Haag's Theorem, Perturbation, Existence and QFT.

[gptejms]

From this formula one can obtain more explicit and physically transparent formulas in each individual sector. For example, in the 2-particle sector

$$H_0 = \sqrt{m^2c^4 + p_1^2c^2} + \sqrt{m^2c^4 + p_2^2c^2}$$

where $p_1$ and $p_2$ are momenta of the two particles.

What I meant was that once an H_0 is defined thus(in a partiular sector),it continues to act only in this sector or subspace.

That's an interesting question. I haven't thought it through. I am not sure whether Haag's theorem requires both interacting and non-interacting theories to have the same vacuum vector. This requirement would exclude all traditional relativistic quantum field theories known to man.

Why would it exclude all traditional relativistic quantum field theories known to man?You may like to read the following from an article on Haag's theorem that I have downloaded.This is as the author says 'a gist of the heuristic version of Haag's original theorem'.

'This argument takes the form of a reductio. Suppose that we are trying to describe both a free scalar field and a self-interacting scalar field using the same Hilbert space $$H$$. Suppose that we demand of the vacuum state that it be the unique (up to phase) normalized state $$|0> \epsilon H$$ that is invariant under Euclidean translations. And suppose that the vacuum state is the ground state in that it is an eigenstate of the Hamiltonian with eigenvalue 0.These suppositions are fullfilled in the case of the free scalar field with mass m > 0, the usual no-particle state $$|0_F>$$ (bare vacuum), and the free field Hamiltonian H_F .
Since the vacuum state $$|0_I>$$ of the interacting field (dressed vacuum or physical vacuum) should also be invariant under Euclidean translations,it follows from the stated assumptions that |0_I> = c|0_F>, |c| = 1, and since |0_I> is annihilated by the Hamiltonian H for the interacting field, it follows that H|0_F> = 0. But the typical Hamiltonians for interacting fields take the form H_F +H_I , where H_I describes the interaction of the field with itself, and such Hamiltonians do not annihilate |0_F> (H polarizes the vacuum)'.

If you want to read more google for 'Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory'.

 

[meopemuk]

From this formula one can obtain more explicit and physically transparent formulas in each individual sector. For example, in the 2-particle sector

$$H_0 = \sqrt{m^2c^4 + p_1^2c^2} + \sqrt{m^2c^4 + p_2^2c^2}$$

where $p_1$ and $p_2$ are momenta of the two particles.

What I meant was that once an H_0 is defined thus(in a partiular sector),it continues to act only in this sector or subspace.

Yes, we agree on this point. Each sector of the Fock space is invariant with respect to the action of the free Hamiltonian H_0. In other words, free time evolution conserves the number of particles, as expected.

Why would it exclude all traditional relativistic quantum field theories known to man?You may like to read the following from an article on Haag's theorem that I have downloaded.This is as the author says 'a gist of the heuristic version of Haag's original theorem'.

'This argument takes the form of a reductio. Suppose that we are trying to describe both a free scalar field and a self-interacting scalar field using the same Hilbert space $$H$$. Suppose that we demand of the vacuum state that it be the unique (up to phase) normalized state $$|0> \in H$$ that is invariant under Euclidean translations. And suppose that the vacuum state is the ground state in that it is an eigenstate of the Hamiltonian with eigenvalue 0.These suppositions are fullfilled in the case of the free scalar field with mass m > 0, the usual no-particle state $$|0_F>$$ (bare vacuum), and the free field Hamiltonian H_F .
Since the vacuum state $$|0_I>$$ of the interacting field (dressed vacuum or physical vacuum) should also be invariant under Euclidean translations,it follows from the stated assumptions that |0_I> = c|0_F>, |c| = 1, and since |0_I> is annihilated by the Hamiltonian H for the interacting field, it follows that H|0_F> = 0. But the typical Hamiltonians for interacting fields take the form H_F +H_I , where H_I describes the interaction of the field with itself, and such Hamiltonians do not annihilate |0_F> (H polarizes the vacuum)'.

If you want to read more google for 'Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory'.

Thank you for the reference. I downloaded this article, and started to read it. It looks interesting.

This "heuristic" version of Haag's theorem uses the fact that all relativistic quantum field theories "known to man" polarize vacuum. This means that the lowest energy eigenstate $|vac \rangle$ of the full interacting Hamiltonian H = H_0 + V (or H = H_F + H_I in your notation) is different from the lowest energy eigenstate $|0 \rangle$ of the free Hamiltonian H_0. In QED, this fact is evident from the presence of tri-linear terms in the interaction Hamiltonian V. For example, there are terms with three creation operators

$$a^{\dag}b^{\dag}c^{\dag}$$...(1)

with a non-trivial action on the vector $|0 \rangle$ (here $a^{\dag}b^{\dag}c^{\dag}$ are creation operators for electrons, positrons, and photons, respectively). The presence of these terms implies that $|0 \rangle$ is not an eigenvector of H.

This version of Haag's theorem does not apply to "dressed particle" theories (including the "dressed particle" version of QED), because in such theories terms of the type (1) are explicitly absent in interaction Hamiltonians and the vacuum is not "polarized".

In my previous posts I discussed the version of Haag's theorem, which is marked as "HWW theorem, Part II" in the paper you referred to. My major objection was related to the use of the covariant transformation law for interacting fields

$$U(\Lambda) \phi(x) U^{-1}(\Lambda) = \phi(\Lambda x)$$

(which, in a different notation, is given in eq. (11) for j=2). My point was that there is no empirical or theoretical evidence for the validity of this transformation law. For example, I am pretty sure that interacting fields in QED do not obey such transformations. However, this doesn't mean that QED is inaccurate or relativistically non-invariant or inappropriate for any other reason.

Eugene.

 

[strangerep]

I think I can now clarify a few more things...

Perhaps the most important thing to know about Haag's thm
is that it's formulated within the framework of (orthodox)
axiomatic QFT. This point is critical because if one
doesn't have solid axioms, one doesn't have a solid
mathematical framework, and therefore cannot prove any
worthwhile theorems.

First, here's a summary of the (orthodox) Axioms...

Axiom-1 postulates the physical existence of Minkowski
spacetime, asserting the "events happen in a 4D Minkowski
spacetime, an event being something about which it makes
sense to say that it does or does not occur".

Axiom-1 also postulates that a quantum state of a physical
system is described by a ray in a separable Hilbert
space "H". To every measurable physical quantity "a"
there corresponds a self-adjoint, generally unbounded,
operator "A" in the separable Hilbert space.

Axiom-2 postulates that we are given a finite number of
operator-valued distributions phi_i(x) over Minkowski
spacetime, called "fields". These fields, smeared out
with test functions and then denoted phi(f), define an
(in general unbounded) operators acting in the Hilbert space.

It is not until Axiom-3 that we come to the postulate that
there exists in H a unitary continuous representation of
the Poincare group. It is here that the usual expression
of the transformation rules for fields enters, expressed
in terms of how these transformations work for Minkowski
spacetime points. (This is the item about which Eugene has
frequently been asking whether a "proof" exists.
Answer: "no, it's an axiom".)

Axiom-4 deals with the spectral conditions of the mass^2
and energy operators.

Axiom-5 deals with causality, using the notion of test
functions with compact support in Minkowski space to
define "local" operator fields which either commute
or anti-commute with each other if the supports of the
respective test functions are spacelike-separated.

Axiom-6 postulates that the vacuum vector (being
the eigenstate of the energy operator with lowest
eigenvalue) is "cyclic" with respect to the field
operators phi(f). This means (loosely) that the whole
Hilbert space H can be generated by the action of the
phi(f) operators on the vacuum. It also involves the
operators being "irreducible" in H, meaning that every
operator in H can be approximated arbitrarily closely
by functions of the phi(f).

Then there's also an "Axiom-0" postulating that the
Hilbert space decomposes in coherent Hilbert spaces
corresponding to the various physically-meaningful
quantum numbers (e.g: spin, charges, etc).

----- (End of Axioms) -----

You don't have to read very far into these axioms to see
that Eugene's approach to QFT is outside this framework.
E.g: the use of Minkowski spacetime points as physical
events, and "attaching" field operators to these points,
is distinctly different from an approach which insists strictly
that physically-measurable positions must correspond
to the eigenvalues of a self-adjoint operator.

Now, about Haag's thm... It doesn't talk specifically
about "interacting" and "non-interacting" representations.
Rather it takes the 1st representation as free and assumes
the existence of a 2nd Poincare representation in the same
Hilbert space described by the above axioms (and with
the same kind of expression for Lorentz transformations
of the fields). It finds that this 2nd representation
must also be free.

Discussions then turn to what should be "done" about
Haag's thm, i.e: which pre-condition(s) or axiom(s)
should be abandoned or modified. Eugene (and Shirokov,
afaict) advocate that the problem lies in the form
of Lorentz boosts acting on the fields (which was
originally motivated by the way the fields are "attached"
to points of Minkowski spacetime). The Earman/Fraser
paper mentioned earlier (and also other authors,
e.g: Barton, as well as some others who start from
an algebraic approach to QFT) advocate the importance
of unitarily inequivalent representations (with their
different vacuum vectors and disjoint Hilbert spaces).
Ironically, this is closely related to the way Haag
originally discussed his theorem in the paper
mentioned earlier in this thread.

Also earlier in this thread, the question arose about whether
Haag's thm excluded "all traditional relativistic QFTs
known to man". I think the key point here is that the
orthodox axiomatic QFT framework does not work at all
well with modern gauge field theories such as the Standard
Model. Hence, Haag's thm is pretty much irrelevant there.
The modern approach is to use Path Integral techniques,
but even this has not yet been rigorously successful
wrt proving existence of a 4D interacting QFT.

[And BTW, the irrelevance of Haag's thm seems also to
apply to the famous Coleman-Mandula thm which motivated
supersymmetry - since the C-M thm is also formulated
under the auspices of axiomatic QFT.]

I'm not sure whether the above sufficiently addresses all the issues
previously raised in this thread, so I'll leave it at that for now and
wait to see if anyone wants to talk further.

 

[Haelfix]

Coleman Mandula is considerably stronger as it requires far less axioms than Haags theorem.. AFAICR, you basically only need finite particle species below some mass scale and analytic cross sections for elastic 2 body scattering.

Strictly speaking I seem to recall it hasn't been proved for all QFTs of interest, and the cases where you can evade the axioms are all thoroughly studied and interesting in their own right.

 

[meopemuk]

Thank you strangerep,

your summary is very good. I agree completely.

For me your analysis means that AQFT (with its present axioms) is a purely formal exercise disconnected from reality.

Eugene.