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The Jacobian Conjecture As a Problem of Perturbative Quantum Field Theory

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The Jacobian Conjecture As a Problem of Perturbative Quantum Field Theory Ann. Henri Poincar´e 4 (2003) 199 – 215 c Birkh¨auser Verlag, Basel, 2003 1424-0637/03/020199-17 DOI 10.1007/s00023-003-0127-7 Annales Henri Poincar´e The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory Abdelmalek Abdesselam Abstract. The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so-called formal inverse approach, that of perturbative quantum field theory. I Introduction The purpose of this modest note, for which we claim no originality except that of connecting apparently unrelated fields, is to draw the attention of theoretical physicists to one of the major unsolved problems of mathematics [29], viz. the Jacobian conjecture. The question is so simple that it was coined in [3] a problem in “high school algebra”. One can formulate it as follows. Let F : Cn → Cn be a map written in coordinates as F (x1,...,xn)=(F1(x1,...,xn),...,Fn(x1,...,xn)) . (1) n One says that F is a polynomial map if the functions Fi : C → C are polynomial. Suppose that the Jacobian determinant def ∂Fi JF(x1,...,xn) =det (x1,...,xn) (2) ∂xj is identically equal to a nonzero constant. Show then that F is globally invertible (for the composition of maps) and that its inverse G def= F −1 is also a polynomial map. Since it was first proposed in [21] (for n = 2 and polynomials with integral coefficients), this problem has resisted all attempts for a solution. In fact, this seemingly simple problem is quite an embarrassment. Indeed, some faulty proofs have even been published (see the indispensable [6] and [11] for a review). We will show here that the Jacobian conjecture can be formulated in a very nice way as a question in perturbative quantum field theory (QFT). We also expect any future progress on this question to be beneficial not only for mathematics, but also for theoretical physics as it would enhance our understanding of perturbation theory. 200 A. Abdesselam Ann. Henri Poincar´e II The formal inverse as a one-point correlation function The most tempting, yet unfortunately least developed, line of attack on the Jaco- bian conjecture is the so-called formal inverse approach. One tries to solve explic- itly for x =(x1,...,xn) in the equation y = F (x), one then finds a power series expression for x in terms of y =(y1,...,yn). By the uniqueness of the power series inverse, all one has to do then is to show that it is in fact a polynomial, that is the terms of high degree in the y variables vanish. One of the many reasons this approach is in its infancy is that it took more than two centuries (say from [22] to [6]) to have a workable formula for the formal inverse in the multivariable case. Early contributions can be found in [23, 19, 9, 30, 24, 15]. An important contri- bution concerning formal inversion is due to Gurjar and Abhyankar [4]. Modern literature on reversion and Lagrange-Good type formulas is huge and we invite the reader to consult [6, 14, 18, 34] for more complete references. The first formula for the coefficients of the formal inverse power series G in terms of those of F is due to J. Towber and was first published in [35] (see also [17]). In physicists’ terms ours is the following. Claim: (A. A., V. Rivasseau) The formal solution of y = F (x), without any assump- tion on F except that its linear part is invertible, is the perturbation expansion of the normalized one-point correlation function 1 −φF (φ)+φy xi = dφdφ φie (3) Z Cn where φ1,...,φn,φ1,...,φn are the components of a complex Bosonic field. The integration is over Cn with the measure n d(Re φi)d(Im φi) dφdφ def= , (4) π i=1 def n def n we used the notation φF (φ) = i=1 φiFi(φ1,...,φn), φy = i=1 φiyi,and Z def= dφdφ e−φF (φ)+φy . (5) Cn We obtained this expression by solving iteratively the equation y = F (x) thereby generating a tree expansion in the same way one expresses the effective action Γ(φ) in terms of the logarithm W (J) of the partition function in QFT (see [36] for instance). We then determined the Feynman rules of this tree expan- sion and finally the “path integral” formulation (3), only to realize that in fact our formula is closely related to the one introduced by G. Gallavotti, following a suggestion of G. Parisi, to express the Lindstedt perturbation series in the context of KAM theory [13]. The right-hand side of (3) is defined, using formal Gaussian integration,asa ratio of formal power series in the y variables with coefficients in C, thanks to the Vol. 4, 2003 The Jacobian Conjecture as a Problem of Quantum Field Theory 201 formula, also known as “Wick’s theorem”, for the moments of a complex Gaussian measure. In the following, we will state and prove a precise theorem, using some analysis, for the case where Fi(x)=xi − Hi(x), with the Hi(x), 1 ≤ i ≤ n,being homogeneous of the same degree d. Indeed, it is enough to treat the cubic case d =3,for all dimensions n, in order to prove the Jacobian conjecture in full generality [6]. However, formula (3) is completely combinatorial in nature and its proper setting is in the ring of formal power series with variables corresponding to the fields φ and φ together with the y’s, over any commutative ring containing the field of rationals. One simply has to define formal Gaussian integration, somewhat in the spirit of [5]. We refer to [1] for a formulation and proof of our claim as a decent mathematical theorem. The latter article will also provide more details on how Feynman diagrams can be useful in algebraic combinatorics and how well they fit in the Joyal theory of combinatorial species [20]. We also refer to [2] for a very simple heuristic proof of the Lagrange-Good multivariable inversion formula, which becomes a fully rigorous and purely combinatorial proof when interpreted using the formalism of [1]. Now let Fi(x)=xi − Hi(x)withHi(x) written in tensorial notation as n 1 Hi(x)= wi,j ...j xj ...xj (6) d! 1 d 1 d j1,...,jd=1 so that the 1-contravariant and d-covariant tensor wi,j1 ...jd is completely symmetric in the j indices. Let us write n n d def φwφ = φiwi,j1...jd φj1 ...φjd (7) i=1 j1,...,jd=1 so that (3) becomes − 1 d dφdφ φi exp φφ + d! φwφ + φy Gi(y)= . (8) − 1 d dφdφ exp φφ + d! φwφ + φy The free propagator is represented as an oriented line = δij (9) i j for the contraction of a pair φiφj. There are two types of vertices: the w-vertices represented by d half lines = φwφd (10) 202 A. Abdesselam Ann. Henri Poincar´e and the y-vertices represented by = φy . (11) As is well known in QFT, the numerator and denominator of (8) can be calculated by expanding 1 exp φwφd + φy d! and integrating term by term with respect to the normalized complex Gaussian measure dφdφ e−φφ. The result is a sum over all possible Feynman diagrams that can be built from the vertices of (10) and (11) (and the source φi for the numerator) by joining the half-lines of compatible directions. A quick look at the vertices shows that the only possible diagrams are trees connected to the source φi, or vacuum graphs made by an oriented loop of say k ≥ 1 w-vertices to which k(d − 1) trees, whose leaves are y-vertices, are attached. When one factors out the denominator, the only diagrams that remain are made of a single tree with the source φi as its root. Therefore, at least formally, we have 1 Gi(y)= Ai(T ) (12) V !N! V ≥0 N≥0 T where T is a Cayley tree (viewed as a set of unordered pairs) on a finite set E = E(V,N). The latter is chosen, noncanonically, once for each pair (V,N), and must be the disjoint union of Eroot of cardinality 1, Einternal of cardinality V and Eleaf of cardinality N. T is constrained by the condition that elements of Eroot ∪ Eleaf have valence 1 while those of Einternal have valence d +1.This automatically enforces the relation (d − 1)V = N − 1 which can be checked by counting the half-lines. Even though we write, in the sequel, seemingly independent sums over V and N, the previous relation is always assumed. We now define the amplitude Ai(T ). One directs the edges of T towards the root in Eroot. For each such edge l ∈T, one introduces an index il in the set {1,...,n}. One then considers the expression Ai(T , (il)l∈T ) which is the product of the following factors. –Foreacha ∈ Eleaf ,ifl(a) is the unique line going from a, we take the factor yil(a) . –Foreachb ∈ Einternal,if{l1(b),...,ld(b)} is the set of lines coming into b and l (b) is the unique line leaving b,wetakethefactorwi ,i ...i . 0 l0(b) l1(b) ld(b) The resulting monomial in the y’s and w’s is Ai(T , (il)l∈T ) by definition. Finally Ai(T )isthesumofAi(T , (il)l∈T ) over all the indices (il)l∈T except the index of the line arriving at the root which is fixed at the value i, the source index.
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