Renormalization and Hopf algebras

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(1)Renormalization and Hopf Algebras. Catalina Gómez Sánchez Advisor: Andrés Reyes. Universidad de los Andes Bogotá D.C January 30, 2008.

(2) Acknowledgements I want to thank my advisor Professor Andrés Reyes for having suggested this subject for my project and whose help and support made it all possible. I also want to thank Professor Alessandra Frabetti for giving me the best introduction to the subject I could have possibly had and for her valuable suggestions.. 1.

(3) Contents 1 Introduction. 3. 2 Hopf Algebras 5 2.1 Algebras, Coalgebras and Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 The Algebra of Representative Functions . . . . . . . . . . . . . . . . . . . . . . 16 3 Quantum Field Theory and Feynman Graphs 3.1 The Propagator . . . . . . . . . . . . . . . . . . . . . . 3.2 Generating Functionals for Scalar Fields . . . . . . . . 3.2.1 Free Fields . . . . . . . . . . . . . . . . . . . . . 3.2.2 Interacting Fields . . . . . . . . . . . . . . . . . 3.2.3 Another Approach to the Generating Functional 3.3 Green Functions and the S Matrix . . . . . . . . . . . . 3.4 φ4 theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Green Functions for φ4 . . . . . . . . . . . 3.4.2 Feynman Rules for φ4 . . . . . . . . . . . . . . 3.4.3 Vertex Functions . . . . . . . . . . . . . . . . . 4 Renormalization 4.1 Divergencies in φ4 . . . . . . . 4.2 Dimensional Regularization . 4.3 Renormalization of φ4 . . . . 4.3.1 Counterterms . . . . . 4.3.2 Bogoliubov’s Recursion. . . . . . . . . . . . . . . . . . . . . Formula. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and Zimmermann’s Solution .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . .. 20 20 22 22 25 25 28 30 33 35 35. . . . . .. 38 38 39 42 43 44. 5 The Hopf Algebra Structure of Renormalization in Perturbative Theories 49 5.1 The Hopf Algebra of Parenthesized Words . . . . . . . . . . . . . . . . . . . . . 49 5.2 The Hopf Algebra of Feynman Graphs . . . . . . . . . . . . . . . . . . . . . . . 54 6 Conclusions and Perspectives. 58. 2.

(4) Chapter 1 Introduction The renormalization process used in Quantum Field Theory (QFT) has proved to be a valuable and succesful tool. Through the redefinition of some physical parameters, it allows us to eliminate the divergencies that come up in the process of evaluating Feynman integrals and yields values for physical quantities that actually agree to an amazing extent with experiment. One of the earliest achievements of renormalization is its prediction of the anomalous magnetic moment of the electron, which is accurate up to the tenth decimal place. The basic idea behind renormalization is one that is used all the time in physics. Parameters are introduced into physical laws in order to reflect how interactions take place, saving us the need of describing each one of these interactions in detail, a process that would be completely impractical in most cases. Some examples of such parameters are the refraction index n in the laws of refraction of light, the electric and magnetic permittivities µ and ǫ in media different from the vacuum, and the typical hydrodynamics example where the inertial mass of a ballon that accelerates upwards in air/water can be redefined in order to predict the right acceleration through Newton’s laws. The difference between these examples and renormalization used in QFT however, is that in the latter parameters must be redefined through a recursive process. The work of theoretical physicists such as Bogoliubov, Hepp and Zimmermann gave a recursive formula for ”hiding” the unwanted divergencies into what they called bare physical parameters, and although this actually worked it left many thinking that the whole process only showed that our ignorance with respect to the theory could be wrapped into a package. Furthermore, the proof that the ”miraculous” recursion formula in fact made divergent integrals finite was thought convoluted and inaccessible by most mathematicians, mainly because of the lack of any mathematical structure to support it. However, this situation has radically changed over the past decade. In 1998 Dirk Kreimer and Alain Connes showed that the renormalization process has a very specific, well defined algebraic structure: a Hopf algebra. The possible consequences could not be any more exciting,“...Hopf algebras provide a natural setting for studying generalized symmetries and an organizing tool in non-commutative geometrical contexts.” [1]. There is no 3.

(5) doubt that the high level of symmetry that this finding inveighs is something worth studying. In [12] Kreimer writes: ”In this paper we want to demonstrate that renormalization is not such an ad hoc procedure, but on the contrary arises in a very natural manner from the properties of Hopf algebras”. It is in fact the antipode of this Hopf algebra that gives us the key to decode the apparently intrincate substraction procedure given by the recursion formula. The implications of Connes and Kreimer’s finding are still to be seen and we all hope will broaden our knowledge of the geometry of space-time and the nature of the quantum fields themselves studied by phycisists all around the world today. Some results involving the automatization process of renormalization via Hopf algebras are already in development [15]. The main interest and purpose of this project is then to understand the Hopf algebra structure of renormalization.. 4.

(6) Chapter 2 Hopf Algebras A Hopf Algebra is a vector space over a field k on which there is both an algebra and coalgebra structure related to each other by certain compatibility conditions. In this chapter we shall give the basic concepts that lead to the definition of such an algebra. The first two sections of this chapter are based on [1] and [2].. 2.1. Algebras, Coalgebras and Bialgebras. Definition 1: A unital associative algebra is a triple (A, m, u) where A is a vector space over a field k, and m and u are k-linear maps: m:A ⊗ A→A m(a1 ⊗ a2 ) = a1 · a2 , u:k → A u(λ) = λ1A ,. (2.1). (2.2). where a1 , a2 ∈ A, λ ∈ k, with “·” denoting the product in the vector space and 1A the identity element in A. The maps m, u are called the product and unit respectively, and are such that the following diagrams commute: A⊗A⊗A. m⊗id. -. A⊗A m. id⊗m. ?. A⊗A. m. Diagram 1 5. -. ?. A.

(7) k⊗A. u⊗id. -. id⊗u A⊗A A⊗k. ∼. ?. -. A. id. m. ∼. ?. ?. A. . A. id. Diagram 2 The commutativity of diagram 1 gives the associativity of the product m. Definition 2: An algebra is commutative if the following diagram commutes: σ. A⊗A. -. @ m@ R @. A⊗A m. A Diagram 3 where σ denotes the twist operator, that is, σ(a ⊗ b) = b ⊗ a Definition 3: The prefix ”co” means inverting the direction of the arrows in the previous diagrams. Thus, a co-algebra is a triple (C, ∆, ε) where C is a vector space over a field k, and ∆ and ε are k-linear maps: ∆:A → A⊗A X ∆(c) = c1i ⊗ c2i. (2.3). f inite. ε:C → k. (2.4). The maps ∆, ε are called the coproduct and the counit respectively, and are defined by the following diagrams: ∆⊗id C ⊗C ⊗C C ⊗C. 6. 6. id⊗∆. ∆. C⊗C . ∆. 6. C.

(8) Diagram 4 ε⊗id k⊗C C⊗C ∼. 6. id⊗ε. -. C ⊗k. 6. 6. ∼. ∆. C . -. C. id. id. C. Diagram 5 Note that these two diagrams are the same that define m and u but the direction in the arrows has been inverted. The commutativity of the first gives the cocommutativity of the coproduct in the coalgebra and can also be expressed as (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆. (2.5). whereas according to the second, the counit must always be such that id = (ε ⊗ id) ◦ ∆ = (id ⊗ ε) ◦ ∆. (2.6). which applied on c ∈ C, and using the above definition for the coproduct ∆(c) can be written as X X ε(c1 )c2 = c1 ε(c2 ) (2.7) where the index i of the sum has been omitted for simplicity. We shall continue to do so as long as this notation (sigma notation) does not become ambiguous. Definition 4: A coalgebra is called cocommutative if the following diagram commutes: C⊗C . σ. C ⊗C . @ I @ ∆ @. ∆. C Diagram 6 or in sigma notation X. c1 ⊗ c2 =. X. c2 ⊗ c1. (2.8). ∀c ∈ C. Note that the tensor product of two unital algebras is also a unital algebra, where 1A⊗A = 1A ⊗ 1A :. 7.

(9) m⊗m′. id⊗σ⊗id. (A ⊗ A′ ) ⊗ (A ⊗ A′ ) −−−−−→ A ⊗ A ⊗ A′ ⊗ A′ −−−→ A ⊗ A′ m(a ⊗ a′ ) ⊗ (b ⊗ b′ ) = ab ⊗ a′ b′ ∈ A ⊗ A′. (2.9). and u⊗u′. k → k ⊗ k −−−→ A ⊗ A′ = (λµ)1A ⊗ 1A uA⊗A′ (λ ⊗ µ). (2.10). Similarly, the tensor product of two coalgebras is also a coalgebra where the coproduct and counit are obtained by inverting the direction of the arrows of (1.9) and (1.10): ∆⊗∆′. id⊗σ⊗id. (C ⊗ C ′ ) −−−→ C ⊗ C ⊗ C ′ ⊗ C ′ −−−−−→ (C ⊗ C ′ ) ⊗ (C ⊗ C ′ ) X c1i ⊗ c′1j ⊗ c2i ⊗ c′2j = ∆C⊗C ′ (c ⊗ c′ ). (2.11). i,j. and ε⊗ε′. C ⊗ C ′ −−→ k ⊗ k → k εC⊗C ′ (c ⊗ c′ ) = ε(c)ε(c′). (2.12). Definition 5: Let (A, mA , uA ) and (B, mB , uB ) be k-algebras. The linear map f : A → B is a morphism of algebras if the following diagrams commute: A⊗A. f ⊗f. -. B⊗B. mA. mB. ?. ?. -. A. B. f. Diagram 7 f. A. . @ I uA @. B. uB. k Diagram 8. 8.

(10) Definition 6: Let (C, ∆C , εC ) and (D, ∆D , εD ) be coalgebras. The linear map g : C → D is a morphism of coalgebras if the following diagrams commute: g. C. -. D. ∆C. ∆D. ?. C ⊗C. g⊗g. ?. D⊗D. Diagram 9 g. C @ εC @ R. -. D. εD. k Diagram 10 Definition 7: The quintet (B, m, u, ∆, ε) is called a Bialgebra if the triple (B, m, u) is a unital algebra, (B, ∆, ε) is a counital coalgebra and ∆ and ε are morphisms of algebras. Proposition 1: The following statements are equivalent: i) m and u are morphisms of coalgebras. ii) ∆ and ε are morphisms of algebras. Proof: Using the diagrams from definitions 5 and 6, we can see that m is a morphism of coalgebras if and only if the following diagrams commute: -. m. B⊗B. B. ∆⊗∆. ?. B⊗B⊗B⊗B. ∆. id⊗σ⊗id. ?. B⊗B⊗B⊗H. m⊗m. Diagram 11. 9. ?. B⊗B.

(11) -. m. B⊗B. B. ε⊗ε. ?. k⊗k. ε. ?. -. k. id. ?. k. Diagram 12 Similarly, u is a morphism of coalgebras if and only if the following diagrams commute: -. B. k. B. 6 ∆. k⊗k. u⊗u. ?. B⊗B. Diagram 13 u. k @. R id @. -. B. ε. k Diagram 14 But ∆ is a morphism of algebras if and only if diagrams 11 and 13 commute, and ε is a morphism of coalgebras if and only if 12 and 14 commute. q.e.d Definition 8: Let H, L be bialgebras. A linear map f : H → L is a morphism of bialgebras if f is a morphism of algebras and coalgebras between the underlying algebras and coalgebras respectively.. 2.2. Hopf Algebras. Let (A, m, u) be an algebra and (C, ∆, ε) be a coalgebra. Let us consider the vector space Hom(C, A), that is, the vector space of all linear maps from C to A.. 10.

(12) Let f, g ∈ Hom(C, A). We define the convolution product ∗ in Hom(C, A) such that f ∗ g ∈ Hom(C, A) by means of the following composition:. f ⊗g. ∆. m. C− → C ⊗ C −−→ A ⊗ A − →A X (f ∗ g)(c) = f (c1 )g(c2). (2.13). ∀c ∈ C. Proposition 2: (Hom(C, a), ∗, u ◦ ε)) is a unital associative algebra. Proof: Starting from the known diagrams that define ∆, m and ∗, we build the following diagram:. C. * ∆ . C⊗C. (f ∗g)⊗h. HH ∆⊗id HH HH j. C ⊗C ⊗C. HH HH ∆ HH j H. f ⊗g⊗h. -. * id⊗∆ . C⊗C. * m⊗id . A⊗A. A⊗A⊗A. HH H id⊗m HH j H -. f ⊗(g∗h). HH HHm HH j H * m . A⊗A. Diagram 15 Note that going from C to A through the top “path” is the same as going through the bottom, meaning that the convolution product ∗ in Hom(C, A) is in fact associative. We now prove that u ◦ ε is the identity element of the algebra: X (f ∗ (u ◦ ε))(c) = m ◦ (f ⊗ (u ◦ ε)) ◦ ∆(c) = m( f (c1 ) ⊗ (u ◦ ε)(c2 )) X X = f (c1 )ε(c2 )1A = f ( c1 ε(c2 )) = f (c). (2.14). The same procedure shows that (u ◦ ε) ∗ f = f . q.e.d Let H be a bialgebra. Let us consider the vector space Hom(H C , H A ) with product ∗ as defined above, and unit u ◦ ε. Note that the identity map I : H → H belongs to Hom(H C , H A ). Definition 9: Let H be a bialgebra. The linear map S : H → H whose inverse with respect to ∗ ∈ 11. A.

(13) Hom(H C , H A ) is the identity map I : H → H is called the antipode, that is, the antipode of H is such that S∗I =I ∗S =u◦ε. (2.15). Proposition 3: Since S is the inverse of I, and I is unique, then so is S. Proof: Suppose that there exist S and S ′ such that S ∗ I = I ∗ S = u ◦ ε and S ′ ∗ I = I ∗ S ′ = u ◦ ε. Multiplying the first of these two equations by S ′ on the left, and given that ∗ is associative, we obtain S ′ ∗ (I ∗ S) (S ′ ∗ I) ∗ S (u ◦ ε) ∗ S S. = = = =. S ′ ∗ (u ◦ ε) S′ S′ S′. Definition 10: A bialgebra H that possesses and antipode S is called a Hopf Algebra. In a Hopf Algebra, the antipode is then given by the following diagram: ∆ H ⊗H . H. ∆-. u◦ε. id⊗S. ?. H ⊗H. m. -. ?. H m. H ⊗H S⊗id. ?. H ⊗H. Diagram 16 which can also be written as m ◦ (id ⊗ S) ◦ ∆ = m ◦ (S ⊗ id) ◦ ∆ = u ◦ ε. (2.16). or in sigma notation X. S(h1 )h2 =. X. h1 S(h2 ) = ε(h)1A. Proposition 4: Let H be a Hopf Algebra with antipode S. Then i) S(hg) = S(g)S(h)∀g, h ∈ H ii) S(1) = 1 12. (2.17).

(14) P iii) ∆(S(h)) = S(h2 ) ⊗ S(h1 ) iv) ε(S(h)) = ε(h) i) and ii) mean that S is an antihomomorphism of algebras, while iii) and iv) mean that S is an antihomomorphism of coalgebras. For proof of this proposition see [2], page 153. Proposition 5: Let H be a Hopf Algebra with antipode S. Then the following assertions are equivalent: P i) PS(h2 )h1 = ε(h)1A ii) h2 S(h1 ) = ε(h)1A iii) S ◦ S = S 2 = I For proof see [2], page 155. Corollary: If H is a commutative or cocommutative Hopf Algebra, then S 2 = I. Proof: P P Let H be commutative. Since S(h1 )h2 = ε(h)1A , then h2 S(h1 ) = ε(h)1A which is part ii) of proposition 4. Now, let H be cocommutative. Since part i) of proposition 4.. P. h1 S(h2 ) = ε(h)1A , then. P. S(h2 )h1 = ε(h)1A which is. Example 1 [2], [4] Let G be a group P and k a field. Let kG be the vector space with basis {g | g ∈ G} such that ∀x ∈ kG, x = g αg g where g ∈ G, αg ∈ k. We shall define a coalgebra structure over kG as follows: ∆(g) = g ⊗ g ε(g) = 1∀g ∈ G. We need only define the coproduct ∆ and counit ε for all elements of g and then extend by linearity to all the elements in the vector space. Let us now check that the definitions given above agree with the definitions of these two algebraic structures. For the coproduct we have: (∆⊗id)◦∆(g) = (∆⊗id)(g ⊗g) = ∆(g)⊗g = g ⊗g ⊗g = (id⊗∆)(g ⊗g) = (id⊗∆)∆(g) (2.18) 13.

(15) so ∆ is in fact coassociative. For the counit we have (ε ⊗ id) ◦ ∆(g) = ε(g) ⊗ g = 1 ⊗ g = g = g ⊗ 1 = g ⊗ ε(g) = (id ⊗ ε) ◦ ∆(g). (2.19). So far, we have shown that kG is a (trivially) commutative coalgebra. Let x, y ∈ kG be such that x = u:. P. αg and y =. m(x, y) = m(x =. X. P. αg, y =. βh. We now define the product m and counit. X. βh) =. XX. αβ(gh). u(k) = ke. (2.20) (2.21). where k ∈ K and e is the neutral element in G. Note that with the product m defined above kG will not in general be a commutative algebra, it will be so only if G is abelian. Using m and u we can see that ∆ and ε are morphisms of algebras, that is:. XX XX XX ∆(x · y) = ∆( αβ(gh)) = αβ(gh) ⊗ (gh) = αβ(g ⊗ g)(h ⊗ h) X X = α(g ⊗ g) β(h ⊗ h) = ∆(x)∆(y) (2.22) ε(x · y) = ε(. X. αβ(gh)) =. X. X X X X αβε(gh) = ( α)( β) = ε( αg)ε( βh) = ε(x)ε(y) (2.23). We now define the antipode S(g) = g −1 ∀ g ∈ G and extend to all kG by linearity. Then m ◦ (S ⊗ id) ◦ ∆(g) = m(S(g) ⊗ g) = g −1 g = e m ◦ (id ⊗ S) ◦ ∆(g) = m(g ⊗ S(g)) = gg −1 = e (U ◦ ε)(g) = u(1) = e. (2.24). and XX XX XX S(x · y) = ( αβ(gh)) = αβS(gh) = αβ(gh)−1 X X X X =( βh−1 )( αg −1) = S( βh)S( αg) = S(y)S(x). (2.25). so S agrees with all the conditions for the antipode, that is, it is the inverse of the identity map with respect to convolution and it is an antihomomorphism of algebras.. 14.

(16) We have now induced a Hopf Algebra structure on the vector space kG, we say that (kG, m, u, ∆, ε, S) is a Hopf Algebra.. Example 2 [4] Let S 1 be the unitary circle made up of all elements of the form eiθ , where θ ∈ [0, 2π). Let 1 the set of continuous complex functions over S 1 that vanish at infinity, that is, RCt02(S ) be 2 |f (t)| dt < ∞ ∀ t1 , t2 such that t2 − t1 = 2π. t1 With the usual function addition and multiplication (pointwise), C0 (S 1 ) is an algebra. The functions f ∈ C0 (S 1 ) are really functions of θ so we can denote them by C0 ([0, 2π)). We can extend f ∈ C0 ([0, 2π)) to all R by associate to R π we can P setting fikθ(2π) = f (0) and so on. Then 1 −ikt f (t)e dt ∈ C. c e where θ ∈ each f its Fourier series f (θ) = ∞ R and c = k k=−∞ k 2π −π. We can now identify C0 (S 1 ) with the set of Fourier series that we will denote C[Z, Z −1] where Z corresponds to the elements of the form eikθ and Z −1 corresponds to the elements of the form e−ikθ . We define ∆(Z) = Z ⊗ Z ε(Z) = 1 S(Z) = Z −1. (2.26). ∆(Z −1 ) = Z −1 ⊗ Z −1 ε(Z −1) = 1 S(Z −1 ) = Z. (2.27). and. It is not hard to see that these definitions give C[Z, Z −1 ] a Hopf Algebra structure. For the coproduct we have: (∆ ⊗ id)∆(eikθ ) = (∆ ⊗ id)(eikθ ⊗ eikθ ) = eikθ ⊗ eikθ ⊗ eikθ = (id ⊗ ∆)(eikθ ⊗ eikθ ) = (id ⊗ ∆)∆(eikθ ) ′. ′. ′. ∆(eikθ eikθ ) = (eikθ eikθ ) ⊗ (eikθ eikθ ) = (eikθ eikθ ) ⊗ (eikθ eikθ ) = ∆(eikθ )∆(eikθ ). (2.28) (2.29). meaning that the coproduct is both coassociative and homomorphism of algebras. For the counit we have (ε ⊗ id)∆(eikθ ) = ε(eikθ ) ⊗ eikθ = 1 ⊗ eikθ = eikθ 15. (2.30).

(17) Similarly one can show that (id ⊗ ε)∆(eikθ ) = eikθ . Also ′. ′. ′. ε(eikθ eikθ ) = ε(eik(θ+θ ) ) = 1 = ε(eikθ )ε(eikθ ). (2.31). so the condition for the counit holds, and it is a homomorphism of algebras. For the antipode we have m(S ⊗ id)∆(eikθ ) = m(S(eikθ ) ⊗ eikθ ) = e−ikθ eikθ = 1. (2.32). Similarly, one shows that m(id ⊗ S)∆(eikθ ) = 1, and since u ◦ ε(eikθ ) = u(1) = 1, then S is in fact the antipode. Also, S(eikθ eikθ ) = S(eik(θ+θ ) ) = e−ik(θ+θ ) = e−ikθ e−ikθ = S(eikθ )S(eikθ ) ′. ′. ′. ′. ′. (2.33). and so S is a homomorphism of algebras (here the prefix anti can be spared because C[Z, Z −1 ] is commutative).. 2.3. The Algebra of Representative Functions. This section is based on [1] and the work of A. Frabetti in [3]. We now turn our attention to a particular kind of commutative Hopf Algebras. Let G be a group and let F (G) denote the set of functions of G over the complex field. The vector space F (G) is a unital associative and commutative algebra with product (f g)(x) = f (x)g(x) for f, g ∈ F (G) and unit given by the constant function 1(x) = 1. We would like to make the identification F (G) ⊗ F (G) = F (G × G) but this is not true in general. The latter is a much larger vector space than the first, that is, F (G) ⊗ F (G) is a strict subalgebra of F (G × G). However, if we restrict ourselves to compact groups, it is possible to find a smaller algebra R(G) ⊂ F (G) such that R(G) ⊗ R(G) = R(G × G), the algebra of representative functions. Definition 11: A function f : G → C is called representative if there exist a finite number of functions f1 , f2 , ..., fk such that any left translate of f can be written as a linear combination of such functions: (Lx f )(y) = f (xy) =. k X i=1. 16. li (x)fi. (2.34).

(18) There exists a natural homomorphism that we shall name π π : R(G) ⊗ R(G) → R(G × G) π(f ⊗ g)(x, y) = f (x)g(y). (2.35). Lemma: π is an isomorphism of algebras. For proof see [1], page 42. This lemma allows us to translate the structure of the group G into a Hopf Algebra structure in R(G). • ∀f ∈ G, the multiplication operation in G ” × ” : G × G → G induces the map Cm : R(G) → R(G × G) which gives us the coproduct ∆ : π −1 ◦ Cm : R(G) → R(G) ⊗ R(G) ∆(f )(x, y) = f (x · y). (2.36). Evidently, ∆ is a homomorphism of algebras: ∆(f g)(x, y) = (f g)(x · y) = f (x · y)g(x · y) = ∆(f )(x, y)∆(g)(x, y). (2.37). One can easily verify that ∆ is coassociative (just like we have done before). This is due to the associativity in G.. • The neutral element e ∈ G induces the counit ε : R(G) → C ε(f ) = f (e). (2.38). Just as we did in the examples from last section, one can verify that the counit so defined is a homomorphism of algebras and agrees with the required condition. • The operation of inversion in G, x → x−1 induces the antipode S : R(G) → R(G) S(f )(x) = f (x−1 ). 17. (2.39).

(19) which is in fact the inverse of the identity map, since X X m(S ⊗ id)∆(f )(x) = m( S(f1 ) ⊗ f2 )(x) = f1 (x−1 )f2 (x) = ∆(f )(x, x−1 ). = f (e) = ε(f ) = u(ε(f )). (2.40). S is a homomorphism of algebras and also, since R(G) is commutative then S should be such that S 2 = I. We can see that this is true, since S(S(f )(x)) = S(f (x−1 )) = f ((x−1 )−1 ) = f (x). Using representation theory, in particular the Peter-Weyl theorem, it is possible to show that • R(G) ⊗ R(G) = R(G × G) • As an algebra, R(G) is generated by the matrix elements of all of the group’s representations of finite dimention. • R(G) is also generated by the matrix elements of one faithfull representation of G. These last matrix elements that generate R(G) are called the coordinate functions. Example 3 (Exercise 1.13 of [3]) Let’s consider the group H3 , the group of all upper triangular 3 × 3 matrices ∈ GL(3, C) whose diagonal entries are all 1. That is, H3 = {M3×3 (C)|mi,i = 1, mi,j<i = 0}. Evidently, any M ∈ H3 is of the form   1 a b M = 0 1 c  0 0 1 R(H3 ) is then generated by three coordinate functions x : H3 → C, y : H3 → C and z : H3 → C defined as follows: x(M) = a y(M) = b z(M) = c. (2.41). so R(H3 ) = C[x, y, z]. We shall now define the Hopf Algebra structure on H3 . • Coproduct:.     1 a b 1 a′ b′ For two matrices M = 0 1 c  and M ′ = 0 1 c′  ∈ H3 , their product (usual matrix 0 0 1 0 0 1   1 a + a′ b′ + ac′ + b 1 c + c′  product) yields M · M ′ = 0 0 0 1 18.

(20) so that ∆x(M, M ′ ) = x(M · M ′ ) = a + a′ = x(M) · 1 + 1 · x(M ′ ) ∆y(M, M ′ ) = y(M · M ′ ) = b + ac′ + b′ = y(M) · 1 + x(M)z(M ′ ) + 1 · y(M ′ ) ∆z(M, M ′ ) = z(M · M ′ ) = c + c′ = z(M) · 1 + 1 · z(M ′ ) (2.42) Finally, we write the expression for the coproduct of the coordinate functions as ∆x = x ⊗ 1 + 1 ⊗ x ∆y = y ⊗ 1 + x ⊗ z + 1 ⊗ y ∆z = z ⊗ 1 + 1 ⊗ z. (2.43). • Counit: Since ε(f ) = f (e), then for H3 ε(x) = x(1) = 0 ε(y) = y(1) = 0 ε(z) = z(1) = 0. (2.44). • Antipode: As defined above, S(x)(M) = x(M −1 ). Inverting M we obtain M −1 so that.   1 −a −a + ac = 0 1 −c  0 0 1. S(x)(M) = x(M −1 ) = −a = −x(M) S(y)(M) = y(M −1 ) = −b + ac = −y(M) + x(M)z(M) S(z)(M) = z(M −1 ) = −c = −z(M). (2.45). and finally for the antipode we can write S(x) = −x S(y) = −y + xz S(z) = −z. 19. (2.46).

(21) Chapter 3 Quantum Field Theory and Feynman Graphs In this chapter we will follow the treatment of L. Ryder in [5] extensively, taking some results for granted for the sake of simplicity, but going deeper into some proceedings which are of most importance for the purpose of this document.. 3.1. The Propagator. In the path formulation of quantum mechanics, given an initial wavefunction ψ(qi ti ) for a particle, it is possible to find the wave function at a later time tf through the propagator K(qf tf ; qi ti ): Z ψ(qf tf ) = K(qf tf ; qi ti )ψ(qi ti )dqi (3.1) It is known that ψ(qf tf ) is the probability amplitude that the particle is found at qf in tf . Thus, the propagator gives the probability amplitude for the transition from qi at ti to qf at tf : P (qf tf ; qi ti ) = |K(qf tf )|2. (3.2). By dividing the time interval from ti to tf into two equal intervals (from ti to t and then from t to tf ) and using (3.2), we obtain Z Z ψ(qf tf ) = K(qf tf ; qt)K(qt; qi ti )ψ(qi ti )dqi dq (3.3) and so K(qf tf ; qi ti ) =. Z. K(qf tf ; qt)K(qt; qi ti )dq. 20. (3.4).

(22) so that the transition from the initial to the final state can be viewed as the result of the transition from the initial state to all available intermediate states followed by the transition from this state to the final one. Also, as one might expect K(qf tf ; qi ti ) = hqf tf |qi ti i. (3.5). where |qti = e−iHt/~|qi Now, by splitting the time interval from ti to tf into n + 1 equal time intervals of length τ we find Z Z Z hqf tf |qi ti i = ... dq1 dq2 ...dqn hqf tf |qn tn ihqn tn |qn−1 tn−1 i...hq1 t1 |qi ti i (3.6) where integration is over all possible “paths”. We will assume that H is of the form p2 + V (q) 2m which is general enough for the current purposes. H=. (3.7). Using (3.7), we can easily compute the propagator over a small segment (from tj to tj+1 ). The result is Z i 1 dpj exp [pj (qj+1 − qj ) − τ H(pj , q̄j )] hqj+1tj+1 |qj tj i = (3.8) h ~ where q̄j = 21 (qj + qj+1 ). Finally, inserting this result into (3.6) and taking the continum limit yields the expression for the complete propagator: hqf tf |qi ti i = lim. n→∞. Z Y n j=1. dqj. n Y dpj j=0. X n i [pj (qj+1 − qj ) − τ H(pj , q̄j )] exp h ~. (3.9). j6=0. which can be written symbolically as Z tf Z DqDp i hqf tf |qi ti i = exp dt[pq̇ − H(p, q)] h ~ ti or replacing H by its explicit form and integrating over p, we can also write Z tf Z i L(q, q̇)dt hqf tf |qi ti i = N Dq exp ~ ti. (3.10). (3.11). Nevertheless, it is not clear that such an expression should be well defined mathematically, or in other words, that it should make any sense (note that the “measure” Dq in the above integral will not be invariant under transaltion). We will talk about this later on, but for now we shall ignore this fact until we can express (3.11) in a particularly useful way. 21.

(23) In the context of QFT we would like to speak of particles. Particles are created, they interact and they are destroyed. This is what we observe everyday in particle physics experiments. The events of creation and anihilation can be represented by the presence of a source, where the boundary conditions q(tf ) = qf and q(ti ) = qi must be redefined as vacuum to vacuum transitions via creation-interaction-destruction through the presence of a source J in the propagator [8]: L → L + ~J(t)q(t). (3.12). We define this transition amplitude as Z[J] ∝ h0, ∞|0, −∞iJ where |0, ti represents the vacuum state at time t. Inserting (3.12) in (3.11) we find Z ∞ Z 1 2 i dt(L + ~Jq + iǫq ) ∝ h0, +∞|0, −∞iJ Z[J] = Dq exp ~ −∞ 2. (3.13). (3.14). The term 21 iǫq 2 has been added in order to obtain a ground state (see [5] pg 175). An important relation concerning Z[J] that will be of most importance later on is Z ∞ Z δ n Z[J] i 1 2 n Dqq(t1 )...q(tn ) exp =i dt(L + ~Jq + iǫq ) δJ(t1 )...J(tn ) ~ −∞ 2. (3.15). Note that this is exactly the expectation value δ n Z[J] |J=0 ∝ in h0, ∞|T [q(t1)...q(tn )]|0, −∞i δJ(t1 )...J(tn ). (3.16). when we set J = 0. T is the time ordering operator.. 3.2. Generating Functionals for Scalar Fields. 3.2.1. Free Fields. For a scalar field φ(xµ ) with a source J(xµ ). Z[J] =. Z. Z 1 2 4 µ mu Dφ exp i d x(L(φ) + J(x )φ(x ) + iǫφ ) ∝ h0, +∞|0, −∞iJ 2. analogously to (3.14), where ~ = 1 and q(t) → φ(xµ ).. 22. (3.17).

(24) Here, Minkowski space has been broken up into a 4-dimensional lattice made up of cells of volume δ 4 , in each of which φ is taken to be constant. In the case of the free particle L = 1 (∂ φ∂ µ φ − m2 φ2 ) so 2 µ Z Z 1 µ 2 2 4 (3.18) Z0 [J] = Dφ exp i { [∂µ φ∂ φ − (m − iǫ)φ ] + φJ}d x 2 which can be written as Z0 [J] =. Z. Dφ exp. . −i. Z. 1 2 4 [ φ( + m − iǫ)φ − Jφ]d x 2. (3.19). Since the exponent of (3.19) is a cuadratic form, we shall use some generalized formulae for gaussian integration to ”fix” the problem of the undefined measure in the above result. We know that for one variable x Z. ∞. 2 − ax2. e. dx =. −∞. so for n variables x1 , ..., xn Z. . exp. Rn. r. 2π a. (3.20). (2π)n/2 1X 2 an xn dx1 ...dxn = Qn 1/2 − 2 n ai. (3.21). i=1. Let A be a diagonal n × n matrix with entries a1 , a2 , ..., an and let x = (x1 , x2 , ..., xn ) be an n-vector. Then, note that X an x2n = (x, Ax) (3.22) and. det A =. n Y. ai. (3.23). i=1. dn x , (2π)n/2. If we define the measure (dx) =. Z. e−. then (3.21) can be written as. (x,Ax) 2. (dx) = (det A)−1/2. (3.24). for any diagonalizable A. We also know that Z. ∞. −ax2 +bx+c. e. dx = exp. −∞. . b2 +c 4a. 1/2 π a. We can generalize (3.25) for any cuadratic form Q(x) = 12 (x, Ax) + (b, x) + c. 23. (3.25).

(25) Z. exp. . . 1 − (x, Ax) + (b, x) + c 2. . . 1 −1 (dx) = exp (b, A b) − c (det A)−1/2 2. (3.26). Even though these formulae are only valid for finite dimensional vector spaces, we will now assume that we can use them for infinite dimensional ones (this would need to be carefully justified, but let us forget about that for now). Comparing (3.19) to (3.26) we see that if we set A = i( + m2 − iǫ), b = −iJ and c = 0 then Z i 2 −1 J(x)( + m − iǫ) J(y)dxdy [det i( + m2 − iǫ)]−1/2 Z0 [J] = exp 2. (3.27). From (3.24) we find the expression −1/2. 2. [det i( + m − iǫ)]. =. Z. . i Dφ exp − 2. Z. φ( + m − iǫ)φdx 2. (3.28). which is in fact a number, so finally Z i Z0 [J] = N exp J(x)∆F (x − y)J(y)dxdy 2. (3.29). ( + m2 − iǫ)−1 = −∆F (x). (3.30). where. The equation satisfied by ∆F (x) is then ( + m2 − iǫ)∆F (x) = −δ 4 (x). (3.31). and its Fourier representation is 1 ∆F (x) = (2π)4. Z. d4 k. e−ikx k 2 − m2 + iǫ. (3.32). Like functions, functionals can also be expanded as power series. Such an expansion of Z0 [J] yields Z0 [J] =. ∞ Z X. dx1 ...dxn. n=0. 1 τn (x1 , ..., xn )J(x1 )...J(xn ) n!. (3.33). where τ (x1 , ..., xn ) =. δ n Z0 [J] 1 |J=0 in δJ(x1 )...δJ(xn ) 24. (3.34).

(26) The functional Z0 [J] is called the generating functional of the functions τ (x1 , ..., xn ). Normalizing Z0 [0] = 1 and comparing this last equation to (3.16) we can see that τ (x1 , ..., xn ) = h0|T [φ(x1 )...φ(xn )]|0i. (3.35). These are the Green functions of the theory. They can (and should) be interpreted as representing the propagation of n scalar particles between sources and so, they are closely related to the elements of the scattering matrix S.. 3.2.2. Interacting Fields. For interacting fields, L = L0 +Lint , where L0 denotes the free field lagrangian and Lint denotes the part belonging to interactions. The normalized generating functional is R R Dφ exp(iS + i Jφdx) R (3.36) Z[J] = DφeiS Note that from (3.30), (3.29) and (3.36) it is possible to find the differential equation that must be satisfied by Z[J] δ 2 1 δZ[J] ′ Z[J] = J(x)Z[J] (3.37) ( + m ) − Lint i δJ(x) iδJ(x). The solution to this equation is (see [5] pg 197) Z δ dx Z0 [J] Z[J] = N exp i Lint iδJ. 3.2.3. (3.38). Another Approach to the Generating Functional. In the past subsections we have seen how to obtain an expression for the generating functional Z[J]. However there is another way of reaching the same expressions for both free and interacting fields which is through field operators. For the sake of completeness we shall outline this approach, starting from the definition of the generating functional (this section is taken from [7]) Z[J] = h0|T ei. R. d4 xφ(x)J(x). |0i. (3.39). Let’s start with the case where Lint = 0. The differential equation for Z0 [J] is ( + m2 ). δ Z0 [J] = J(x)Z0 [J] iδJ(x). Here is what we do to calculate the right hand side of this equation starting from (3.39) First we differentiate Z0 [J] once with respect to J(x) to obtain. 25. (3.40).

(27) R 4 δ Z0 [J] = h0|T [φ(x)ei d xφ(x)J(x) ]|0i iδJ(x). = h0|(T ei. R∞ t. dt′. R. d3 x′ φ(x′ )J(x′ ). − )φ(t, → x )(T ei. Rt. −∞. dt′. R. d3 x′ φ(x′ )J(x′ ). )|0i. (3.41). Next we differentiate twice with respect to x0 , that is, with respect to t to obtain R 3 ′ Rt R∞ ′R 3 ′ δ ′ ′ ′ ′ ′ → → x ) + J(t, − x ))(T ei −∞ dt d x φ(x )J(x ) )|0i Z0 [J] = h0|(T ei t dt d x φ(x )J(x ) )(φ̈(t, − iδJ(x) (3.42) By differentiating (3.40) twice with respect to each of the spatial coordinates we obtain. ∂t ∂ t. Rt R 3 ′ R∞ ′R 3 ′ δ ′ ′ ′ ′ ′ → Z0 [J] = h0|(T ei t dt d x φ(x )J(x ) )(∂xi ∂ xi φ(t, − x ))(T ei −∞ dt d x φ(x )J(x ) )|0i iδJ(x) (3.43) which when added to the previous equation gives. ∂xi ∂ xi. ( + m2 ) h0|(T ei. R∞ t. dt′. δ Z0 [J] = iδJ(x). R. d3 x′ φ(x′ )J(x′ ). ). → → [ ( + m2 )φ(t, − x ) + J(t, − x )](T ei → = J(t, − x )Z0 [J]. Rt. −∞. dt′. R. d3 x′ φ(x′ )J(x′ ). )|0i (3.44). Now, when there is an F (x) satisfying a differential equation of the form ( + m2 )F (x) = f (x). (3.45). and it is such that lim F (x). gives only positive frequency part. lim F (x). gives only negative frequency part. t→+∞. and t→−∞. the the solution to its differential equation is given by Z F (x) = d4 y∆F (x − y)f (y). (3.46). From (3.44) and knowing that free field must satisfy the Klein-Gordon equation, we can see δ that the above conditions are satisfied by F (x) = ( + m2 ) Z01[J] iδJ(x) Z0 [J]. The solution for this F(x) is then given by Z δ 1 2 Z0 [J] = d4 y∆F (x − y)J(y) (3.47) ( + m ) Z0 [J] iδJ(x) 26.

(28) Integrating this result (with respect to J(x)) we finally obtain Z i J(x)∆F (x − y)J(y)dxdy Z0 [J] = N exp 2. (3.48). which is exactly (3.29).. For the case of interacting fields with Lint = −V [φ], φ(x) will not satisfy the Klein-Gordon equation but ( + m2 )φ(x) = −. δV = −V ′ [φ] δφ. (3.49). A similar procedure as that followed for free fields can be done here except that Rt R 3 ′ R∞ ′R 3 ′ δ ′ ′ ′ ′ ′ → x )(T ei −∞ dt d x φ(x )J(x ) )|0i Z[J] = h0|(T ei t dt d x φ(x )J(x ) )(φ̈(t, − iδJ(x) Z Rt R 3 ′ R∞ ′R 3 ′ − → ′ ′ ′ → → i t dt d x φ(x′ )J(x′ ) +ih0|(T e ) d3 x′ [φ̇(t, − x ), φ(t, x′ )]J(t, − x ))(T ei −∞ dt d x φ(x )J(x ) )|0i(3.50). ∂t ∂ t. so we must use the commutation relation for the field operators → − − → − → [φ̇(t, → x ), φ(t, x′ )] = −iδ(− x − x′ ). (3.51). to obtain. ( + m2 ). R 3 ′ Rt R∞ ′R 3 ′ δ ′ ′ ′ ′ ′ Z[J] = h0|(T ei t dt d x φ(x )J(x ) )( + m2 )φ(x)(T ei −∞ dt d x φ(x )J(x ) )|0i iδJ(x) → + J(t, − x )Z[J]. = J(x)Z[J] − h0|T V ′ [φ]ei. R. d4 xφ(x)J(x). |0i. (3.52). If we assume that V ′ [φ] is of the form φm (which is the form that will interest us in the future), we can use the following trick R 4 R δ ′ i d4 xφ(x)J(x) ′ h0|T ei d xφ(x)J(x) |0i (3.53) h0|T V [φ](x)e |0i ≡ V iδJ(x) which allows us, upon performing induction on m (the degree of the monomial), to write (3.52) as δ δ 2 ′ ( + m ) Z[J] (3.54) Z[J] = J(x)Z[J] − V iδJ(x) iδJ(x) Using the following two known relations. 27.

(29) Z 4 J(x), d yV i. e. R. d4 yLint [ iδJδ(y) ]. −i. J(x)e. R. δ iδJ(x). . d4 y ′ Lint [ iδJδ(y ′ ) ]. . δ = iV iδJ(x) Z 4 = J(x) − i J(x), d yLint ′. (3.55) δ iδJ(y ′ ). . (3.56). we obtain (x + m2 ). 1 e−i. R. d4 yLint [ iδJδ(y) ]. R 4 δ δ e−i d yLint [ iδJ (y) ] Z[J] = J(x) Z[J] iδJ(x). (3.57). which is of the form (3.45). From the above expression one can also verify that it satisies the required asymptotic conditions, and so the solution is given by Z R δ −i d4 yLint [ iδJδ(y) ] Z[J] = d4 y∆(x − y)J(y) (3.58) ln e iδJ(y) Integrating and solving for Z[J] we obtain Z δ Z[J] = N exp i Lint dx Z0 [J] iδJ. (3.59). which is the same expression obtained in (3.38).. 3.3. Green Functions and the S Matrix. Since the whole point of quantum field theory is to be able to predict observable physical phenomenae, it is in order to dedicate a section of this document to how generating functionals and Green functions, which are mere mathematical expressions, are related to quantities that can be measured through experiment. The most common processes concerning particle phycisists are scattering processes where cross sections or lifetimes are measured. In order to calculate these quantities the quantum mechanical amplitude for the process must be computed first. After the amplitude is obtained the remaining procedure is quite straightfoward, although tiresome in general. Let’s suppose that the system we are interested in studying is found in an initial state α. This initial state can be, for example, two incoming free particles of masses m1 and m2 with momentum p1 and p2 respectively. After some kind of scattering process takes place, the system is found to be in a final state β. The amplitude associated to this process is called the scattering amplitude and is denoted by Sαβ . Assuming that there are no long range forces (which is not true in general, as for example in QED, but true enough for our current study purposes) we can consider the initial and final states α and β to be asymptotic states consisting of free particles, denoted by 28.

(30) |αiin = |α, t → −∞i |βiout = |β, t → +∞i. (3.60). In this case, the in and out states are related to each other through a unitary operator S, whose matrix elements are the transition amplitudes Sαβ . For the case of our scalar field φ(x), this can be written as φout (x) = S † φin S. (3.61). We wish to find a practical expression for calculating this S. To avoid wandering from our original purpose, we will only tell the main idea of how an expression for S is obtained, a detailed account is found in (ref. Ryder, Itzykson). Considering φ(x) at an intermediate time where interaction is present, φ(x) must obey the equation Kφ(x) =. ∂Lint ∂φ(x). (3.62). where K is the Klein-Gordon operator. We also need to impose some sort of asymptotic condition on φ(x) so that it will be in accordance with φout/in in the limit t → ±∞. However, if this condition is imposed as it is on the field operators, no scattering will ever take place! The correct way of imposing the condition we need is lim ha|φ(x)|bi = h|φout/in (x)|i. t→±∞. (3.63). where |ai and |bi are arbitrary states (in Hilbert space). This condition known as the weak asymptotic condition was found by Lehmann, Symanzik and Zimmermann in 1995. Using (3.62) and (3.63), an expression for S is found [5] Z δ dz Z[J]|J=0 S = exp φin (z)K δJ(z). (3.64). Note that upon expansion of the exponential, we will be left with terms consisting of partial derivatives of Z[J] with respect to J(z) and evaluated for J = 0, which are the Green functions of our theory! In fact, to find a particular element of S one must simply follow the following steps: i) Calculate the Green function G(x1 , ..., xn ) as prescribed in the previous section. ii) Multiply by function.. Qn i. (xi + m2 ), which eliminates all external propagators from the Green. 29.

(31) iii) Multiply by. Qn i. φ(xi ).. Finally we obtain Sn (x1 , ..., xn ) =. Y. φ(xi )(xi + m2 )G(x1 , ..., xn ). (3.65). i. which relates the Green functions directly to the scattering amplitude of a process.. 3.4. φ4 theory. We will now consider the case of a specific type of interaction for scalar particles. 1 1 g L = ∂µ φ∂ µ − m2 φ2 − φ4 (3.66) 2 2 4! so clearly Lint = − 4!g φ4 . Using (3.38) and keeping in mind that Z[J] must be equal to one when J = 0 we obtain . −ig 4!. R. . δ iδJ(z). 4. . −ig 4!. . . −i 2. R. J(x)∆F (x − y)J(y)dxdy exp dz exp Z[J] = 4 R R −ig δ −i dz exp 2 J(x)∆F (x − y)J(y)dxdy exp 4! iδJ(z) Now, the only way of treating exp. . R. δ iδJ(z). 4. . (3.67). J=0. . dz is by expanding the exponential into its. Taylor series. This yields a perturbative series, in both the numerator and denominator, that can be used to the desired order in powers of g. The numerator would then look like this: 4 2 Z 4 2 Z ig δ δ 1 ig 3 1− dz + dz + O(g ) Z0 [J] (3.68) 4! iδJ(z) 2! 4! iδJ(z) In order to compute Z[J] to order g, we must differentiate Z0 [J] four times with respect to J(z). The result is Z 2 Z 4 δ4 2 Z0 [J] = −3∆F (0) +6i∆F (0) ∆F (z −x)J(x)dx + ∆F (z −x)J(x)dx (3.69) δJ(z)4 so to order g 1. Z[J] = R ig. . . . ig 1− 4!. Z . − 3∆F (0)2. − 3∆F (0)2 dz Z 2 Z 4 +6i∆F (0) ∆F (z − x)J(x)dx + ∆F (z − x)J(x)dx dz Z0 [J] 1−. 4!. 30. (3.70).

(32) Note that if we use the binomial theorem to bring the denominator to the numerator, all terms without any J’s will dissappear. So Z 2 Z 4 Z ig 6i∆F (0) ∆F (z − x)J(x)dx + ∆F (z − x)J(x)dx dz Z0 [J] Z[J] = 1 − 4! (3.71) This last fact is common to all properly normalized generating functionals, to all orders. It is useful to represent the terms inside the integral by diagrams. For example, since ∆F (0) represents propagation from some point x to some point y where x = y then we shall represent it by. Figure 3.1. Multiplication of various terms would simply mean joining them at some point (if possible) keeping in mind that because we are dealing with φ4 theory, every vertex (point where lines meet) should join 4 lines. The first term in the integrand of (3.70) would then look like this:. Figure 3.2. The first term in the integrand of (3.71) means that there are two free propagators coming from a common point in space-time z each arriving at a different point x where there is a source, this multiplied by fig. (3.4). This could then be represented by. Figure 3.3. where the x represents a source J. The second term would be then represented by. 31.

(33) Figure 3.4. Now, to calculate Z[J] to order g 2 we need to differentiate Z0 [J] four times with respect to J(z1 ) and four times with respect to J(z2 ). Since we have already performed differentiation with respect to J(z1 ), we can continue from there in order to find the integrand of the second order term of (3.68). δ4 δ4 Z0 [J] = ∆2F (z1 − z2 )∆2F (0) + ∆4F (z1 − z2 ) + ∆4F (0) (3.72) δJ(z2 )4 δJ(z1 )4 Z 2 2 ∆F (z2 − x)J(x)dx + ∆F (0)∆F (z1 − z2 ) Z Z 3 + ∆F (z1 − z2 ) ∆F (z1 − x)J(x)dx ∆F (z2 − x)J(x)dx Z Z 2 + ∆F (0)∆F (z1 − z2 ) ∆F (z1 − x)J(x)dx ∆F (z1 − x)J(x)dx Z 3 Z + ∆F (0)∆F (z1 − z2 ) ∆F (z1 − x)J(x)dx ∆F (z2 − x)J(x)dx 2 Z 2 Z 2 ∆F (z2 − x)J(x)dx + ∆F (z1 − z2 ) ∆F (z2 − x)J(x)dx 3 Z 3 Z ∆F (z1 − x)J(x)dx ∆F (z2 − x)J(x)dx + ∆F (z1 − z2 ) 2 Z 4 Z 3 2 ∆F (z1 − x)J(x)dx + ∆F (0) ∆F (z1 − x)J(x)dx + ∆F (0) 2 Z 2 Z 2 ∆F (z1 − x)J(x)dx ∆F (z2 − x)J(x)dx + ∆F (0) Z 2 Z 4 + ∆F (0) ∆F (z1 − x)J(x)dx ∆F (z2 − x)J(x)dx Z 4 Z 4 + ∆F (z1 − x)J(x)dx ∆F (z2 − x)J(x)dx where all numerical constants (or symmetry factors) have been ommited. The first three terms in (3.73) will dissapear upon normalization so that the term to order g 2 in Z[J] is diagramatically written as 32.

(34) and finally. It is important to note that the number of vertices in these diagrams equals the order in powers of g to which the perturbative series was taken to.. 3.4.1. The Green Functions for φ4. We can now calculate the Green functions. The two point function to order g 2 is τ = + + +. δ2 Z[J]|J=0 δJ(x2 )δJ(x1 ) Z ∆F (x1 − x2 ) + g ∆F (0) ∆F (z − x1 )∆F (z − x2 )dz (3.73) Z 2 g ∆F (0) ∆2F (z1 − z2 )∆F (z1 − x1 )∆F (z1 − z2 )dz1 dz2 Z ∆3F (z1 − z2 )[∆F (z1 − x1 )∆F (z2 − x2 ) + ∆F (z1 − x2 )∆F (z2 − x1 )]dz1 dz2 Z 2 ∆F (0) ∆F (z1 − z2 )[∆F (z1 − x1 )∆F (z2 − x2 ) + ∆F (z1 − x2 )∆F (z2 − x1 )]dz1 dz2 (x1 , x2 ) =. (numerical constants have also been ignored here).. 33.

(35) As we did before for the generating functional, we can also represent the Green functions through diagrams in a natural way. Off course, in these diagrams we no longer have any x representing J’s, we only have z’s, x’s and propagators that go from one to another. The first are common (internal) starting points for propagators, the second are the propagators’ end points. Note that the number of external lines (or legs) that a diagram will have is determined by the Green function (2,4,6... point function will give diagrams with 2,4,6... external legs) and the number of z’s is determined, as before, by the highest order taken into acount in the perturbative series of Z[J]. Diagramatically. The four point function is (calculated up to order g, not g 2 for obvious practical reasons). Here we have written the numerical constants acompanying each term in order to illustrate how these numbers can be easily calculated without having to drag them along in the process of calculating the Green functions (this is why we have ignored them before). These numbers are symmetry factors that arise from the fact that lines in a diagram can be interchanged without changing the diagram itself, that is, there are different schemes that lead to the same diagram. Take fig. 3.5 for example:. Figure 3.5. We have one vertex with four available spots. We have 4 legs that must be joined to this vertex. The number of different ways this can be done is 4!, so the symmetry factor accompanying this diagram must be 24. This can be done for all diagrams. Looking at the 4-point function above, we can see that there are two different types of graphs: connected (the ones where all lines are joined to each other) and disconnected. The latter represent particles that propagate independently from each other (there is no interaction). These graphs don’t contribute to the non trivial part of the S matrix and so they are not interesting.. 34.

(36) 3.4.2. Feynman Rules for φ4. From the two Green functions calculated above we can deduce the Feynman rules for φ4 theory. 1. To each internal line with end verices zi and zj , associate a propagator ∆F (zi − zj ). 2. To each internal vertex (joining 4 lines) associate. −ig . 4!. 3. Count the number of possible configurations that yield this diagram (symmetry factor). 4. Multiply all the above factors and integrate over the positions of all internal vertices. We have seen how to deduce the Feynman rules for φ4 theory in position representation [5]. Computation of Feynman integrals is simplest in momentum representation. We can then write these rules in momentum representation [6] by performing a simple fourier transform. 1. Denote external lines of incoming momenta by p1 , p2 , ..., p2n , denote by k1 , ..., kI the momenta of internal lines. 2. Associate a factor of. i pj −m2. 3. Associate a factor of. d4 kl i (2π)4 kl2 −m2. to the jth external line. to each internal line.. P 4. Associate a factor of −ig(2π)4 δ 4 ( l ǫv,l ql ), where ǫv,l is -1 if ql is coming out of the vertex and is +1 if it is coming out of the vertex. 5. Multiply all the above factors, integrate over all internal momenta and divide by the symmetry factor of the diagram.. 3.4.3. Vertex Functions. The two point function (that was calculated above to order 2) can be diagramatically written to all orders as. 35.

(37) Since external propagators are common to all graphs and are not relevant for calculating quantum mechanical amplitudes (Sαβ ), we shall multiply both external lines in the above graphs by inverse propagators to obtain truncated graphs, graphs where external legs have been “amputated”. The amputated legs will be denoted by dotted lines. In the above equation there are two types of graphs. The first type of graph is called ”one particle irreducible” or 1PI, these graphs cannot be written as products of other graphs. For example. are 1PI while. are not. These last two graphs belong to the second type, the ones that are not 1PI. P If we define 1i (p) to be the sum of all 1PI graphs in the 2-point function to all orders, then (2) GC can be written as (2) GC. P P P (p) (p) (p) 1 P = G0 (p)+G0 (p) G0 (p)+G0 (p) G0 (p) G0 (p)+... = 2 (3.74) 2 i i i p − m − (p). where G0 (p) =. i p2 −m2. is the free propagator.. Let us define the 2-point vertex function Γ(2) (p) by (2). GC Γ(2) (p) = i 36. (3.75).

(38) so that Γ(2) (p) = p2 − m2 −. X. (p). Vertex functions can be analogously defined for any n-point function.. 37. (3.76).

(39) Chapter 4 Renormalization Our proceedings in this chapter will follow [5] in sections 1 and 2. The first part of the last section is based on the work of L. Ryder in [5] and of V. Rivasseau in [9], the second part is based on [9] and the on the work of Itzykson and Zuber in [6].. 4.1. Divergencies in φ4. R 1 From (3.32) we can see that ∆F (0) ∝ d4 k k2 −m 2 diverges quadratically as k → ∞. This is the first divergence that comes up when we attempt to compute the Feynman integrals for the 2-point function to first order and higher. The mathematical nature of the problem lies in the fact that in perturbative computations distributions are multiplied without care resulting in ill defined quantities. More complicated divergencies eventually come up along the expansion, so it is useful to establish some kind of criteria that will tell us at first glance if a certain diagram diverges and how it does so. This criteria is called the superficial degree of divergence (D) and can be obtained from the Feynman rules by counting the power to which the momentum is raised in both numerator and denominator of the corresponding Feynman integral [5]. The resulting expression is D = dL − 2I. (4.1). where d is the dimension of the space, L is the number of loops and I is the number of internal lines in a diagram. Being n the number of internal vertices and E the number of external lines, one can also verify that L = I − n + 1 and 4n = E + 2I so that d D =d− − 1 E + n(d − 4) (4.2) 2 Note that for d=4, the dimension of Minkowski space-time, the superficial degree of divergence for φ4 is independent of n: D = 4 − E.. 38. (4.3).

(40) In fact, it only depends on the number of external lines so that the diagrams contributing to the same Green function to all orders are all divergent to the same degree. If by redefining a finite number of physical quantities that appear in the lagrangian all divergent diagrams in a theory can be made finite, the theory is called renormalizable. In non-renormalizable theories, D actually grows with the order of perturbation which means that every Green function is divergent to a high enough order. The fact that such theories are called non-renormalizable does not mean that they cannot be made finite. It means that if renormalization where to be performed, the renormalized integral would depend on an infinite number of parameters without any established relationships between them. Renormalizable theories are those in which only a finite number of functions lead to divergencies, whereas in super-renormalizable theories D decreases with the order of perturbation so that only a finite number of diagrams diverge.. 4.2. Dimensional Regularization. Regularization is the process used to isolate the divergencies in Feynman integrals. It is usually performed before renormalization in order to identify where the divergence happens and make the renormalization process more explicit. There are various types of regularization techniques, but here we will only use one called dimensional regularization [5]. In dimensional regularization, Feynman integrals are computed for a space of arbitrary dimension d (where d is off course an integer). U.V divergencies appear as poles at certain values of d. After a result is obtained for a space of dimension d, d can be made 4 (or whatever the desired value is). First of all, we must generalize the four dimensional lagrangian (3.66) to a d-dimensional space. Doing a bit of dimensional analysis on L, we can see that the coupling constant g is dimensionless (in 4D) and that in order for it to remain so for d 6= 4, we must multiply it by µ4−d where µ is an arbitrary mass parameter: g → µ4−d g Let us take as an example fig. 4.1. Figure 4.1. 39. (4.4).

(41) The expression corresponding to this diagram is now Z dd p 1 1 4−d gµ d 2 2 (2π) p − m2. (4.5). which can be computed using integral tables, to obtain 2−d/2 −igm2 4πµ2 Γ(1 − d/2) 32π 2 −m2. (4.6). Knowing that the gamma function has poles for negative integer arguments, we can see that d = 4 gives a simple pole for this integral. We shall now expand Γ(1 − d/2) using 1 1 (−1)n 1 + 1 + + ... + − γ + O(ε) (4.7) Γ(−n + ε) = n! ε 2 n where γ = 0, 577 is the Euler-Mascheroni constant. Setting ǫ = 4 − d (or d = 4 − ǫ) gives −2 − 1 + γ + O(ǫ) (4.8) ǫ 2−d/2 ǫ/2 4πµ2 4πµ2 ε = −m2 in (4.6) and obtain Finally we use a = 1 + ε ln a + ... to expand −m2 Γ(1 − d/2) = Γ(−1 + ǫ/2) =. 1 4−d gµ 2. Z. igm2 igm2 dd p 4πµ2 1 1 + γ + ln + O(ǫ) = + (2π)d p2 − m2 16π 2 ǫ 32π 2 −m2 igm2 + f inite = 16π 2 ǫ. (4.9). The first term in the above equation obviously goes to infinity as d approaches 4. Let us now consider the following diagram. Figure 4.2. It obviously belongs to the second order contribution to the 4-point function and its corresponding d-D integral is Z dd p 1 1 1 2 2 4−d g (µ ) (4.10) 2 (2π)d p2 − m2 (p − q)2 − m2 A similar procedure to that done with (4.5) can be performed here. The regularized expression for (4.10) is then 40.

(42) ig 2 µǫ + finite 16π 2 ǫ As before, the first term in the above expression diverges as d aproaches 4.. (4.11). We will now see what happens to the diagrams of figures 4.1 and 4.2 as parts of the 2-point and 4-point vertex functions respectively. Note that to first order, equation (4.9) is vertex function to first order is. P i. so that, ignoring the finite parts of. Γ (p) = p − m 1 − (2). 2. 2. g 16π 2 ǫ. . P , the 2-point (4.12). Now for the 4-point vertex function, note that additional to fig. 4.2 that represents the contribution from the s channel to the second order term, there will also be contributions from the u and t channels, whose corresponding diagrams are. respectively [5]. The expresions for u and t channel contributions are essentially the same as that for the s. The only difference is that whenever s = (p1 + p2 )2 appears in (4.10), u = (p1 + p4 )2 and t = (p1 + p3 )2 will appear instead. The 4-point vertex function to second order would then look somehting like this:. Note that the term corresponding to the first graph is Z ǫ ∆F (x1 − z)∆F (x2 − z)∆F (x3 − z)∆F (x4 − z)dz −igµ. (4.13). (which was its contribution in the original Green function) divided by the 4 inverse propagators. The resultant contribution to the vertex function is then −igµǫ .. 41.

(43) Since the pi ’s don’t appear in the divergent term of (4.10), the divergent parts of the last 3 graphs in the 4-point vertex function are exactly the same so that g ig 2 µǫ ǫ (4) ǫ + finite (4.14) + finite = −igµ 1 + 3 Γ (pi ) = −igµ + 3 16π 2 ǫ 16π 2 ǫ Note that the regularized expressions obtained here for Γ(2) and Γ(4) are not to the same order of g, but they do have the same number of loops. It can be easily shown that if instead of expanding in powers of g we expand in L (the number of loops), this expansion is really one in powers of ~ around the classical theory, which has more physical relevance than the first.. 4.3. Renormalization of φ4. In this section we will approach renormalization by two different techniques. The first will be only used initially in order to make the second one (the more formal one that we shall use until the end) easier to follow. To the 1-loop aproximation Γ(2) and Γ(4) should obviously be finite quantities. Hence, we will write the 2-point vertex function as Γ(2) (p) = p21 − m21. (4.15). where m1 is a new finite parameter representing the physical mass, and depends on the original mass parameter m found in the lagrangian: g 2 2 m1 = m 1 − (4.16) 16π 2 ǫ The original mass m is then taken to be infinite and to have no direct physical meaning, it’s m2 g only function is to make the term 16π 2 ǫ a finite quantity. Having defined the new mass m1 we say that we have imposed a renormalization condition on the vertex function that reads (2). ΓR (m1 ) = 0. (4.17). or (2). ΓR (0) = −m21. (4.18). The renormalization condition for the 4-point vertex function is (4). ΓR (0, 0, 0, 0) = g1 (2). (4.19). where, as in ΓR , g1 is a new parameter defined by this condition that represents the physical coupling constant that depends on the original parameter g found in the lagrangian. This “original” g is infinite and has no direct physical meaning.. 42.

(44) 4.3.1. Counterterms. The more formal method for renormalizing is called the method of counterterms. Counterterms are terms added to the original lagrangian in order to make Feynman integrals finite. For igm2 example, since the contribution from the diagram in figure 4.1 is 16π 2 ǫ + f inite, then the counterterm that must be added to the lagrangian will be of the form −gm2 2 −δm2 φ2 δL1 = φ ≡ (4.20) 32π 2 ǫ 2 This counterterm belongs to the φ2 term in the lagrangian, so in a way the fact that φ is a self-interacting field modifies the value of the physical mass m. What we are actually doing is incluiding a term to the interaction lagrangian that we had not taken into account before and that must be added if an interaction is present. The use of this counterterm gives the new Feynman rule. =⇒ Figure 4.3. −igm2 16π 2ǫ. meaning that the divergence has been cured. The counterterm that must be added in order to make Γ(4) (pi ) finite is δL2 = −. 1 −ig 2 µǫ 4 −Bgµǫ 4 φ = φ 4! 16π 2 ǫ 4!. (4.21). giving the Feynman rule. −ig 2 µǫ =⇒ 16π 2ǫ Figure 4.4. The lagrangian to which no counterterms have been added and that leads to infinite Feynman integrals is called the “bare” lagrangian, and the parameters herein are called bare parameters because they are the parameters when no interaction has been taken into account. Using the Feynman rules, we can write the amplitude for any diagram G we wish. The general expression is AG (x1 , ..., xN ) =. Z Y n i=1. 43. dzi. Y. I∈G. ∆F (zI − xI ). (4.22).

(45) or in momentum space AG (p1 , ..., pN ) =. X d d pI Y δ εv,l pl 2 2 p − m I l I∈G. Z Y. (4.23). where εv,l is -1 if p is coming out of the vertex v and +1 if it is coming into v. Let’s go back to the example of (4.2) and call this diagram G. We can use (4.23) to see that Z 1 AG = d4 p (4.24) 2 2 (p − m )((p − q)2 − m2 ) as we found in (4.10). This integral is logarithmically divergent as p → ∞. But if instead of this we write Z 1 1 q2 4 d p ∽ = − d4 p (p2 − m2 )((p − q)2 − m2 ) (p2 − m2 )2 (p2 − m2 )2 ((p − q)2 − m2 ) (4.25) we obtain a convergent integral. This might seem a little out of the blue at first, but we will later see that this mysterious term we have added is exactly the counterterm we seek. AR G. Z . It is important to keep in mind that: i) when a counterterm is added to L it renormalizes G every time it appears in the expansion. ii) local counterterms of φ2 , φ4 or (∂φ)2 can always be absorbed by parameters of L. iii) since counterterms only affect bare parameters which are not physically observable, they do not make the process physically inconsistent. iv) When a local counterterm is defined for a diagram G with E external legs, the modified lagrangian gives a new vertex with E external lines.. 4.3.2. Bogoliubov’s Recursion Formula and Zimmermann’s Solution. As we saw in the last subsection, every time that we define a counterterm for a given graph we are shrinking the graph into a single vertex. For example. 44.

(46) Figure 4.5. Note that the renormalization of the subdiagram G1 is nothing more than what we did for Γ(4) (pi ) in (4.21). So for every G1 that is a subgraph of G this corresponds to doing the following sum AG1 + cG1 AG/G1. (4.26). where AG/G1 is the resultant (reduced) graph after reducing G1 to a vertex [9]. By using (4.26), if a graph G has more than one subgraph, we can only define the reduced graphs for families S of disjoint graphs γ’s, that is for subgraphs that don’t overlap with one another. The definition of counterterms is inductive, starting with the smaller 1PI graphs and going outward towards the bigger ones. This is consistent with the workings of perturbation theory. Assuming that we have already introduced all counterterms to a given order l, then the renormalized amplitude for a graph G of order l + 1 is X Y R AG/S AG = cγ + cG (4.27) S. γ∈S. where cG is the counterterm attached to G itself and the sum is over all families S of disjoint primitively divergent subgraphs. This recursive pattern was first found by Bogoliubov. Hepp was the first to prove that Bogoliubov’s recursion actually leads to finite amplitudes and the first explicit solution was given by Zimmermann. In the BPHZ formalism, where the renormalization conditions are taken at a point of zero momentum as we did for Γ(2) and Γ(4) , the contribution of a counterterm to the renormalized integrals is −T D IG. (4.28). dk(1 − T D )IG (p, k). (4.29). and so AR G (p). =. Z. where T D is the taylor operator that expands the argument into its Taylor series and cuts at order D (the superficial degree of divergence for the graph), and IG is the integrand of the 45.

(47) bare amplitude of the graph G; note that the term we mysteriously added to (4.24) in (4.25) is exactly this! Such definition of counterterms actually makes a lot of sense if we look at it in terms of D: since the superficial degree of divergence is already telling us how a given integral diverges, we need to substract a term that cures this kind of divergence. And what simpler choice than T D IG can we possibly find? A sketch of Zimmermann’s solution to Bogoliubov’s recursion is as follows. Let’s say that we wish to find the amplitude of a graph G. Let’s also say that γ is a subgraph of G with D(γ) ≥ 0, IG is the integrand of G, R̄G is the integrand of G where all counterterms of lower order have already been included, and RG is the integrand of G that leads to a finite integral. As we saw above RG = (1 − T D(G) )R̄G. (4.30). For γ ⊂ G, the contribution of cγ is (−Tγ R̄γ ) so inserting this in the place of γ we must sum the term IG/γ (−Tγ R̄γ ). (4.31). IG/{γ1 ,γ2 } (−Tγ1 R̄γ1 )(−Tγ2 R̄γ2 ). (4.32). For γ1 , γ2 ⊂ G we sum the term. so that in general R̄G = IG +. X. {γ1 ,...,γs },γi ∩γj =∅. s Y IG/{γ1 ,...,γs} (−Tγi R̄γi ). (4.33). i=1. Before we go on, we shall introduce the term of ”forest” that will be of most importance from now on. Definition A forest F is a subset of subgraphs such that for any pair of subgraphs γ1 , γ2 ∈ F one of the following conditions applies: γ1 ⊂ γ2 γ2 ⊂ γ1 γ1 ∩ γ2 = ∅. (4.34). Since R̄γi =. X Y. Fγ1 γ ′ ∈Fγi. then by inserting this into (4.33) we obtain 46. (−Tγ ′ Iγi ). (4.35).

(48) R̄G = IG +. X. {γ1 ,...,γs},γi ∩γj =∅. X Y s Y IG/{γ1 ,...,γs} (−Tγi ) (−Tγ ′ Iγi ) i=1. (4.36). Fγ1 γ ′ ∈Fγi. This equation gives a recursive solution for renormalizating any graph. Even though it was found by assuming divergent subgraphs in a graph G did not overlap with each other, it has been shown [10] that this formula actually works even if they do. As an example to illustrate the procedure, let’s look at fig. 4.6:. Figure 4.6. It has 12 possible forests, namely. Figure 4.7. Note that the sum over disjoint graphs in (4.36) generates each forest only once. The sum in square brackets must be taken over all possible forests associated to the family of disjoint graphs taken in the first sum and the product must be taken over all elements of each forest, starting with the smaller graphs towards the bigger ones. If G is the graph from fig. 4.6, the first sum is taken over each individual subgraph because all subgraphs of G overlapp. Starting with γ1 we obtain the first forest, γ2 generates the second forest, γ3 generates the fourth and sixth forests, γ4 generates the fifth and seventh forests, and finally γ5 generates the third, eight, ninth, tenth, eleventh and twelveth forests. When implementing this formula it is important to know that if the graph G is by itself superficially divergent then the sum over all forests of subgraphs of G must be restricted to forests 47.

(49) where G does not appear [6].. 48.

(50) Chapter 5 The Hopf Algebra Structure of Renormalization in Perturbative Theories As we mentioned earlier, the link between renormalization and Hopf Algebras lies in Bogoliubov’s recursion formula. Following the work first of Kreimer in [12] and then of Connes and Kreimer in [13] we will first define all pertinent algebraic structures, and then we will show the main results of these authors’ work, through examples. The mentioned results are: i) that the coproduct generates the terms that are needed to cancel subdivergencies, ii) that the antipode S[R[X]] is exactly the counterterm needed to make the integral of the diagram X finite, and iii) that equation (2.16) which defines the antipode, acting on graph X, gives the complete renormalized expression for the amplitude of X. In order to define this Hopf Algebra, we will consider φ3 theory in d = 6 [11]. As φ4 in d = 4, this theory is also renormalizable and works pretty much the same except that its interaction 3 lagrangian is Lint = −g φ3 and so its internal vertices are where 3 instead of 4 lines meet. 3!. 5.1. The Hopf Algebra of Parenthesized Words. The first formulation of the Hopf Algebra structure of renormalization was done on the algebra generated by Feynman graphs written as parenthesized words [12]. This formulation is equivalent to that of defining the Hopf Algebra of Feynman graphs as rooted trees [14]. Parenthesized words (PWs) are chains of letters from an alphabet xi , each surroundend by parenthesis in a special way. Here are some facts of PWs and PWs of Feynman graphs: - Each letter in the PW represents a Feynman graph without any subdivergencies, although the diagram is by itself superficially divergent.. 49.

(51) - Letters can have one and only one closing parenthesis at its right hand side but they can have more than one opening parenthesis at its left hand side. If the first opening parenthesis of a PW is closed by the last closing parenthesis, the PW is called irreducible (iPW), if not, it is called reducible. - The number of letters in a PW is called the lenght of the PW. - The maximum number of nested divergencies in an iPW is called the PW’s depht. We will now see how to write a Feynman graph as a PW.. Figure 5.1. The graph above is made up of two subgraphs, namely γ3 and γ4 . γ4 is at the same time made up of two divergent subgraphs γ1 and γ2 which are nested in γ0 . Using the alphabet of the xi s we will let x1 represent γ1 , x2 represent γ2 , x3 represent γ0 (obtained by reducing γ1,2 to a point), x4 represent γ3 , and x5 represent γ0 (obtained by reducing each subdivergence of Γ to a point). The total graph can be written as Γ = (((x1 )(x2 )x3 )(x4 )x5 ). (5.1). This is a PW of length 5 and depth 3. Following the notation used in [12] we will denote words by uppercase letters X and Y and individual letters by xi s. The expression (Xxi ) simply means that if the word X is, say ((xa )xb ) then (Xxi ) = (((xa )xb )xi ).. 50.