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I'll have to think a bit about what you've written, but just to note this is not true due to Haag's theorem.maline said:But realistic Hilbert spaces are direct sums of many such irreps
I'll have to think a bit about what you've written, but just to note this is not true due to Haag's theorem.maline said:But realistic Hilbert spaces are direct sums of many such irreps
Why would Haag's theorem change things here?DarMM said:I'll have to think a bit about what you've written, but just to note this is not true due to Haag's theorem.
I'm not saying it as something that corrects or defends Moretti & Oppio's result, just as an interesting fact in itself. The Hilbert spaces of interacting quantum field theories are not sums of irreps of the Poincaré group with definite invariant masses. The hydrogen atom in QED for example won't belong to a Poincaré irrep of such a form.maline said:Why would Haag's theorem change things here?
I think it is fair to assume that there is at least one good Hilbert space representation for the realistic observables in Nature, with the appropriate unitaries for symmetries. It may be non-unique, but it will predict expectation values to any required degree of precision. If such a Hilbert space is Poincare-covariant (i.e. if physics on flat spacetime is internally consistent) then that Hilbert space can be decomposed into Poincare irreps. We may not know what these will look like, but there should be some nice ones, like say, the set of states (position wavefunctions) of a single ground-state hydrogen atom. Each irrep will have a definite invariant mass, so any operator that changes the mass of the system, such as a smeared electric field operator, will not be covered by the Moretti/Oppio result.
Really? This is very surprising to me! Of course we don't really know what the Hilbert space looks like, but the states of ground-state hydrogen are clearly physical and clearly go only to each other under Poincare transformations. The invariant mass is also well defined. Why would this not be an irrep?DarMM said:I'm not saying it as something that corrects or defends Moretti & Oppio's result, just as an interesting fact in itself. The Hilbert spaces of interacting quantum field theories are not sums of irreps of the Poincaré group with definite invariant masses. The hydrogen atom in QED for example won't belong to a Poincaré irrep of such a form.
Actually, on a Hilbert space of two-scalar-particle states, the Pauli-Lubanski vector is nontrivial (except in the case where they share a single "wavefunction"). So we need to decompose the space further, to states that also have fixed values of the Casimir, i.e. fixed squared orbital angular momentum in the rest frame, along with the fixed scalar product of momenta.maline said:For instance, for a free real scalar field, some irreps are: the vacuum (for which we will need to invent an extra dimension as its "imaginary partner"); the space of one-particle states; the spaces of N-particle states where all the particles are in the same mode; the spaces of two-particle states that, when expressed at wavefunctions on momentum space, have support only for pairs of momenta with some fixed scalar product; etc.
This would really take another thread, but you have example constructions of interacting Hilbert spaces from the works of Magnen, Glimm, Jaffe and others.maline said:Really? This is very surprising to me! Of course we don't really know what the Hilbert space looks like, but the states of ground-state hydrogen are clearly physical and clearly go only to each other under Poincare transformations. The invariant mass is also well defined. Why would this not be an irrep?
Can you (or someone else) give a one-line general idea of what goes wrong?DarMM said:This would really take another thread, but you have example constructions of interacting Hilbert spaces from the works of Magnen, Glimm, Jaffe and others.
For the QED case the book of Othmar Steinmann gives a good construction of some aspects of the Hilbert space.
In general Haag's theorem prevents the decomposition which you are referring to.
There's a variety of effects that lead to it. The fact that interacting Hilbert spaces are not Fock, that the interactions in gauge theories prevent the formation of asymptotic Hilbert spaces that permit Wigner reps (infraparticle problem), at the most extreme end you'd have that there is no such thing as a pure state for a system confined to a region of spacetime, e.g. a hydrogen atom.maline said:Can you (or someone else) give a one-line general idea of what goes wrong?
DarMM said:There's a variety of effects that lead to it. The fact that interacting Hilbert spaces are not Fock, that the interactions in gauge theories prevent the formation of asymptotic Hilbert spaces that permit Wigner reps (infraparticle problem), at the most extreme end you'd have that there is no such thing as a pure state for a system confined to a region of spacetime, e.g. a hydrogen atom.
I understood the question as being more generally about sums of (tensor products) of irreps, hydrogen isn't even that.stevendaryl said:Well, the specific question was why can't a hydrogen atom be considered an "elementary particle". In the sense of irreducible representations, I think it's almost by definition that a composite object can't have an irreducible representation.
maline said:Why would Haag's theorem change things here?
I think it is fair to assume that there is at least one good Hilbert space representation for the realistic observables in Nature, with the appropriate unitaries for symmetries. It may be non-unique, but it will predict expectation values to any required degree of precision. If such a Hilbert space is Poincare-covariant (i.e. if physics on flat spacetime is internally consistent) then that Hilbert space can be decomposed into Poincare irreps. We may not know what these will look like, but there should be some nice ones, like say, the set of states (position wavefunctions) of a single ground-state hydrogen atom. Each irrep will have a definite invariant mass, so any operator that changes the mass of the system, such as a smeared electric field operator, will not be covered by the Moretti/Oppio result.
DarMM said:The Hilbert spaces of interacting quantum field theories are not sums of irreps of the Poincaré group with definite invariant masses.
DarMM said:In general Haag's theorem prevents the decomposition which you are referring to.
What goes wrong is that the irreps of the Poincare group are labelled among others by a continuous mass parameter. Therefore one has to take into account direct integrals of irreps rather than just direct sums. This is reflected by the Kallen-Lehmann formula, which contains an (in general Stieljes) integral over the mass.maline said:Can you (or someone else) give a one-line general idea of what goes wrong?
Does it not tell you it can't be sums of tensor products of irreps?A. Neumaier said:However, Haag's theorem only forbids a Fock space structure of interacting theories, and says nothing about the representation theory of the Poincare group involved.
No. Where is a formulation of the theorem that would say so? It only says that it cannot be the Fock sum of tensor products, and indeed it is something more complicated.DarMM said:Does it not tell you it can't be sums of tensor products of irreps?
Take the 1-particle space of a quasifree scalar field theory. It is not a Fock space and the Poincare group acts reducibly (but as a direct integral).DarMM said:Oh that's interesting, what's an example of a Hilbert space formed from a sum of irreps that's not the Fock sum?
Yes, I'm not saying Haag's theorem forbids integrals of irreps. As you said the representation theory of the Poincaré group is known, but what's an example of a sum of irreps that's not Fock? Since it forbids Fock sums, unless there is an example of a sum that is not Fock that means it forbids sums.A. Neumaier said:Take the 1-particle space of a quasifree scalar field theory. It is not a Fock space and the Poincare group acts reducibly (but as a direct integral).
Well, take the sum of nonsymmetric tensor products, or the sum of symmetric tensor products up to some degree, or the tensor product of a Fock space with a 1-particle space, etc. Fock space is a very special situation, defined by the presence of annihilation and creation operators satisfying CCR/CAR, which is absent in general.DarMM said:what's an example of a sum of irreps that's not Fock?
Well, not these (you were talking in more general terms to which my examples applied) but others.DarMM said:Are those reps possible for a quantum field theory?
I would have thought the only possible summed rep for a field theory is the Fock one, which is then eliminated by Haag's theorem.
No, the definition of a composite object is that its field doesn't appear in the Lagrangian.stevendaryl said:Well, the specific question was why can't a hydrogen atom be considered an "elementary particle". In the sense of irreducible representations, I think it's almost by definition that a composite object can't have an irreducible representation.
I am not discussing tensor products of irreps, but only single irreps. Even in the free case, it makes sense to decompose the Fock space in this way, labelling each irrep according to the scalars that appear in it, such as total mass, particle number, scalar products of momenta, and spin (squared). In the interacting case, the irreps will look very different and ugly, because we need to use the full Hamiltonian for time translations. But some nice, stable cases, like the ground-state hydrogen atom, should still appear.DarMM said:Does it not tell you it can't be sums of tensor products of irreps?
It cannot be a QED electron, since these are infraparticles and have a continuous mass spectrum.maline said:the single particles need not be elementary. All that is required is an object that can be specified without continuous parameters - as bound states usually are - and that is stable under the full Hamiltonian. It can be an electron,
My response with Haag's theorem was originally in response to this:maline said:I am not discussing tensor products of irreps, but only single irreps.
Can you tell me what I'm misunderstanding. I took it to be referring to the whole of the QFT's Hilbert space which by Haag's theorem cannot be just a direct sum, but must involve integrals over irreps, although the integral might involve point masses.maline said:But realistic Hilbert spaces are direct sums of many such irreps
Ground state hydrogen will involve electrons which are infraparticles, so it's a least not clear to me that it's an irrep as opposed to an integral over irreps.maline said:But some nice, stable cases, like the ground-state hydrogen atom, should still appear.
maline said:No, the definition of a composite object is that its field doesn't appear in the Lagrangian.
An irreducible representation of a group is, practically speaking, a Hilbert subspace that is closed under the group's action and that has no proper subspaces that are similarly closed.
Just to say it is difficult to maintain this definition of composite when it comes to gauge theories like QCD. There none of the states in the Hilbert space have a field present in the Lagrangian. Using your definition all states in QCD are composite.maline said:No, the definition of a composite object is that its field doesn't appear in the Lagrangian.
But there seems to be agreement that these states all describe composite objects...DarMM said:Just to say it is difficult to maintain a definition of composite when it comes to gauge theories like QCD. There none of the states in the Hilbert space have a field present in the Lagrangian.
True indeed. I guess they're composites without constituents or however one can express the structure of the QCD Hilbert space in English.A. Neumaier said:But there seems to be agreement that these states all describe composite objects...
Yes, I was using the word "sums" as including integrals. I didn't think the distinction would contribute to the discussion, but apparently this caused some confusion, so I apologize for that.DarMM said:Were you in fact referring to direct sums of integrals of irreps as @A. Neumaier said.
I chose the hydrogen atom specifically because it is electrically neutral and spherically symmetric. As long as it encounters no other particles, it shouldn't generate any photons, including the infrared ones that cause problems.DarMM said:Ground state hydrogen will involve electrons which are infraparticles, so it's a least not clear to me that it's an irrep as opposed to an integral over irreps.
Indeed, the particles we usually call "elementary" in QCD are the quarks and gluons, which do not appear as states of their own.DarMM said:Just to say it is difficult to maintain this definition of composite when it comes to gauge theories like QCD. There none of the states in the Hilbert space have a field present in the Lagrangian. Using your definition all states in QCD are composite.
Here is a paper on a relativistic model for the hydrogen atom:maline said:I chose the hydrogen atom specifically because it is electrically neutral and spherically symmetric.
Because hydrogen has bound and scattering states, the mass spectrum is a combination of discrete and continuous. The discrete part leads to a direct sum; the continuous part ot an direct integral. The total Hilbert space is the direct sum of these two parts.maline said:Anyway, the only way it could be an integral over irreps
Well I was only ever discussing the ground state. The aroused states do not form simple irreps, even though they are discrete, because under time evolution they emit photons and decay.A. Neumaier said:Because hydrogen has bound and scattering states, the mass spectrum is a combination of discrete and continuous. The discrete part leads to a direct sum; the continuous part ot an direct integral. The total Hilbert space is the direct sum of these two parts.
Haag's Theorem is a mathematical theorem in quantum field theory that states that there is no unique, self-consistent way to combine the principles of quantum mechanics and special relativity.
The Poincare Group is a mathematical group that describes the symmetries of Minkowski space, which is the mathematical model of spacetime used in special relativity. It includes translations, rotations, and boosts (Lorentz transformations).
Haag's Theorem is closely related to the Poincare Group because it shows that there is no unique way to combine quantum mechanics and special relativity, which are both fundamental principles of the Poincare Group.
Haag's Theorem has important implications for quantum field theory, as it shows that a consistent theory that combines quantum mechanics and special relativity may not exist. This has led to ongoing research and debate in the field.
There have been various proposed solutions to Haag's Theorem, such as string theory and loop quantum gravity, which attempt to reconcile quantum mechanics and special relativity. However, these theories are still under development and have not been fully accepted by the scientific community.