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Time evolution in quantum field theories
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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.