I Bare QFT theory with cutoff and multi-particle state....

[asimov42]

Hi all,

I was reading Arnold Neumaier's excellent article on the Vacuum Fluctuation Myth, and ran upon one part I have a question about: he notes that "in bare quantum field theory with a cutoff, the vacuum is a complicated multiparticle state depending on the cutoff – though in a way that it diverges when the cutoff is removed, so that nothing physical remains. "

I'm wondering a) why the introduction of a cutoff leads to a multiparticle state? I thought the ground state even in the bare theory was a state with zero particles, even with a cutoff, and b) since we don't observe this multiparticle state (do we?), is this evidence that there should not be a cutoff?

Thanks!

Reference https://www.physicsforums.com/insights/vacuum-fluctuation-myth/

 

[A. Neumaier]

With cutoff one can work in a Fock space created in terms of bare particles, and with a well-defined normally ordered Hamiltonian that describes the interacting dynamics. The physical vacuum is then the ground state of this Hamiltonian. It contains zero physical particles, but expressed in terms of the particles of the bare theory, it is a very complicated multi-bare-particle state. The only state with zero bare particles is the bare vacuum state, but this is not even an eigenstate of the physical Hamiltonian.

 

[asimov42]

Ah thanks Arnold - ok, so the observed state of the physical vacuum contains zero physical particles, but if one refers to particles of the bare theory, you have the complex multiparticle state. And the bare vacuum state is not a physical state (since not eigenstate of physical Hamiltonian)?

 

[A. Neumaier]

Ah thanks Arnold - ok, so the observed state of the physical vacuum contains zero physical particles, but if one refers to particles of the bare theory, you have the complex multiparticle state. And the bare vacuum state is not a physical state (since not eigenstate of physical Hamiltonian)?

The bare vacuum state is (in the cutoff theory) a very complex physical multiparticle state. In the renormalization procedure it disappears and has no longer a physical interpretation.

 

[asimov42]

And if one does not apply the cutoff or renormalization, one is left with a divergent, non-physical result, then.

I guess my final query is when one should expect to see a physical vacuum with zero physical particles - either with an applied cutoff, or in the renormalized case?

 

[A. Neumaier]

And if one does not apply the cutoff or renormalization, one is left with a divergent, non-physical result, then.

I guess my final query is when one should expect to see a physical vacuum with zero physical particles - either with an applied cutoff, or in the renormalized case?

The only physically correct theory is the renormalized one, after having removed the cutoff. The physical vacuum is of course an idealization, as the real world is filled with lots of fields (not only stuff describable in terms of particles).

 

[RockyMarciano]

The physical vacuum is of course an idealization, as the real world is filled with lots of fields (not only stuff describable in terms of particles).

Then I'd suggest not to call it physical.

 

[A. Neumaier]

Then I'd suggest not to call it physical.

If one is not allowed to make idealizations one cannot do physics until a theory of everything is available...

Many physical systems are idealizations. There is no isolated system, but systems are often treated as isolated.

 

[RockyMarciano]

If one is not allowed to make idealizations one cannot do physics until a theory of everything is available...

Many physical systems are idealizations. There is no isolated system, but systems are often treated as isolated.

Yes, all this doesn't change the fact that calling physical what is ideal could be more confusing than callint it ideal. Just a practical suggestion.

 

[A. Neumaier]

Yes, all this doesn't change the fact that calling physical what is ideal could be more confusing than calling it ideal. Just a practical suggestion.

One cannot change this. Physical means tractable in terms of a dynamical model based on physical principles.

 

[asimov42]

One more question about renormalization: once the cutoff has been removed, if I want a more and more accurate answer, can I sum up to an arbitrary loop order?

 

[asimov42]

Any takers?

 

[A. Neumaier]

One more question about renormalization: once the cutoff has been removed, if I want a more and more accurate answer, can I sum up to an arbitrary loop order?

The series is only asymptotic, so at first the accuracy improves but at some point it will get worse. For arbitrarily accurate answers you'll need to sum infinite families of terms, using for example the renormalization group. It is unknown whether this is enough.

 

[asimov42]

Thanks Arnold! So the answers should still remain finite, with the cutoff removed, but of course it's not possible to sum infinite families of terms.

 

[A. Neumaier]

Thanks Arnold! So the answers should still remain finite, with the cutoff removed, but of course it's not possible to sum infinite families of terms.

Yes, the answers must be finite if the theory is well-defined. But summing infinitely many diagrams of a certain kind (ladder, or rainbow, etc.) is a common activity when evaluating series derived in terns of Feynman diagrams.

 

[asimov42]

The bare vacuum state is (in the cutoff theory) a very complex physical multiparticle state. In the renormalization procedure it disappears and has no longer a physical interpretation.

@A. Neumaier (or others): the renormalization procedure must also cause the complicated bare-multiparticle-state to disappear, correct?

 

[A. Neumaier]

@A. Neumaier (or others): the renormalization procedure must also cause the complicated bare-multiparticle-state to disappear, correct?

Yes. The Hilbert space changes. The renormalized Hilbert space is no longer a Fock space, contains no number operator, and hence has no notion of particles. Particles (physical ones) appear instead asymptotically, in states that satisfy cluster separability.

 

[asimov42]

Got it - thank you @A. Neumaier! So as with the bare vacuum state, the bare multiparticle state no longer has a physical interpretation (just so I'm clear). That is, through renormalization, one has taken an un-physical, divergent result with the cutoff removed, and 'gotten rid of the infinity.''...

Clearly, as a physical electron moves through the physical vacuum, it does not bump into a bunch of bare particles ... is it possible to describes more fully what renormalization does in this case? @A. Neumaier had mentioned that the divergence in the original Fock space occurs such that nothing physical remains. Does this carry over as the Hilbert space for the physical vacuum being 'empty' (save fields)?

 

[A. Neumaier]

through renormalization, one has taken an un-physical, divergent result with the cutoff removed, and 'gotten rid of the infinity.''...

Yes, renormalization removes the basic problems with the bare particles.

the Hilbert space for the physical vacuum being 'empty' (save fields)?

The Hilbert space contains all states, not only the vacuum state. Only the vacuum state corresponds to an empty universe.