MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 1. Generalities on quantum field theory 1.1. Classical mechanics. In classical mechanics, we study the motion of a particle. This motion is described by a (vector) function of one variable, q = q(t), representing the position of the particle as a function of time. This function must satisfy the Newton equation of motion, q¨= −U (cid:1)(q), where U the potential energy, and the mass of the particle is 1. Another way to express this law of motion is to say that q(t) must be a solution of a certain variational problem. Namely, one introduces the Lagrangian q˙2 L(q)= −U (q) 2 (the difference of kinetic and potential energy), and the action functional � b S(q)= L(q)dt a (for some fixed a < b). Then the law of motion can be expressed as the least action principle: q(t) must be a critical point of S on the space of all functions with given q(a) and q(b). In other words, the Newton equation is the Euler-Lagrange equation for the solution of the variational problem defined by S. Remark 1. The name “least action principle” comes from the fact that in some cases (for example when U (cid:1)(cid:1) ≤ 0) the action is not only extremized but also minimized at the solution q(t). In general, however, it is not the case, and the trajectory of the particle may not be a minimum, but only a saddle point of the action. Therefore, the law of motion is better formulated as the “extremal (or stationary) action principle”; this is the way we will think of it in the future. Remark 2. Physicists often consider solutions of Newton’s equation on the whole line rather than on a fixed interval [a, b]�. In this case, the naive definition of an extremal does not make sense, since the action integral S(q)= L(q)dt is improper and in general diverges. Instead, one makes the following R “correct” definition: a function q(t) on R is an extremal of S if the expression � � d ∂L ∂L ds |s=0 L(q + sε)dt := ( ∂q ε˙+ ∂q˙ε¨+ ···), R R where ε(t) is any compactly supported perturbation, is identically zero. With this definition, the extremals are exactly the solutions of Newton’s equation. 1.2. Classical field theory. In classical field theory, the situation is similar. In this case, we should think not of a single particle, but of a “continuum of particles” (e.g. a string, a membrane, a jet of fluid); so the motion is described by a classical field – a (vector) function φ(x, t) depending on both space and time coordinates (x ∈ Rd , t ∈ R). Consequently, the equation of motion is a partial differential equation. For example, for a string or a membrane the equation of motion is the wave equation �φ =0, where � is the D’Alambertian ∂2 −v2∆ (here ∆ is the Laplacian with respect to the space coordinates, t and v the velocity of wave propagation). As in mechanics, in c�lassical field theory there is a Lagrangian L(φ) (a differential polynomial in φ), whose integral S(φ) = L(φ)dxdt over a region D in space and time is called the action. The law of D motion can be expressed as the condition that the action must be extremized over any closed region D and fixed boundary conditions; so the equations of motion (also called the field equations) are the 1 2 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY Euler-Lagrange equations for this variational problem. For example, in the case of string or membrane, the Lagrangian is 1 L(u)= (φ2 − v 2(∇φ)2). 2 t Remark. Like in mechanics, solutions of the field equations on the whole space (rather than a closed region D) are extremals of the action in the sense that � d ds|s=0 L(u+ sε)dxdt =0, Rd+1 where ε is a compactly supported perturbation. 1.3. Brownian motion. One of the main differences between classical and quantum mechanics is, roughly speaking, that quantum particles do not have to obey the classical equations of motion, but can randomly deviate from their classical trajectories. Therefore, given the position and velocity of the particle at a given time, we cannot determine its position at a later time, but can only determine the density of probability that at this later time the particle will be found at a given point. In this sense quantum particles are similar to random (Brownian) particles. Brownian particles are a bit easier to understand conceptually, so let us begin with them. The motion of a Brownian particle in Rd in a potential field U : Rd → R is described by a stochastic process q = q(t), q =(q1,...,qd) ∈ Rd. That is, for each real t we have a random variable q(t)(position of the particle at a time t), such that the dependence of t is regular in some sense. The random dynamics of the particle is “defined” as follows: 1 if y : [a,b] → Rd is a continuously differentiable function, then the density of probability that q(t)= y(t)for t ∈ [a,b] is proportional to e−S(y)/κ, where � S(y):= b( 1y(cid:1)2 − U(y))dt is the action for the corresponding classical mechanical system, and κ is the a 2 diffusion coefficient. Thus, for given q(a) and q(b), the likeliest q(t) is the one that minimizes S (in particular, solves the classical equations of motion q¨= −U(cid:1)(q)), while the likelihood of the other paths decays exponentially with the deviation of the action of these paths from the minimal possible. Remark. This discussion assumes that the extremum of S at q is actually a minimum, which we know is not always the case. All the information we can hope to get about such a process is contained in the correlation functions <qi1(t1) ...qin (tn) >, which by definition are the expectation values of the products of random variables qi1(t1) ...qin (tn) (more specifically, by Kolmogorov’s theorem the stochastic process q(t) is completely determined by these functions). So such functions should be regarded as the output, or answer, of the theory of the Brownian particle. So the main question is how to compute the correlation functions. The definition above obviously gives the following answer: given t1,...,tn ∈ [a,b], we have � (1) <qj1(t1) ...qjn (tn) >= qj1(t1) ...qjn (tn)e −S(q)/κDq, where integration is carried o�u t over the space of paths [a,b] → Rn, and Dq is a Lebesgue measure on the space of paths such that e−S(q)/κDq = 1. Such an integral is called a path integral, since it is an integral over the space of paths. It is clear, however, that such definition and answer are a priori not satisfactory from the mathe- matical viewpoint, since the infinite dimensional integration that we used requires justification. In this particular case, such justification is possible within the framework of Lebesgue measure theory, and the corresponding integration theory is called the theory of Wiener integrals. (To be more precise, one cannot define the measure Dq, but one can define the measure e−S(q)/κDq for sufficiently nice potentials U(q)). 1 We put the word “defined” in quotation marks because this definition is obviously heuristic and not rigorous; see below for more explanations MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 3 1.4. Quantum mechanics. Now let us turn to a quantum particle. Quantum mechanics is notoriously difficult to visualize, and the randomness of the behavior of a quantum particle is less intuitive and more subtle than that of a Brownian particle; nevertheless, it was pointed out by Feynman that the behavior of a quantum particle in a potential field U(q) is correctly described by the same model, with the real positive parameter κ replaced by the imaginary number i(cid:1),where (cid:1) > 0 is the Planck constant. In other words, the dynamics of a quantum particle can be expressed (we will discuss later how) via the correlation functions � (2) <qj1(t1) ...qjn (tn) >= qj1(t1) ...qjn (tn)e iS(q)/(cid:1)Dq, � where Dq is normalized so that eiS(q)/(cid:1)Dq =1. 1.5. Quantum field theory. The situation is the same in field theory. Namely, a useful theory of quantum fields (used in the study of interactions of elementary particles) is obtained when one considers correlation functions � (3) <φj1(x1,t1) ...φjn (xn,tn) >= φj1(x1,t1) ...φjn (xn,tn)e iS(φ)/(cid:1)Dφ, � where Dφ is normalized so that eiS(φ)/(cid:1)Dφ =1. Of course, from the mathematical point of view, this setting is a priori even less satisfactory than the one for the Brownian particle, since it involves integration with respect to the complex valued measure eiS(q)/(cid:1)Dq, which nobody knows how to define. Nevertheless, physicists imagine that certain integrals of this type exist and come to correct and interesting conclusions (both physical and mathematical). Therefore, making sense of such integrals is an interesting problem for mathematicians, and will be one of our main occupation during the course.2 2to be more precise, we will make sense of path integrals as power series in κ or (cid:1). 4 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 2. The steepest descent and stationary phase formulas Now, let us forget for a moment that the integrals (1,2,3) are infinite dimensional and hence problem- atic to define, and ask ourselves the following question: why should we expect that when the parameter κ or (cid:1) goes to zero, we recover the usual classical mechanics or field theory? The answer is that this expectation is based on the steepest descent (respectiv� ely, stationary ph�a se) principle from classical analysis: if f(x) is a function in Rd then the integrals g(x)e−f (x)/κdx, g(x)eif (x)/�dx “localize” to minima, respectively critical points, of the function f. As this classical fact is of central importance to the whole course, let us now discuss it in some detail. 2.1. The steepest descent formula. Let f, g :[a, b] → R be smooth functions. Theorem 2.1. (The steepest descent formula) Assume that f attains a global minimum at a unique point c ∈ (a, b), such that f(cid:1)(cid:1)(c) > 0. Then one has � b (4) g(x)e −f (x)/�dx = (cid:1)1/2 e −f (c)/�I((cid:1)), a √ where I((cid:1)) extends t o a smooth function on [ 0, ∞) such that I(0) = 2π √g(c) . f (cid:1)(cid:1) (c) Proof. Let I((cid:1)) be defined by the equation (4). Let (cid:3) be a real number, such that 1 >(cid:3) > 0, and let I ((cid:1)) be defined by the same equation, but with 2 1 integration over [c − (cid:1) 12−(cid:2),c + (cid:1) 12−(cid:2)]. It is clear that I((cid:1)) − I1((cid:1)) is “rapidly decaying in (cid:1)” (i.e. it is O((cid:1)N ), (cid:1) → 0 for any N). So it suffices to prove the theorem for I1((cid:1)). √ Further, let us make in the integral defining I ((cid:1)) the change of variables y =(x − c)/ (cid:1). Then we 1 get � �−(cid:1) √ √ (5) I ((cid:1))= g(c + y (cid:1))e(f (c)−f (c+y �))/�dy. 1 −�−(cid:1) √ Now, note that the integrand is a smooth function with respect to (cid:1) for (cid:1) ≥√0. Let I2((cid:1)) be the same integral as in (5) but with integrand replaced by its Taylor expansion in (cid:1) at 0 modulo (cid:1)N. Then |I ((cid:1)) − I ((cid:1))|≤ C(cid:1)N −(cid:2) . 1 2 Finally, let I ((cid:1)) be defined by the same integral as I ((cid:1)) but with limits from −∞ to ∞. Then 3 2 I ((cid:1)) − I ((cid:1)) is rapidly decaying in (cid:1). 2 3 Thus, it suffices to show that I3((cid:1)) admits a Taylor expansion in (cid:1)1/√2 modulo (cid:1)N −(cid:2) , and that the value at zero is as stated. But we know that I ((cid:1)) is a polynomial in (cid:1). Also, the integrals giving 3 coefficients of non-integer powers of (cid:1) are integrals over R of odd functions, so they are zero. So the first statement (existence of the Taylor expansion) is proved. The value I (0) is given by the integral � 3 g(c) ∞ e − f (cid:1)(cid:1) (2c )y 2 dy, −∞ so it is computed from the well known Poisson integral: � ∞ e − y2 2 dy =√2 π. −∞ The theorem is proved. � 2.2. Stationary phase formula. This theorem has the following imaginary analog, called the sta- tionary phase formula. Theorem 2.2. Assume that f has a unique critical point c ∈ (a, b), with f(cid:1)(cid:1)(c) (cid:7)= 0, and g vanishes with all derivatives at a, b. Then � b g(x)e if (x)/�dx = (cid:1)1/2 e if (c)/�I((cid:1)), a √ where I((cid:1)) extends t o a smooth function on [0, ∞) such that I(0) = 2πe±πi/4 √g(c) , where ± is the | f (cid:1)(cid:1) (c)| sign of f(cid:1)(cid:1)(c). MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 5 Remark. It is important to assume that g vanishes with all derivatives at the ends of the the integration interval. Otherwise we will get some additional boundary contributions. Proof. (sketch). The proof is analogous to the real case, but slightly more subtle. The differences are as follows. First of all, the Poisson integral is replaced with the (conditionally convergent) Fresnel integral � ∞ e iy22 dy =√2 πeπi/4 . −∞ Further, one should partition g in a sum of two smooth functions, one localized around c on an interval of size 2(cid:1)1/2−(cid:2), and the other vanishing near c. Next, one needs to show that only the first summand � matters, by using Riemann’s lemma: if f has no critical points in the support of g then b g(x)eif (x)/� a is rapidly decaying (prove this!). Finally, for g localized around c, ones makes the change of variable like in the real case. The statement about existence of Taylor expansion is proved as in the real case, and the value at 0 is calculated using Fresnel integral. � 2.3. Non-analyticity of I((cid:1)) and Borel summation. It is very important to note that the Taylor series for I((cid:1)) is usually not convergent and is only an asymptotic expansion, so that the function I is smooth but not analytic at zero. To illustrate this, consider the integral � ∞ e − x 22+�x 4 dx =√2 π(cid:1)1/2I((cid:1)), −∞ where � I((cid:1))= √1 ∞ e − y 2+2� y 4 dy. 2π −∞ The latter integral expands asymptotically as �∞ I((cid:1))= a (cid:1)n , n n=0 where � an = (√−2 1π)n −∞∞ e −y 2/2 2yn4nn! dy = (√−21π)n 2n+ 21Γ(2n + 21)/n!. It is clear that this sequence has superexponential growth, so the radius of convergence of the series is zero. Remark. In fact, the non-analyticity of I((cid:1)) is related to the fact that the integral defining I((cid:1)) is divergent for (cid:1) < 0. Let us now discuss the question: to what extent does the asymptotic expansion of the function I((cid:1)) (which we can find using Feynman diagrams as explained below) actually determines this function? � Suppose that I˜((cid:1)) = a (cid:1)n is a series with zero radius of convergence. In general, we cannot n≥0 n uniquely determine a function I on� [0,ε) whose expansion is given by such a series. However, assume that a are such that the series g((cid:1))= a (cid:1)n/n! is convergent in some neighborhood of 0, analytically i n≥0 n continues to [0, ∞), and has at most exponential growth as (cid:1) →∞. In this case there is a “canonical” way to construct a smooth function I on [0,ε) with (asymptotic) Taylor expansion I˜, called Borel summation of I˜. Namely, the function I is defined by the formula � ∞ I((cid:1))= g((cid:1)u)e −udu 0 The fact that I has the Taylor expansion I˜follows from the fact that for t> 0 one has � ∞ xn e −xdx = n!. � 0 � For example, consider the series I˜= (−1)nn!(cid:1)n. Then g((cid:1))= (−1)n(cid:1)n = 1 . Hence, the � n≥0 n≥0 1+� Borel summation yields I((cid:1))= ∞ e−u du. 0 1+�u Physicists expect that in many situations perturbation expansions in quantum field theory are Borel summable, and the actual answers are obtained from these expansions by Borel summation. The Borel summability of perturbation series has actually been established in a few nontrivial examples of QFT. 6 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 2.4. Application of steepest descent. Let us give an application of Theorem 2.1. Consider the integral � ∞ ts e −tdt, s > 0. 0 It is well known that this integral is equal to the Gamma function Γ(s + 1). By doing a change of variable t = sx, we get � � Γ(s +1) ∞ ∞ = xse −sxdx = e −s(x−logx)dx. ss+1 0 0 Thus, we can apply Theorem 2.1 for (cid:1) = 1/s, f (x) = x −log x, g(x) = 1 (of course, the interval [a, b] is now infinite, and the function f blows up on the boundary, but one can easily see that the theorem is still applicable). The function f (x)= x −log x has a unique critical point on [0, ∞), which is c =1, and we have f (cid:1)(cid:1)(c) = 1. Then we get √ Γ(s +1) = ss e −s 2πs(1 + a /s + a /s2 + ···). 1 2 This is the celebrated Stirling’s formula. 2.5. Multidimensional versions of steepest descent and stationary phase. Theorems 2.1,2.2 have multidimensional analogs. To formulate them, let V be a real vector space of dimension d with a fixed volume element dx, and let f, g be smooth functions in a closed box B ⊂ V . Theorem 2.3. Assume that f has global minimum on B at a unique interior point c, such that f (cid:1)(cid:1)(c) > 0. Then � (6) g(x)e −f (x)/�dx = (cid:1)d/2 e −f (c)/�I((cid:1)), B where I((cid:1)) extends t o a smooth function on [ 0, ∞) such that I(0) = (2π)d/2 √ g(c) . det f (cid:1)(cid:1) (c) Theorem 2.4. Assume that f has a unique critical point c in B, such that det f (cid:1)(cid:1)(c) (cid:7)= 0, and that g vanishes with all derivatives on the boundary of the box. Then � (7) g(x)eif (x)/�dx = (cid:1)d/2 e if (c)/�I((cid:1)), B where I((cid:1)) extends to a smooth function on [0, ∞) such that I(0) = (2π)d/2eπiσ/4 √ g(c) , where σ | det f (cid:1)(cid:1) (c)| is the signature of the symmetric bilinear form f (cid:1)(cid:1)(c). Remark. In presence of a volume element on V , the determinant of a symmetric bilinear form is well defined. The proofs of these theorems are parallel to the proofs of their one dimensional versions. Namely, the 1-dimensional Poisson and Fresnel integrals are replaced with their multidimensional versions – the Gaussian integrals � e −B(x,x)/2dx =(2π)d/2(det B)−1/2 , V for a symmetric bilinear form B > 0, and � e iB(x,x)/2dx =(2π)d/2 e πiσ(B)/4|det B|−1/2 , V for nondegenerate-B. These integral formulas are easily deduced from the one-dimensional ones by diagonalizing the bilinear form-B. MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 7 3. Feynman calculus 3.1. Wick’s theorem. Let V be a real vector space of dimension d with volume element dx. Let S(x) be a smooth function on a box B ⊂ V which attains a minimum at x = c ∈ Interior(B), and g be any smooth function on B. In the last section we have shown that the function � I((cid:1))= (cid:1)−d/2 e S(c)/� g(x)e −S(x)/�dx B admits an asymptotic power series expansion in (cid:1): (8) I((cid:1))= A0 + A1(cid:1) + ···+ Am(cid:1)m + ··· Our main question now will be: how to compute the coefficients A ? i It turns out that although the problem of computing I((cid:1)) is transcendental, the problem of computing the coefficients A is in fact purely algebraic, and involves only differentiation of the functions S and g i at the point c. Indeed, recalling the proof of equation 8 (which we gave in the 1-dimensional case), we see that the calculation of A reduces to calculation of integrals of the form i � P(x)e −B(x,x)/2dx, V where P is a polynomial and B is a positive definite bilinear form (in fact, B(v,u)= (∂ ∂ S)(c)). But v u such integrals can be exactly evaluated. Namely, it is sufficient to consider the case when P is a product of linear functions, in which case the answer is given by the following elementary formula, known to physicists as Wick’s theorem. For a positive integer k, consider the set {1,...,2k}. By a pairing σ on this set we will mean its partition into k disjoint two-element subsets (pairs). A pairing can be visualized by drawing 2k points and connecting two points with an edge if they belong to the same pair (see Fig. 1). This will give k edges, which are not connected to each other. 1 3 1 3 2 4 2 4 1 3 2 4 Figure 1. Pairings of the set {1,2,3,4} Let us denote the set of pairings on {1,...,2k} by Π . It is clear that |Π | = (2n)! . For any σ ∈ Π , k k 2n·n! k we can think of σ as a permutation of {1,...,2k}, such that σ2 =1 and σ has no fixed points. Namely, σ maps any element i to the second element σ(i) of the pair containing i. Theorem 3.1. Let B−1 denote the inverse form on V∗, and let (cid:3)1,...,(cid:3)m ∈ V∗. Then, if m is even, we have � (2π)d/2 � � (cid:3)1(x) ...(cid:3)m(x)e −B(x,x)/2dx= √ B−1((cid:3)i,(cid:3)σ(i)) det B V σ∈Πm/2 i∈{1,...,m}/σ If m is odd, the integral is zero. 8 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY Proof. If m is odd, the statement is obvious, because the integrand is an odd function. So consider the even case. Since both sides of the equation are symmetric polylinear forms in (cid:3)1,...,(cid:3)m, it suffices to prove the result when (cid:3)1 = ··· = (cid:3)m = (cid:3). Further, it is clear that the formula to be proved is stable under linear changes of variable (check it!), so we can choose a coordinate system in such a way that B(x,x) = x21 + ···+ xd2, and (cid:3)(x) = x1. Therefore, it is sufficient to assume that d = 1, and (cid:3)(x) = x. In this case, the theorem says that � ∞ x2k e −x 2/2dx =(2π)1/2 (2k)! , 2kk! −∞ which is easily obtained from the definition of the Gamma function by change of variable y = x2/2. � Examples. � (2π)d/2 (cid:3)1(x)(cid:3)2(x)e −B(x,x)/2dx = √ B−1((cid:3)1,(cid:3)2). det B V � (cid:3)1(x)(cid:3)2(x)(cid:3)3(x)(cid:3)4(x)e −B(x,x)/2dx = V (2π)d/2 √ (B−1((cid:3)1,(cid:3)2)B−1((cid:3)3,(cid:3)4)+ B−1((cid:3)1,(cid:3)3)B−1((cid:3)2,(cid:3)4)+ B−1((cid:3)1,(cid:3)4)B−1((cid:3)2,(cid:3)3)). det B Wick’s theorem shows that the problem of computing A is of combinatorial nature. In fact, the i central role in this computation is played by certain finite graphs, which are called Feynman diagrams. They are the main subject of the remainder of this section. 3.2. Feynman’s diagrams and Feynman’s theorem. We come back to the problem of computing the coefficients A . Since each particular A depends only on a finite number of derivatives of g at i i c, it suffices to assume that g is a polynomial, or, more specifically, a product of linear functions: g = (cid:3)1 ...(cid:3)N , (cid:3)i ∈ V∗ . Thus, it suffices to be able to compute the series expansion of the integral � <(cid:3)1 ...(cid:3)N >:= (cid:1)−d/2 e S(c)/� (cid:3)1(x) ...(cid:3)N (x)e −S(x)/�dx B Without loss of generality we may assume that c = 0, and S(c) = 0. Then the (asymptotic) Taylor � expansion of S at c is S(x) = B(x,x) + Br (x,x,...,x) , where B = dr f(0). Therefore, regarding 2 r≥3 r! r √ the left hand side as a power series in (cid:1), and making a change of variable x → x/ (cid:1) (like in the last section), we get � P <(cid:3)1 ...(cid:3)N >= (cid:1)N/2 (cid:3)1(x) ...(cid:3)N (x)e − B(x2, x) − r≥3 �r/2−1 Br (xr,!...,x) dx. V (This is an identity of expansions in (cid:1), as we ignored the rapidly decaying error which comes from replacing the box by the whole space). The theorem below, due to Feynman, gives the value of this integral in terms of Feynman diagrams. This theorem is easy to prove but is central in quantum field theory, and will be one of the main theorems of this course. Before formulating this theorem, let us introduce some notation. Let G≥3(N) be the set of isomorphism classes of graphs with N 1-valent “external” vertices, labeled by 1,...,N, and a finite number of unlabeled “internal” vertices, of any valency ≥ 3. Note that here and below graphs are allowed to have multiple edges between two vertices, and loops from a vertex to itself (see Fig. 2). For each graph Γ ∈ G≥3(N), we define the Feynman amplitude of Γ as follows. 1. Put the covector (cid:3) at the j-th external vertex. j 2. Put the tensor −B at each m-valent internal vertex. m 3. Take the contraction of the tensors along edges of Γ, using the bilinear form B−1 . This will produce a number, called the amplitude of Γ and denoted FΓ((cid:3)1,...,(cid:3)N ). Remark. If Γ is not connected, then FΓ is defined to be the product of numbers obtained from the connected components. Also, the amplitude of the empty diagram is defined to be 1. MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 9 Theorem 3.2. (Feynman) One has (2π)d/2 � (cid:1)b(Γ) <(cid:3)1 ...(cid:3)N >= √ det B |Aut(Γ)|FΓ((cid:3)1,...,(cid:3)N ), Γ∈G≥3 (N ) where b(Γ) is is the number of edges minus the number of internal vertices of Γ. (here by an automorphism of Γ we mean a permutation of vertices AND edges which preserves the graph structure, see Fig. 3; thus there can exist nontrivial automorphisms which act trivially on vertices). Remark 1. Note that this sum is infinite, but (cid:1)-adically convergent. Remark 2. We note that Theorem 3.2 is a generalization of Wick’s theorem: the latter is obtained if S(x) = B(x,x)/2. Indeed, in this case graphs which give nonzero amplitudes do not have internal vertices, and thus reduce to graphs corresponding to pairings σ. Let us now make some comments about the terminology. In quantum field theory, the function <(cid:3)1 ...(cid:3)N > is called the N-point correlation function, and graphs Γ are called Feynman diagrams. The form B−1 which is put on the edges is called the propagator. The cubic and higher terms B /m!in m the expansion of the function S are called interaction terms, since such terms (in the action functional) describe interaction between particles. The situation in which S is quadratic (i.e., there is no interaction) is called a free theory; i.e. for the free theory the correlation functions are determined by Wick’s formula. Remark 3. Sometimes it is convenient to consider normalized correlation functions <(cid:3)1 ...(cid:3)N >norm = < (cid:3)1 ...(cid:3)N > / < ∅ > (where < ∅ > denotes the integral without insertions). Feynman’s theorem Γ0 N =0 Γ1 Γ2 Γ3 1 1 2 N =0 N =1 N =2 Γ4 1 2 N =2 Figure 2. Elements of G≥3(N) ? 1 = Figure 3. An automorphism of a graph 10 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY implies that they are given by the formula � (cid:1)b(Γ) <(cid:3)1 ...(cid:3)N >norm = |Aut(Γ)|FΓ((cid:3)1,...,(cid:3)N ), Γ∈G∗ (N ) ≥3 where G∗≥3(N) is the set of all graphs in G≥3(N) which have no components without external vertices. 3.3. Another version of Feynman’s theorem. Before proving Theorem 3.2, we would like to slightly modify and generalize it. Namely, in quantum field theory it is often useful to consider an interacting theory as a deformation of a free theory. This means that S(x)= B(x,x)/2+ S˜(x), where S˜(x) is the � perturbation S˜(x) = g B (x,x,...,x)/m!, where g are (formal) parameters. Consider the m≥0 m m m partition function � Z = (cid:1)−d/2 e −S(x)/�dx V as a series in gi and (cid:1) (this series inv�o lves only positive powers of gi but arbitrary powers of (cid:1); however, the coefficient of a given monomial gni is a finite sum, and hence contains only finitely many powers i i of (cid:1)). Let n =(n0,n1,...) be a sequence of nonnegative integers, almost all zero. Let G(n) denote the set of isomorphism classes of graphs with n0 0-valent vertices, n1 1-valent vertices, n2 2-valent vertices, etc. (thus, now we are considering graphs without external vertices). Theorem 3.3. One has (2π)d/2 ��� � � (cid:1)b(Γ) Z = √d et B gin i |Aut(Γ)| FΓ, n i Γ∈G(n) where FΓ is the amplitude defined as before, and b(Γ) is the number of edges minus the number of vertices of Γ. We will prove Theorem 3.3 in the next subsection. Meanwhile, let us show that Theorem 3.2 is in fact a special case of Theorem 3.3. Indeed, because of symmetry of the correlation functions with respect to (cid:3)1,...,(cid:3)N , it is sufficient to consider the case (cid:3)1 = ··· = (cid:3)N = (cid:3). In this case, denote the correlation function < (cid:3)N > (expectation value of (cid:3)N ). Clearly, to compute < (cid:3)N > for all N, it is � sufficient to compute the generating function < et(cid:2) >:= < (cid:3)N > tN. But this expectation value is N ! exactly the one given by Theorem 3.3 for gi =1, i ≥ 3, g0 = g2 =0, g1 = −(cid:1)t, B1 = (cid:3), B0 =0, B2 =0. Thus, Theorem 3.3 implies Theorem 3.2 (note that the factor N! in the denominator is accounted for by the fact that in Theorem 3.3 we consider unlabeled, rather than labeled, 1-valent vertices – convince yourself of this!). 3.4. Pr√o ofofFeynman’s theorem. Now we will prove Theorem 3.3. Let us make a change of variable y = x/ (cid:1). Expanding the exponential in a Taylor series, we obtain � Z = Zn, n where � � g ni Zn = V e −B(y,y)/2 i (i!)ni i ni!(−(cid:1)i/2−1Bi(y,y,...,y))nidy Writing B as a sum of products of linear functions, and using Wick’s theorem, we find that the value i of the integral for each n can be expressed combinatorially as follows. 1. Attach to each factor −B a “flower” — a vertex with i outgoing edges (see Fig. 4). i 2. Consider the set T of ends of these outgoing edges (see Fig. 5), and for any pairing σ of this set, consider the corresponding contraction of tensors −B using the form B−1 . This will produce a number i F(σ). 3. The integral Zn is given by (9) Zn = √(2 dπe)td /B2 � (i!)gnin iin i! (cid:1)ni( 2i −1) � Fσ i σ