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Relativistic invariance and Wightman axioms
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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.