The linearity (and thus the fields superposition) are violated for the
fields evolution in Barut's self-field approach (these are non-linear
integro-differential equations of Hartree type). One can approximate
these via piecewise-linear evolution of the QED formalism.
It was indeed this discussion that made me decide you didn't make the
distinction between the linearity of the dynamics and the linearity of
the operators over the quantum state space.
Note that Barut has demonstrated this linearization explicitly for finite
number of QM particles. He first writes the action S of, say, interacting
Dirac fields F1(x) and F2(x) via EM field A(x). He eliminates A
(expressing it as integrals over currents), thus gets action
S(F1,F2,F1',F2') with current-current only interaction of F1 and F2.
Regular variation of S via dF1 and dF2 yields nonlinear Hartree equations
(similar to those that already Schrodinger tried in 1926, except that
Schr. used K-G fields F1 & F2). Barut now does an interesting ansatz. He
defines a function:
G(x1,x2)=F1(x1)*F2(x2) ... (1)
Then he shows that the action S(F1,F2..) can be rewritten as a function
of G(x1,x2) and G' with no leftover F1 and F2. Then he varies S(G,G')
action via dG, and the stationarity of S yields equations for G(x1,x2),
also non-linear. But unlike the non-linear equations for fermion fields
F1(x) and F2(x), the equations for G(x1,x2) become linear if he drops the
self-interaction terms. They are also precisely the equations of standard
2-fermion QM in configuration space (x1,x2), thus he obtains the real
reason for using the Hilbert space products for multi-particle QM
The price paid for the linearization is that the evolution of G(x1,x2)
contains non-physical solutions. Namely the variation in dG is weaker
than the independent variations in dF1 and dF2. Consequently, the
evolution of G(x1,x2) is less constrained by its dS=0, thus the G(x1,x2)
can take paths that the evolution of F1(x) and F2(x), under their dS=0
for independent dF1 and dF2, cannot.
In particular, the evolution of G(x1,x2) can produce states such
Ge(x1,x2)=F1a(x1)F2a(x2) + F1b(x1)F2b(x2), corresponding to the entangled
two particle states of QM. The mapping (1) cannot reconstruct any more
the exact solution F1(x) and F2(x) uniquely from this Ge, thus the
indeterminism and entanglement arise as the artifacts of the
linearization approximation, without adding any physical content which
was not already present in the equations for F1(x) and F2(x) ("quantum
computing" enthusiasts will likely not welcome this fact since the power
of qc is result of precisely the exponential explosion of paths to evolve
the qc "solutions" all in parallel; unfortunately almost all of these
"solutions" are non-solutions for the exact evolution).
All this is not amazing in fact. It only means that the true solution of
the classical coupled field problem gives different solutions than the
quantum theory of finite particle number. That's not surprising at all,
for the basic postulates are completely different: a quantum theory of a
finite number of particles has a totally different setup than a classical
field theory with non-linear interactions. If by coincidence, in certain
circumstances, both ressemble, doesn't mean much.
It also means that you cannot conclude anything about a quantum theory of
a finite number of particles by studying a classical field theory with
non-linear terms. They are simply two totally different theories.
The von Neumann's projection postulate is thus needed here as an ad hoc
fixup of the indeterministic evolution of F1(x) and F2(x) produced by the
approximation. It selects probabilistically one particular physical
solution (those that factorize Ge) of actual fields F1(x), F2(x) which
the linear evolution of Ge() cannot. The exact evolution equations for
F1(x) and F2(x), don't need such ad hoc fixups since they always produce
only the valid solutions (whenever one can solve them).
No, a quantum theory of a finite number of particles is just something
different. It cannot be described by a linear classical field theory,
nor by a non-linear classical field theory, except for the 1-particle
case, where it is equivalent to a linear classical field theory.
A quantum theory of a finite number of particles CAN however, be
described by a linear "field theory" in CONFIGURATION SPACE. That's
simply the wave function. So for 3 particles, we have an equivalent
linear field theory in 9 dimensions. That's Schroedinger's equation.
However, von Neumann's postulate is an integral part of quantum theory.
So if you have another theory that predicts other things, it is simply
that: another theory. You cannot conclude anything from that other
theory to talk about quantum theory.
The same results hold for any finite number of particles, each particle
adding 3 new dimensions to the configuration space and more
indeterminism. The infinite N cases (with the anti/-symmetrization
reduction of H(N), which Barut uses in the case of 'identical' particles
for F1(x) and F2(x), as well) are exactly the fermion and boson Fock
spaces of the QED. For all values of N, though, the QM description in 3N
dimensional configurations space (the product H^N, with
anti/symmetrization reductions) remains precisely the linearized
approximation (with the indeterminism, entanglement and projection price
paid) of the exact evolution equations, and absolutely nothing more. You
can check the couple of his references (links to ICTP preprints) on this
topic I posted few messages back.
It is in fact not amazing that the linear field theory in 3 dimensions is
equivalent to the "non-interacting" quantum theory... up to a point you
point out yourself: the existence, in quantum theory, of superpositions
of states, which disappears, obviously (I take your word for it), in the
non-linear field theory.
In quantum theory, their existence is EXPLICITLY POSTULATED, so this
already proves the difference between the two theories.
But all this is about "finite-number of particle" quantum theory, which
we also know, can only be non-relativistic.
Quantum field theory is the quantum theory of FIELDS. So this time, the
configuration space is the space of all possible field configurations,
and each configuration is a BASIS VECTOR in the Hilbert space of states.
This is a HUGE space, and it is in this HUGE SPACE that the superposition
principle holds, not in the configuration space of fields.
For ANY non-linear field equation, (such as Barut's, which simply sticks
to the classical equations at the basis of QED) you can set up such a
corresponding Hilbert space. If you leave the field equations linear,
this corresponds to the free field situation, and this corresponds to a
certain configuration space, and to it corresponds a quantum field
hilbert space called Fock space. If you now assume that the
*configuration space* for the non-linear field equations is the same (not
the solutions, of course), this Fock space will remain valid for the
interacting quantum field theory.
There is however, not necessary a 1-1 relation between the solutions of
the classical non-linear field equations, and the evolution equations in
the quantum theory, even if starting from the quantum state that
corresponds to a classical state to which the classical theory can be
applied.
Indeed, as an example: in the hydrogen atom, there is not necessary an
identity between the classically calculated Bohr orbits and the solutions
to the quantum hydrogen atom. But of course, there will be links, and
the Feynman path integral formulation makes this rather clear, as is well
explained in most QFT texts. Note that the quantum theory has always
MANY MORE solutions than the corresponding classical field theory,
because of the superposition principle.
However, all this is disgression, through I've been already through this
with you. At the end of the day, it is clear that classical (non-linear)
field theory, and its associated quantum field theory, ARE DIFFERENT
THEORIES.
The quantum field theory is a theory which has, by postulate, a LINEAR
behaviour in an INFINITELY MUCH BIGGER space than the non-linear
classical theory. It allows (that's the superposition principle) much
more states as physically distinct states, than the classical theory.
The non-linearity of the interacting classical field theory is taking
into account fully by the relationships between the LINEAR operators over
the Hilbert space.
In the case h->0, all the solutions of the non-linear field equations
correspond to solutions of the quantum field theory. However, the
quantum field theory has many MORE solutions, because of the
superposition principle.
Because of the hugely complicated problem (much more complicated than the
non-linear classical field equations) an approach is by Feynman diagrams.
But there are other techniques, such as lattice QFT.
QED is such a theory, and it is WITHIN that theory that I've been giving
my answers, which stand unchallenged (and given their simplicity it will
be hard to challenge them :-)
The linearity over state space (the superposition principle) together
with the correspondence between any measurement and a hermitean operator,
as set out by von Neumann, are an integral part of QED. So I'm allowed
to use these postulates to say things about predictions of QED.
We can have ANOTHER discussion over Barut's approach. But it is not THIS
discussion. This discussion is about you denying that standard QED
predicts anti-correlations in detector hits between two detectors, when
the incoming state is a 1-photon state.
I think I have demonstrated that this cannot be right.
cheers,
Patrick.