---------------------------------------- Time evolution in quantum field theories ---------------------------------------- In an introductory textbook treatment of relativistic quantum field theory (QFT) one only finds a discussion of how to compute scattering information, since this is the most elementary and important output for studying the subnuclear world. Since scattering only concerns how things in the infinite past turn into things in the infinite future, it looks as if QFT had nothing to say about the finite-time evolution of quantum fields. But this is an illusion. For example, the QED field equations are used in http://arxiv.org/pdf/1007.1099 for the derivation of a quantum kinetic equation relating (in an appropriate approximation involving expectation values) the electromagnetic field and the Wigner matrix of the electron field. Quantum kinetic equations are the basis for more phenomenological work on semiconductor equations, hydrodynamic equations, etc.. Quantum field theory is the theory of computing and interpreting expectations of products of the basic fields at different space-time arguments. In traditional renormalized perturbation theory, this is done by means of representing these expectations as infinite series of terms representable as weighted sums over multi-momentum integrals, commonly expressed through various Feynman diagrams. The terms in this series are given a well-defined mathematical sense by a renormalization process consisting in a carefully taken limit of the appropriate sums of integrals. This looks untidy in most textbook treatments of renormalization (for a careful treatment, see the books by Salmhofer or by Scharf), where there is talk about subtracting infinities, which gives the whole procedure an air of arbitrariness. But renormalization is nothing but a more complex version of the elementary stuff we learn early in our science education about how to subtract infinities arising when setting x=1 in an expression such as x/(1-x) - x^2/(1-x): We simply simplify the expression to (x-x^2)/(1-x)=x before taking the limit x->1, and get the perfectly well-defined value 1. For simplicity, let me restrict to the case of a single massive Hermitian scalar field Phi(x) only - everything extends without difficulties to arbitrarily massive fields of arbitrary spin, and most of it applies also to massless fields (which may have additional infrared complications, though). Writing Phi(f)=integral dx f(x) Phi(x) for arbitrary test functions f (x) (from Schwartz space), one can define the multilinear Wightman functionals W(f_1,...,f_N)=, where the expectation is with respect to the vacuum state of the theory. W(f_1,...,f_N) can be written as a formal integral over Wightman distributions W(x_1,...,x_n), which are the limiting cases when the f_k tend to delta distributions centered at x_k. The formal properties of the Wightman functions of massive fields are expressed by the so-called Wightman axioms, http://en.wikipedia.org/wiki/ The Wightman axioms can be derived at the level of rigor conventional in theoretical physics for all renormalizable quantum field theories with a Poincare-invariant action. (In dimensions d<4, there are also mathematically fully rigorous constructions of interacting quantum field theories derived from a large class of Poincare-invariant actions; this branch of mathematical physics is called constructive field theory. But in the most important dimension d=4, there are technical obstacles that haven't been overcome so far in full rigor. Therefore, in this post, I shall argue only on the level of rigor as defined, e.g., by Weinberg's QFT treatise.) For an important step in this derivation, see the posts #125-#140 of the thread http://www.physicsforums.com/showthread.php?t=388556 . From the Wightman functions, one can construct their time-ordered version and from these the usual S-matrix elements. However, one can do much more! Given Wightman distributions satisfying the Wightman axioms, it is not difficult to construct the physical Hilbert space. It consists of all limits of linear combinations of terms |f_1,...,f_N> with inner product :=. It is an instructive exercise to show that, given the properties of the Wightman distributions, this indeed defines a Euclidean space whose closure is the required Hilbert space. This Hilbert space carries a unitary representation of the Poincare group, in which the group element U acts as U|f_1,...,f_N>:=|Uf_1,...,Uf_N>, where Uf is the action of the Poincare group on the single particle space. The time translations form a 1-parameter group whose infinitesimal generator H is (by standard functional analysis) a densely defined, self-adjoint linear operator. In particular, in a covariant position representation of the single p article space, the dynamics is very simple and explicit: The 2-particle wave function |psi(x_1,x_2)> at time t=0 evolves to |psi(x_1+tu,x_2+tu)> at time t (where u=(1,0,0,0^T). The only nontrivial thing is the inner product, since it requires the knowledge of the Wightman functions. These are known explicitly only for generalized free fields and for a number of explicitly solvable models in 1+1 dimensions. The Wightman axioms guarantee that the spectrum of H is nonnegative, and that |> (the case N=0 of |f_1,...,f_N>) is the unique pure state annihilated by H. This is the physical vacuum state. Thus everything required by standard quantum mechanics is in place - except that the massless case needs extra considerations (which figure under the heading ''infraparticles''). The physical 1-particle states are the states |f_1>. Product states such as |f_1,f_2> can be viewed as the interacting analogue of 2-particle states (which they are in the free case). However, in a fixed reference frame (of the center of mass of the scattering system, where a Hamiltonian picture makes sense), these 2-particle states are no longer composed of exactly two in/out particles, giving rise to nontrivial scattering. In any theory that does not conserve particle number, pure 2-particle systems can exist only for a fleeting moment, since the Schroedinger dynamics immediately generates admixtures of N-particle states for other N. Pure 2-particle states arising in practice are always asymptotic states. Here the particles are inside two beams with fairly definite momentum and transversal position, which form a commuting set of operators in 1-particle space, hence can be prepared simultaneously. While they are far away from each other, one can treat them as states in the asymptotic in-space of scattering theory, i.e., at time t=-inf. (This space is constructed in the Wightman framework by means of Haag-Ruelle theory.) But once they come close enough that the beams begin to overlap, the particle view becomes meaningless, and in the collision region of the beams the quantum field behaves more like a fluid described by hydrodynamic or kinetic equations than like particles. At the current state of knowledge, it is unknown how to make the derivation sketched above fully rigorous when the dimension is 1+3. However, on the level of rigor commonly used in theoretical physics, things are fully adequate. In practice, the closed time path (CTP = Schwinger-Keldysh) formalism is the most common way to construct the Wightman functions and their time-ordered version in a nicely arranged way that makes it comparatively simple to derive quantum kinetic equations. See, e.g., http://arxiv.org/pdf/hep-th/9504073 for an introduction, Phys. Rev. D 37, 2878-2900 (1988) for a derivation of the Boltzmann equation.