------------------------------------------- Relativistic invariance and Wightman axioms ------------------------------------------- The Wightman axioms are generally viewed to be the correct axiomatic basis for quantum field theory, at least for the case where there are no massless asymptotic states and the ground state is Lorentz invariant (representing the vacuum). The necessity for the Wightman axioms stems from the belief in fundamental physical principles - relativity, causality, the existence of fields and a vacuum, and a separable Hilbert space accommodating all these. These together make the Wightman axioms essentially (namley in the bsence of gauge invariance, which complicates the picture) unescapable. A transformation law that does not satisfy the commutation rules of the Poincare algebra has no representation of the Poincare group in which H=P_0 (the interacting Hamiltonian) generates the physical time translations. The theory is relativistic if the physical creation operators (that create physical particles from the vacuum and satisfy causal commutation relations) generate n-point vacuum expectation values that are Poincare covariant. But this is just what the Wightman axioms require. In the following, I refer to Vol. I of Weinberg's excellent book on quantum field theory. Weinberg proves relativistic invariance in a heuristic fashion in Section 3.3 (there for the S-matrix, which, in view of the the LSZ-formula p.430 proves it for the time-ordered expectation values, which is only little weaker than the Wightman axioms. Using closed-time-path integrals, one can extend the argument to contour-ordered expectation values, which include the Wightman functions. Of course, this ''proof'' is only in perturbation theory, and not a mathematical proof but only one according to the usual standards of theoretical physics. Note that the transformations are more fundamental and must be present in any relativistic quantum theory, whether with or without fields. (Indeed, as long as one works in the Schroedinger representation, one can completely dispense with the fields; but they are nevertheless there, as shown by my construction below.) Later he simply take this for granted and specializes it to quantum fields. This specialization is done first tentatively in Section 3.5 (see the middle of p.144), and further justified in Chapter 4 (see p.169); later it is assumed without further ado. Note that the first few chapters are in the Schroedinger picture. The translation to the Heisenberg picture is as follows: For an arbitrary observable A_0 in the Schroedinger picture, the corresponding quantum field A(x) satisfies A(x) = U(x) A_0 U(-x) ... (1) A(Lambda x) = U(Lambda) A(x) U(Lambda^{-1}) ... (2) where the translations U(x) and the Lorentz transforms U(Lambda) are the physical (interacting) ones. This transformation law appears for the special case of the interaction part of the energy density field in (3.5.11) - with the free representation U_0 in place of U, but is a general property of the Heisenberg picture. (Proof: Take (1) as the definition of the field, and deduce (2) from (1) and the properties of arbitrary unitary representations of the Poincare group. The proof doesn't depend on whether the representation is given in the instant form or any other form.) On p. 144f, H_{curly}(x,t), the interaction part of the energy density is a function of free (asymptotoc) fields. Later chapters specialize the expression for H_{curly}(x,t) to those corresponding to Lagrangian field theories, expressing it in terms of the corresponding free field operators. Section 3.5 discusses the needed properties of H_{curly}(x,t) for creating a good interacting representation of the Poincare group, resulting in the requirement of causal commutation rules (with caveats in the footnote for contact terms; cf. p. 277ff). This is the reason why Chapter 5 bothers to construct free fields, since it is with their help that this condition can be satisfied if the interaction is represented as a sum of integrals of local products of free fields. Ignoring infrared issues in case of massless particles (where the Wightman axioms seem to be defective), the situation is in a bit more detail the following: In order to be able to talk about the S-matrix, one needs to have asymptotic 1-particle states whose tensor product describes the possible input to a scattering event. Clearly, we can prepare independently beams of any kind of free physical particles (elementary or bound states) in the theory and bring them to a collision. I'll call these particles asymptotic particles. For example, in QED, we can prepare photons, electrons, and positrons, which are the only asymptotic particles of the theory. In QCD, we can prepare mesons and baryons, but not quarks or gluons as - due to confinement - the latter are not asymptotic particles. (However, because they are gauge theories, QED and QCD do not quite satisfy the Wightman axioms.) Weinberg now assumes (on p.110) that the unperturbed Hamiltonian describes the free motion of all these asymptotic particles, with their observable quantum numbers (mass, spin, charges). Asymptotic in-states are therefore elements of a Fock space generated from the asymptotic vacuum by means of free creation operators, one for each asymptotic particle species. These creation operators define the free quantum fields introduced on p.144 and used in the remainder of the chapter and in Chapter 4. According to (3.1.8), the interaction is defined as the difference of the actual Hamiltonian and this free Hamiltonian. There is, however, a difficulty that Weinberg does not directly discuss in the book: The asymptotic particles need not correspond one to one to the bare particles in which the Hamiltonian is derived from an action. This is most obvious in case of QCD, where the action involves quarks and gluons only, while the asymptotic particles are mesons and baryons. The assumptions break down, and perturbation theory is meaningless - a nonperturbative approach is called for, about which Weinberg is silent in Volume 1. He only says that the bound state problem is poorly solved in QFT (p.560), though with some trickery he is able to consider bound states for QED in an external field (needed to get the Lamb shift). This breakdown of perturbation theory is the formal reason why low energy predictions from QCD are very hard - it is part of the unsolved confinement problem of QCD. (The derivation of effective actions for mesons on baryons from QCD is still a web of guesswork, with few hard results and much input of phenomenology in addition to intuition derived from QCD proper.) Even in case of QED (and all other field theories without bound states), the problem remains that the masses of the asymptotic particles don't match the corresponding coefficients of the action from which the Hamiltonian is derived (the so-called bare masses and charges) - rather they are complicated functions of these, determined only as part of the solution process. The simplest instance of this is the anharmonic oscillator, which can be viewed as a 1+0-dimensional quantum field theory. Here the mass corresponds to the difference between the first two eigenvalues, and this difference changes as a function of the interaction strength. This is the origin of the need for renormalization. Renormalization is a technique for parameterizing the bare parameters as a function of the observable parameters (or parameters related to these in a fairly insensitive fashion). For my views on this, see my paper A. Neumaier Renormalization without infinities - an elementary tutorial http://www.mat.univie.ac.at/~neum/ms/ren.pdf An additional problem in QFTs of dimension 1+d (d>0) is that perturbation theory is infinitely sensitive to changes in the bare parameters, leading to divergent integrals in second-order perturbation theory. Fortunately, renormalization cures this defect automatically, at the cost of making the bare parameters tend to infinity in a particular, fairly well-understood fashion. This was the breakthrough that earned Feynman, Tomonaga and Schwinger the Nobel prize. But the computations become quite technical.... Returning to Weinberg, it is fortunate that (because of the LSZ formula) the formal S-matrix contains essentially the same information as more rigorous approaches that work with the Wightman axioms. Therefore his derivations in Chapter 3 and 4 remain plausible (though not at the level of a mathematical proof) even in the face of the above difficulties. The main insight from Chapter 3.5 is the need for the causal commutation rules for the interaction density to get Lorentz invariance (which is not dependent on a particular representation of it in terms of the asymptotic Fock space), and from Chapter 4 hints for the particular structure of the interaction from the cluster decomposition principle. The result is that one should represent the interaction as a Lorentz invariant scalar in terms of integrals over products of local field operators satisfying causal commutation relations and carrying an irreducible representation of the Poincare group. Chapter 5 describes the possibilities for the free part. Interacting fields are introduced only in Chapter 7. Section 7.1 discusses the standard Hamiltonian approach in the instant form and the Schroedinger picture, and introduces in (7.1.28/29) the interacting field operators in the Heisenberg picture. Since in the instant form, space translations are implemented kinematically, these equations imply that A(x) = U(x) A_0 U(-x) (1) for all Operators A_0=F(Q,P), where - unlike in (3.5.12) - the translations U(x) are the physical (interacting) ones. Moreover, (7,1,27) defines the form of the free Lagrangian in terms of the physical parameters. As in the Hamiltonian case discussed in Chapter 3, the interaction is defined as the difference V=L_0-L where L is the full action (with bare parameters). The fact that bare and physical parameters are generally different leads to the observation that the so defined interaction automatically has counterterms (for QED, this is done on p.473). Section 7.2 then reviews the construction of a Hamiltonian from the Lagrangian. Sections 7.3 and 7.4 verify that there is a unitary representation of the Poincare group in which P_0 is the interacting Hamiltonian defined in Section 7.2. The most important commutation relations (those needed to derive the Lorentz invariance of the S-matrix in Section 3.3) are verified on p.p. 316-317. To see that the interacting quantum fields are Poincare-covariant with respect to the interacting representation of the Poincare group, we need to show that A(Lambda x) = U(Lambda) A(x) U(Lambda^{-1}). (2) This follows from (1) and the fact that the translations are part of an (interacting) unitary representation of the Poincare group, provided that A_0 is Lorentz-invariant, which is the case, e.g., for A_0:=Q(0) and local, derivative-free interactions: Expand both sides using (1) and simplify the result using the rules of the group representation and the fact that A_0 is Lorentz invariant. To prove that Q(0) is Lorentz invariant for Phi^4 theory, one needs to show that the Lorentz generators commute with Q(0). This can be seen directly for the free representation of the Lorentz group, using the definition (5.2.11) of the field in terms of the creation and annihilation operators, and their transformation law (5.1.11). For the interacting repesentation, the boost generators differ from that of the free case by an additional term W defined in (3.5.17). Since H(x,0) for a real scalar field is, according to (7.1.35), a polynomial in Q(x)=Phi(x,0), namely the normally ordered product g:Phi(x,0)^4: . Thus it suffices to verify that [Phi(x,0),Phi(0,0)]=0, which is straightforward given (5.2.11). To verify the causal commutation rules for the interacting field, [Phi(x),Phi(y)]=0 if x and y are spacelike separated, (3) we note that there is a Poincare transformation that, because of the space-like assumption, moves x to 0 and y to (r,0) for some r. Indeed, a translation moves x to 0, and y to y-x. Now (see Weinberg's table on p.66) that the restricted Lorentz groups has six orbits on R^4: the open future cone, the open past cone, the future light cone, the past light cone, the complement of the closed, 2-sided causal cone, and the zero point. In particular, y-x can be mapped by a Lorentz transformation to any point outside the causal cone, and hence to a point of the form (r,0). Therefore (3) follows from the transformation properties of the fields and the the causal commutation rules for the free field at t=0, where it coincides with the interacting field. At various points in the developments of Chapter 3 and 7, Weinberg points out problems due to singularities at equal times, which may complicate matters (but not in Phi^4 theory or QED). These must be resolved on the basis of more detailed investigations involving the cohomology of the representations, and lead (sometimes) to anomalies, a quite advanced subject that doesn't alter the basic correctness of his analysis (on the level of rigor customary for theoretical physics) and the importance of his conclusions. Above, we verified on the formal level the Wightman axioms relating to relativity and causality. The other Wightman axioms - the spectral boundedness assumptions and the existence of the vacuum - are necessary in order to be able to interpret the theory. (Among others, this excludes the covariant field theory of Horwitz and Piron.) They cannot be proved that easily, not even on the formal level, since these properties emerge only in the renormalized limit. So a lot of technicalities need to be considered.