I have always been annoyed by the standard introduction to quantum field theory in which one imposes those commutation relations on classical fields. This always seemed very strange to me. What the heck are those classical fields that one quantizes? (no one has ever seen a classical approximation of a meson field or of a fermion field!)
As far as I know, only Weinberg presents things in a way that really makes sense, in which the starting point is not the quantization of classical fields but the need to allow the number of particles to vary, so the need to introduce annihilation and creation operators. Then , imposing Lorentz invariance, one is led to quantum fields obeying the usual commutation relations (postulated from the start in standard presentations).
Hi Patrick,
20 years ago I was introduced ,for the first time, to the classical field theories and almost immediately I asked my professor the following question:
"what do Dirac wavefunctions have in common with the classical fields of EM and gravity?"
My reason to ask about fermions and not about bosons was (I thought) obvious, because in principle (Pauli) only fermions have no classical limit.
Any way, His answer was:
1) it (Dirac spinor) is a
C-number, smooth function over spacetime M^4 .
2) it has a
definite transformation law with respect to Poincare' group.
3) a
scalar action can be constructed out of Dirac spinors.
4) (1)+(2) define a relativistic classical field, (a general classical field is defined to be a C-number map from spacetime to some target space):
\phi : M^{n} \rightarrow F
5) classical field theory is nothing but (1)+(2)+(3), i.e the pair (M^{n}, F) together with \delta S[\phi] = 0 .
Defined this way, even classical mechanics is a field theory on (1+0)-dimensional spacetime (affine time):
(t=M^{1}, \math R^{3} )
\delta S[x] = 0
The point is this; In the linear GR, the classical field is given by the symmetric 2nd rank
tensor representation of Poincare' group. We also use the
vector representation to describe the classical EM field, eventhough it (the vector potential) is
not an observable.
So, at least mathematically, why not include the other irreducible representations of Poincare' group, that is the
scalar and the
spinor representations, especially when the aim is formulating QFT using the quantization "law"
{A,B} \rightarrow i \left[\hat{A},\hat{B}\right]
So when you quantize the "fields" (X,P), you get a "QFT" on the affine time "spacetime". This is just QM. Quantizing the fields \left( \phi ,\pi \right) will give us QFT on M^{n} .
Notice that the above mathematical structure has no room for (the often misleading) 2nd quantization.
Let me summarise by saying that the canonical formalism consists of:
i) postulating a
global symmetry group
G.
ii) working out all irreducible representations of
G.
iii) constracting a
real,
G-scalar Lagrangian density out of these IRR's and their 1st order derivatives at the same point. And
iv) making this Lagrangian invariant under the
local G group. This gives rise to interactions.
Up to this point, everything is classical and the theory is CFT.
Now, bringing in Hilbert space together with { , } --> i[ , ] will automatically produce a
G-invariant as well as
Poincare'-invariant interactive QFT. So you almost have everything. you see, there are good reasons for the saying:
"
give me the Lagrangian, I give you everything"
Regarding what you called "the Weinberg's approach" :
a) I would like to remined you that Weinberg (in the first few chapters of his book) was following the so-called
constractive (axiomatic) QFT, also known as the
S-matrix formalism. I know 2 old but classic books on this subject:
BOGOLUBOV, LOGUNOV & TODOROV, "INTRODUCTION TO AXIOMATIC QUANTUM FIELD THEORY", 1969.
JOST, "THE GENERAL THEORY OF QUANTIZED FIELDS", 1965.
and one relatively new "good" book
JAN TOPUSZANSKI, "AN INTRODUCTION TO SYMMETRY AND SUPERSYMMETRY IN QUANTUM FIELD THEORY", 1991.
b) the current form of axiomatic QFT shows that we still can not go too far without a Lagrangian. In chapter 7, where Weinberg is forced to introduce the canonical formalism, he explains the main difficulty of the approach, namely, guessing the form of the Hamiltonian;
(read the last 2 paragraphs on page 292).
c) while the Lagrangian formalism is always possible, the constractive approach is absolutely hopeless in each of the cases where;
1) the pair (M^n , F) is equipped with
local (Lie or graded-Lie) algebraic structure; as in the
local (symmetric or supersymmetric) non-abelian theories.
2) both M^n and F are Riemannian manifolds; string theory and non-linear sigma model.
3) different combinations of (1) and (2) like superstring and supergravity.
4) M^n is non-commutative spacetime.
5) different combinations of (4) with the rest.
6) all the above with nontrivial topology on F; topological field theories.
regards
sam