x, ϕ ) dε, δ e x jα ( x, ϕ ) = b jα ( x, ϕ ) dε be the corresponding infinitesimal transformation of the variables ( x j , ϕ i ). Let x j ( s, t ) be arbitrary functions, let ϕ i ( s, t ) = ϕ i ( x ( s, t )) , π i ( s, t ) = π i ( x ( s, t ))be the solution of the canonical Hamilton equations (19) correspondingto a solution ϕ i ( x ) of the Euler–Lagrange equations (4), and let C t bethe surface x = x ( s, t ). Let us apply formula (2) with the boundarysurface C t ∪ C t and with( x j ( s, t, ε ) , ϕ i ( x ( s, t, ε ) , ε ) , π i ( x ( s, t, ε ) , ε ))= P ( ε )( x ( s, t ) , ϕ ( s, t ) , π ( s, t )) ,δx j ( s ) = δ e x jα ( x ( s, t σ ) , ϕ ( s, t σ )) ,δϕ i ( s ) = δ e ϕ iα ( x ( s, t σ ) , ϕ ( s, t σ )) , σ = 1 , . Since the functional J is preserved by the transformation P ( ε ), we have δJ = 0. This implies that the functional(27) Φ α = Φ α ( x j ( · ); ϕ i ( · ) , π i ( · ))= Z C (cid:16)X π i ( s ) a iα ( x ( s ) , ϕ ( s )) − X H j ( s ) b jα ( x ( s ) , ϕ ( s )) (cid:17) ds satisfies the equalityΦ α ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) = Φ α ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) , i. e. it satisfies equation (20). In other words, Φ α is an observable.This observable generates the symmetry Hamiltonian flow, in thesense that(28) ∂ϕ i ( s,t,ε ) ∂ε (cid:12)(cid:12)(cid:12) ε =0 = δ Φ α δπ i ( s ) ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) , ∂π i ( s,t,ε ) ∂ε (cid:12)(cid:12)(cid:12) ε =0 = − δ Φ α δϕ i ( s ) ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) . Equivalently, for an arbitrary observable Φ( x j ( · ); ϕ i ( · ) , π i ( · )) and forany P ∈ G , the functional(29) P Φ( x j ( · ); ϕ i ( · ) , π i ( · )) = Φ( x j ( · ); e ϕ i ( · ) , e π i ( · ))is also an observable, where(30) ( e x j ( · ) , e ϕ i ( · ) , e π i ( · )) = P ( x ( · ) , ϕ ( · ) , π ( · )) , ALEXANDER ROI STOYANOVSKY and one has(31) ∂P ( ε )Φ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = { Φ α , Φ } ( x j ( · , t ); ϕ i ( · , t ) , π i ( · , t )) . For α, β ∈ g , it is not difficult to check that(32) { Φ α , Φ β } = Φ [ α,β ] . Examples. 1) For the scalar field with self-action (10, 12, 25) suchthat V ( x, ϕ ) = V ( ϕ ) is independent of x , the symmetry group G is thenonhomogeneous Lorentz group O ( n, e × R n +1 .2) For the Plateau problem (16), the symmetry group is the group O ( N ) e × R N of isometries of the Euclidean space R N .2. Mathematical quantization of fields In this Section we construct mathematical quantization of classicalfield theories.2.0. Differential calculus for functionals. Let Ψ( ϕ ( · )) be a func-tional depending on a smooth function ϕ ( s ), s = ( s , . . . , s n ). We shallconsider functions ϕ ( s ) from a topological vector space V of functionswhich is a nuclear space or a union of nuclear spaces [12], for example,the Schwartz space of smooth functions rapidly decreasing at infinityor the space of smooth functions with compact support. We shall callthe elements of the space V by main functions, and the elements of thedual space V ′ by distributions. One has the inclusion V ⊂ V ′ .The functional Ψ is called differentiable if for an arbitrary mainfunction ϕ ( s, ε ) smoothly depending on a small parameter ε , one has(33) δ Ψ( ϕ ( · )) = Z δ Ψ δϕ ( s ) ( ϕ ( · )) δϕ ( s ) ds for some distribution δ Ψ /δϕ ( s ) called the functional or variational de-rivative of Ψ at the point ϕ ( · ), where δ Ψ and δϕ ( s ) denote the differ-ential of Ψ( ϕ ( · , ε )) and ϕ ( s, ε ) with respect to ε at ε = 0.The functional Ψ is called infinitely differentiable if for any k onehas(34) δ k Ψ( ϕ ( · )) = Z δ k Ψ δϕ ( s ) . . . δϕ ( s k ) ( ϕ ( · )) δϕ ( s ) . . . δϕ ( s k ) ds . . . ds k for some symmetric distribution δ k Ψ /δϕ ( s ) . . . δϕ ( s k ) of s , . . . , s k call-ed the k -th functional or variational derivative of Ψ at the point ϕ ( · ), MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 9 where δ k Ψ denotes the k -th differential of Ψ( ϕ ( · , ε )) with respect to ε at ε = 0.2.1. Renormalization map and noncommutative multiplicationof field theory Hamiltonians. The idea of this Subsection has beenproposed in the papers [6–8].Let V be the space of smooth main functions ( ϕ i ( s ), π i ( s ), i =1 , . . . , m ). Denote π i ( s ) = ρ i ( s ), ϕ i ( s ) = ρ m + i ( s ), i = 1 , . . . , m . Let H = H ( ρ ( · )) be an infinitely differentiable functional on the space V , where ρ ( s ) = ( ρ i ( s ), i = 1 , . . . , m ). We shall call H a field theory Hamiltonian . A Hamiltonian H is called regular if for any k and for any ρ ( s ) = ( ρ i ( s )) ∈ V , the k -th functional derivatives δ k Hδρ i ( s ) ...δρ ik ( s k ) ( ρ ( · ))are main functions of ( s , . . . , s k ). A Hamiltonian H is called classical if it generates a well defined Hamiltonian flow(35) ( ∂ϕ i ( s,t ) ∂t = δHδπ i ( s ) ( ϕ i ( · , t ) , π i ( · , t )) , ∂π i ( s,t ) ∂t = − δHδϕ i ( s ) ( ϕ i ( · , t ) , π i ( · , t ))on the space V , i. e. if δHδρ i ( s ) ( ρ ( · )) is a main function of s smoothlydepending on ρ ( · ). In other words, denote by SV ′ = ∞ M k =0 S k V ′ , SV = ∞ M k =0 S k V the topological symmetric algebra of the space V ′ and respectively of V . Then a Hamiltonian H is regular if and only if for any k and any ρ ( · ) one has δ k Hδρ i ( s ) ...δρ ik ( s k ) ( ρ ( · )) ∈ S k V , and H is classical if and onlyif for any k and any ρ ( · ) one has δ k Hδρ i ( s ) . . . δρ i k ( s k ) ( ρ ( · )) ∈ S k V ′ ∩ ( S k − V ′ ⊗ V ) . An example of classical Hamiltonian is the Hamiltonian H ( t ) given by(18) (for any t ).Classical Hamiltonians form a Poisson algebra with respect to mul-tiplication of Hamiltonians and the Poisson bracket (22),(36) { H , H } = Z X i,j ω ij δH δρ i ( s ) δH δρ j ( s ) ds, where ω ij = δ i,j − m − δ i − m,j . This Poisson algebra appeared in a non-rigorous form in the book [11], and in the rigorous form in [6–8]. Hamil-tonian regularization, i. e. approximation of field theory Hamiltoniansby regular Hamiltonians, appeared in the book [14]. Define the Weyl–Moyal algebra as the associative algebra SV of reg-ular polynomial Hamiltonians with the Moyal product(37) H ∗ Moyal H ( ρ ( · )) = exp − ih Z X i,j ω ij δδρ (1) i ( s ) δδρ (2) j ( s ) ds ! H ( ρ (1) ( · )) H ( ρ (2) ( · )) (cid:12)(cid:12) ρ (1) ( · )= ρ (2) ( · )= ρ ( · ) . Here ih =(imaginary unit) × (the Planck constant (a small parameter)).Let(38) G N = G i ,...,i N N ( h, s , . . . , s N ) = ∞ X k =1 h k G i ,...,i N N,k ( s , . . . , s N ) ∈ S N V ′ , for N = 2 , , . . . , be real symmetric distributions depending on h . Fora regular polynomial Hamiltonian H , put(39) e H = exp ∞ X N =2 N ! Z X i ,...,i N G i ,...,i N N ( h, s , . . . , s N ) × δ N δρ i ( s ) . . . δρ i N ( s N ) ds . . . ds N (cid:19) H. The Hamiltonian e H is called the renormalized (regular) Hamiltonian ,and e H − H is called the counterterm (regular) Hamiltonian . The trans-form H → e H (39) is called the renormalization map .Let H → b H be the inverse transform to the transform H → e H , i.e. b H is given by formula (39) with G N replaced by − G N .Define the ∗ -product of regular polynomial Hamiltonians by the for-mula(40) H ∗ H = ( b H ∗ Moyal b H ) e . This is an associative product on the space SV of regular polynomialHamiltonians.Explicitly, one has(41) H ∗ H = exp − ih Z X i,j ω ij δδρ (1) i ( s ) δδρ (2) j ( s ) ds + ∞ X N =2 N − X l =1 G ( l,N − l ) N ! H ( ρ (1) ( · )) H ( ρ (2) ( · )) (cid:12)(cid:12) ρ (1) ( · )= ρ (2) ( · )= ρ ( · ) , MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 11 where(42) G ( l,N − l ) N = 1 l !( N − l )! X i ,...,i N Z G i ,...,i N N ( h, s , . . . , s N ) × δ l δρ (1) i ( s ) . . . δρ (1) i l ( s l ) δ N − l δρ (2) i l +1 ( s l +1 ) . . . δρ (2) i N ( s N ) ds . . . ds N . One has(43) [ H , H ] def = H ∗ H − H ∗ H = − ih { H , H } + o ( h ) . Definition. The transform (39) is called a finite renormalization if all the distributions G N are main functions. Two renormalizationscorresponding to distributions G ′ N and G ′′ N are called equivalent if theydiffer by a finite renormalization (39) with G N = G ′ N − G ′′ N . Finiterenormalizations form a commutative group called the renormalizationgroup .2.2. Quantization of classical field theory equations. Consider aclassical field theory with the action functional (1) and with the gen-eralized Hamilton–Jacobi equation (9). Let (40) be the ∗ -product cor-responding to a renormalization (39). Let us call by the renormalizedTomonaga–Schwinger equation the system of equations(44) ih P j x js k δ Φ δx j ( s ) = P i ϕ is k π i ( s ) ∗ Φ + c k ( s, h )Φ , k = 1 , . . . , n, H (cid:16) x j ( s ) , ϕ i ( s ) , x js k ( s ) , ϕ is k ( s ) , π i ( s ) , − ih δδx j ( s ) (cid:17) ∗ Φ+ C ( s, x j ( s ) , ϕ i ( s ) , π i ( s ) , x js k ( s ) , ϕ is k ( s ) , π is k ( s ) , . . . , h ) ∗ Φ = 0for a functional Φ( x j ( · ); ϕ i ( · ) , π i ( · )), where(45) C ( . . . , h ) = C ( s, x j ( · ) , ϕ i ( · ) , π i ( · ) , h )is a function of s , x j ( s ), ϕ i ( s ), π i ( s ) and their partial derivatives of finiteorders at the point s with C ( . . . , 0) = 0, and c k ( s, h ), k = 1 , . . . , n ,are distributions in s with c k ( s, 0) = 0. We assume, without loss ofgenerality, that(46) H = H ( . . . , − H j ( s )) = n X j =0 C j ( s ) H j ( s ) + A ( s ) , where C j ( s ) and A ( s ) do not depend on H j ′ ( s ), and for any j thecoefficient C j ( s ) before H j ( s ) does not depend on x j ( · ). For the scalarfield with self-action (10–15) this condition is satisfied. because the ( n + 1)-th equation H = 0 in the generalized Hamilton–Jacobiequation is defined not uniquely but only modulo the first n equations. Agreement. The system of equations (44), as well as the subse-quent constructions and formulas, are understood as the formal limitof regularized formulas, with the Hamiltonians H ( ρ ( · )) replaced by reg-ular Hamiltonians H Λ reg ( ρ ( · )), Λ > Λ → H Λ reg = H . We saythat a formal formula including irregular Hamiltonians is well defined if the limit as Λ → ih δ Φ δx j ( s ) = H j ( s, h ) ∗ Φ , where H j ( s, h ) is a deformation of the function (17), i. e. to the equa-tion(48) ih ∂ Φ( x ( · , t ) , ϕ i ( · ) , π i ( · )) ∂t = H ( t, h ) ∗ Φfor any function x ( s, t ), where H ( t, h ) is a deformation of the Hamil-tonian (18). Example. For the scalar field (10, 12), the renormalized Tomonaga–Schwinger equation is(49) ( ihx µs k ( s ) δ Φ δx µ ( s ) = ϕ s k ( s ) π ( s ) ∗ Φ + c k ( h, s )Φ , k = 1 , . . . , n,ih vol · δ Φ δ n ( s ) = (cid:0) π ( s ) + vol (cid:0) dϕ ( s ) + V ( x ( s ) , ϕ ( s )) (cid:1)(cid:1) ∗ Φ . Let Φ( x ( · ); ρ ( · )) = U ( x ( · ) , x ( · ); ρ ( · )) be the (formal) evolution op-erator for equation (48), with the initial condition(50) U ( x ( · ) , x ( · ); ρ ( · )) = 1 . Explicitly, for two surfaces x ( s ) = x ( s, t ) and x ( s ) = x ( s, t ), onehas(51) U ( x ( · ) , x ( · )) = T exp ∗ t Z t ih H ( τ, h ) dτ def = ∞ X k =0 ih ) k Z t ≤ τ ≤ ... ≤ τ k ≤ t H ( τ k , h ) ∗ . . . ∗ H ( τ , h ) dτ . . . dτ k . Definition. The renormalized Tomonaga–Schwinger equation (44)is called integrable if the operator U ( x ( · ), x ( · ); ρ ( · )) is correctly definedup to multiplication by a constant (depending on x ( · ) and x ( · )), i. e.if it depends, up to a constant, not on the function x ( s, t ) but only onthe initial surface x = x ( s ) and the final surface x = x ( s ). MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 13 Theorem 1. For any renormalization (39) , the renormalized To-monaga–Schwinger equation for the scalar field (49) is integrable.Proof. The system of equations (47) is equivalent to the system(52) ih δ b Φ δx j ( s ) = b H j ( s ) ∗ Moyal b Φ . We must check the Frobenius integrability (constant curvature) condi-tion(53) ih δ b H j ( s ) δx j ′ ( s ′ ) − ih δ b H j ′ ( s ′ ) δx j ( s ) + [ b H j ( s ) , b H j ′ ( s ′ )] Moyal = const. For the scalar field with self-action, b H j ( s ) is the sum of a quadraticexpression in ϕ ( · ), π ( · ), a constant, and a function of b V ( x ( s ) , ϕ ( s )).This implies that(54) [ b H j ( s ) , b H j ′ ( s ′ )] Moyal = − ih { b H j ( s ) , b H j ′ ( s ′ ) } , because the terms with h , h , etc. in the Moyal commutator vanish.Hence condition (53) follows from the classical Frobenius integrabilitycondition (24) with V ( x ( s ) , ϕ ( s )) replaced by b V ( x ( s ) , ϕ ( s )). Q.E.D.The function U ( x ( · ) , x ( · ); ρ ( · )) also satisfies the equations(55) ih δU ( x ( · ) , x ( · )) δx j ( s ) = − U ( x ( · ) , x ( · )) ∗ H j ( s, h, x ( · )) , (56) U ( x ( · ) , x ( · )) = U ( x ( · ) , x ( · )) ∗ U ( x ( · ) , x ( · )) . The function U ( x ( · ) , x ( · ); ρ ( · )) is covariant with respect to the leftand right actions of a central extension of the Lie algebra of smoothvector fields(57) Z X i,k a k ( s ) ϕ is k ( s ) π i ( s ) ds, with main functions a k ( s ), k = 1 , . . . , n , on the space of functionsΦ( ρ ( · )). Problem. For the scalar field, formula (51) for U ( x ( · ) , x ( · ); ρ ( · ))makes sense only for space-like surfaces x ( s, τ ) for any τ , because fornon-space-like surfaces the Hamiltonian H ( τ ) is not well defined. How-ever, equations (49) given by formulas (13–15) make sense for an ar-bitrary surface x ( s ). Can the operator U ( x ( · ) , x ( · ); ρ ( · )) be extendedto a (formal) solution of equations (47–50, 55, 56) for non-space-likesurfaces x ( s ), x ( s )? Definition. A quantization of a classical field theory given by thevariational principle (1) is an integrable renormalized Tomonaga–Schwi-nger equation (44). Two quantizations are called equivalent if theydiffer by a finite renormalization.Let us call by a (formal) quantum field theory observable a functionalΦ( x ( · ); ρ ( · )) satisfying the generalized Heisenberg equation (58) ih δ Φ δx j ( s ) = [ H j ( s, h ) , Φ] . For these observables, one has(59) Φ( x ( · )) = U ( x ( · ) , x ( · )) ∗ Φ( x ( · )) ∗ U ( x ( · ) , x ( · )) − . Observables form an associative algebra with respect to the ∗ -product.For any surface x ( s ), this algebra is identified with the algebra ofHamiltonians H ( ρ ( · )) = Φ( x ( · ); ρ ( · )). For equivalent quantizationsof a classical field theory, the algebras of observables are naturallyidentified.Define the conjugate observable Φ ∗ to an observable Φ as the complexconjugate functional. One has(60) (Φ ∗ Φ ) ∗ = Φ ∗ ∗ Φ ∗ . Any quantum field theory observable Φ is covariant with respectto the natural action of the group of diffeomorphisms of the variables( s , . . . , s n ), with the Lie algebra consisting of smooth vector fields (57).2.3. Quantization on surfaces of constant time. Denote x = t ,( x , . . . , x n ) = x . Denote by U ( t , t ) the evolution operator for flatsurfaces of constant time,(61) U ( t , t ) = U ( x ( · ) , x ( · )) , x ( s ) = ( t , s ) , x ( s ) = ( t, s ) . The function U ( t , t ) = Φ( t ) is the evolution operator for the Schr¨odingerequation(62) ihd Φ /dt = H ( t, h ) ∗ Φ , where H ( t, h ) = H ( t, ϕ ( · ) , π ( · ) , h ) is the Hamiltonian for the surface x ( s ) = ( t, s ) of constant time.One has(63) U ( t , t ) = T exp ∗ t Z t ih H ( τ, h ) dτ. MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 15 Example. For the scalar field (10, 49), one has(64) H ( t, h ) = Z π ( x ) + 12 n X j =1 ϕ x j ( x ) + V ( t, x , ϕ ( x )) ! d x . Any quantum field theory observable Φ( x ( · ); ρ ( · )) is uniquely deter-mined by its values for flat surfaces of constant time,(65) Φ( t , ρ ( · )) = Φ( x ( · ); ρ ( · )) , x ( s ) = ( t , s ) . The function Φ( t , ρ ( · )) satisfies the Heisenberg equation(66) ihd Φ /dt = [ H ( t , h ) , Φ] . One has(67) Φ( t ) = U ( t , t ) ∗ Φ( t ) ∗ U ( t , t ) − . Mathematical expressions for the Bogoliubov S -matrix, the Wightman functions, and the Greenfunctions The half S -matrix. Consider the ϕ d model of scalar field in R n +1 with variable interaction, i. e. the Lagrangian F ( x j , ϕ, ϕ x j ) (10) with(68) V ( x, ϕ ) = m ϕ + g ( x ) d ! ϕ d , where g ( x ) is a real smooth function with compact support called theinteraction cutoff function.Let(69) H ( t, ϕ ( · ) , π ( · )) = H ( ϕ ( · ) , π ( · )) + Z g ( t, x ) d ! ϕ ( x ) d d x be the Hamiltonian (64) of scalar field, where(70) H ( ϕ ( · ) , π ( · )) = Z π ( x ) + 12 n X j =1 ϕ x j ( x ) + m ϕ ( x ) ! d x is the Hamiltonian of free scalar field.Consider an arbitrary renormalization (39) and the quantized scalarfield theory (49). Let Φ( t ) be an observable for this quantum fieldtheory. For t = t << 0, Φ( t ) is identified with an observable Φ ( t )of the free quantum field theory (i. e. the theory with g ( x ) ≡ t = t >> 0, Φ( t ) is identified with another observable Φ ( t ) of the free quantum field theory. The observables Φ ( t ) and Φ ( t ) are relatedby the equality(71) Φ ( t ) = U ( t , t ) − ∗ Φ ( t ) ∗ U ( t , t )= S / ( t , t ) − ∗ Φ ( t ) ∗ S / ( t , t ) , where U ( t , t ) is the evolution operator (63) for scalar quantum fieldtheory,(72) S / ( t , t ) = U ( t , t ) − ∗ U ( t , t ) , and(73) U ( t , t ) = exp ∗ t − t ih H is the evolution operator for free quantum field theory. The functional S / ( t , t ) = Φ( t ) is a (formal) free quantum field theory observabledepending on t called the half S -matrix .3.2. Lorentz covariance property. It is the following assumption. The quantization (49) of the free classical field theory (10, 12, 25,68) with g ( x ) ≡ , given by renormalization (39) , is Lorentz covariant,i. e. for any observable Φ( x j ( · ); ϕ ( · ) , π ( · )) of free quantum field theoryand for any element P of the nonhomogeneous Lorentz symmetry group G = O ( n, e × R n +1 , one has (74) P Φ( x j ( · ); ϕ ( · ) , π ( · )) = Φ( P − x ( · ); ϕ ( · ) , π ( · )) , where P Φ is given by (29) . Corollary. For any element α of the Lie algebra g of the group G and for any functional Φ( ϕ ( · ) , π ( · )) , one has (75) [Φ α ( t ) , Φ] = − ih { Φ α ( t ) , Φ } , where Φ α ( t ) is the generator (27) of the symmetry α . In particular,for the Hamiltonian H (70) , one has (76) [ H , Φ] = − ih { H , Φ } . This Corollary implies that any free quantum field theory observableΦ can be considered as a functional of a solution ϕ ( x ) of the Euler–Lagrange equation with g ( x ) ≡ ∂ ϕ ∂t − n X j =1 ∂ ϕ ∂x j + m ϕ = 0 , i. e. the Klein–Gordon equation, with the initial conditions(78) ϕ ( t , x ) = ϕ ( x ) , ∂ϕ /∂t ( t , x ) = π ( x ) . MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 17 More concretely, consider the Schwartz space of functions ( ϕ ( x ) , π ( x )).By (78) this space is identified with the space V of smooth solutions ϕ ( x ) of the Klein–Gordon equation (77) rapidly decreasing in the spacedirections. The free quantum field theory observables Φ( t , ϕ ( · ) , π ( · )) =Φ( ϕ ( x )) are polynomial Hamiltonians on V . The space of observablesΦ has a family of product operations depending on the Planck constant h . For h = 0 it is the usual commutative product, and for h = 0 it isthe noncommutative ∗ -product.The Lorentz covariance property means that the distributions G N ∈ S N V ′ , N = 2 , , . . . , are Lorentz invariant.3.3. The Bogoliubov S -matrix, the Wightman and Green func-tions. The half S -matrix S / ( t , t ) (72) is a (formal) free quantumfield theory observable given by the formula(79) S / = S / [ H int ]( t ) = T exp ∗ t Z −∞ ih H int ( τ ) dτ, where(80) H int ( t ) = U ( t , t ) − ∗ Z g ( t, x ) d ! ϕ ( x ) d d x ∗ U ( t , t )= Z g ( t, x ) d ! ϕ ( t, x ) d d x is the interaction Hamiltonian.Let F be the Fock Hilbert space for free scalar field with mass m inthe space-time R n +1 . Normally ordered linear operators in F can beidentified with their Wick symbols, which are functionals Φ( ϕ ( · )) ofa solution ϕ ( x ) of the Klein–Gordon equation (77). The product ofnormally ordered operators in the space F corresponds to the productΦ ∗ F Φ of their symbols Φ , Φ , given by formula (40) for the trans-form (39) with the trivial functions G F = G F = . . . = 0 and with anontrivial function G F . Hence we have a G -invariant not everywheredefined (discontinuous) homomorphism Φ → b Φ from the algebra of freequantum field theory observables corresponding to a Lorentz covariantrenormalization G , G , . . . (39) to the algebra of Wick symbols oflinear operators in F . The inverse homomorphism Φ → e Φ is given byformula (39) with the functions G − G F , G , G , . . . . We shall callthis homomorphism by renormalization.Denote by(81) h Φ i = Φ( ϕ ≡ F with the Wick symbol Φ. The linear operator in the Fock space(82) S [ g ( · )] = d S / [ e H int ]( t ) = T exp ∗ ∞ Z −∞ ih g ( t, x ) d ! ^ ϕ ( t, x ) d dtd x c , where t >> 0, is called the Bogoliubov S -matrix (with variable inter-action) for the ϕ d model in R n +1 .Let j ( x ) be a real smooth function with compact support called thesource function. Put(83) Z [ H int , j ( · )] = S / [ H int ]( t ) − ∗ S / (cid:20) H int + Z j ( t, x ) ϕ ( t, x ) d x (cid:21) ( t )= T exp ∗ ∞ Z −∞ ih j ( t, x ) ϕ ( t, x ) dtd x , where t >> ϕ ( t, x ) = U ( t , t ) − ∗ ϕ ( x ) ∗ U ( t , t )= S / [ H int ]( t ) − ∗ ϕ ( t, x ) ∗ S / [ H int ]( t ) . The distribution(85) W N ( x , . . . , x N )= lim g ( x ) → g * δZ [ e H int , j ( · )] δ j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ( x ) ≡ ∗ . . . ∗ δZ [ e H int , j ( · )] δ j ( x N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ( x ) ≡ c + is called the Wightman function , and the distribution(86) V N ( x , . . . , x N ) = lim g ( x ) → g * δ N Z [ e H int , j ( · )] δ j ( x ) . . . δ j ( x N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j ( x ) ≡ c + is called the Green function .The following Theorem is obvious. Theorem 2. If the Lorentz covariance assumption holds and if theBogoliubov S -matrix (82) , the Wightman functions (85) , and the Greenfunctions (86) are well defined, then they satisfy the axioms of quantumfield theory [9, 10].Actually the purpose of perturbative quantum field theory is to con-struct a quantized field theory power series (38, 39, 49) such that the MATHEMATICAL APPROACH TO QUANTUM FIELD THEORY 19 S -matrix (82), the Wightman functions (85), and the Green functions(86) are well defined as power series, while axiomatic quantum fieldtheory studies the properties of these functions.3.4. Discussion. The problem with quantities (82, 85, 86) is that theyare, in general, ill-defined, because the renormalization Φ → e Φ and theinverse map Φ → b Φ are discontinuous not everywhere defined maps.To obtain an expression for these quantities, one should choose a reg-ularization H Λ int of the interaction Hamiltonian H int = lim Λ → H Λ int . Thelimit as Λ → The quantum field theory S -matrix describes scattering amplitudes of colliding particles. They areassumed to be non-interacting at large distances. However, the parti-cles always interact with vacuum. In quantum field theory there is nonatural notion of particles or vacuum but there is only the universalform of matter consisting of interacting quantum fields. Hence the def-inition of a quantum field theory from § Conjecture. Axioms of quantum field theory can be, in general,neither mathematically realized nor disproved, like Continuum Hypoth-esis. This Conjecture means that “final mathematical theory of every-thing” does not exist.Mathematical quantum field theory in the Minkowski space-time is ahyperbolic theory. It studies the solutions U ( x ( · ) , x ( · )) of the Cauchyproblem and the related well defined quantities (i. e. quantities inde-pendent of regularization). The following three questions are concernedwith elliptic theories, with a different behavior. The natural problemfor them is the boundary problem. Problem 1. Construct and study mathematical quantization offield theory in the Euclidean space-time. Problem 2. Construct and study mathematical quantization of thePlateau problem (16). This argument is taken from [10]. Problem 3. Construct and study mathematical quantization of theDirichlet principle.