Unruh, Haag et al.: No Room for Particles in Quantum Field Theory?

[lindberg]

TL;DR Summary

The Reeh-Schlieder theorem, the Unruh effect and Haag's theorem don't seem to leave much room for particles in QFT. Or do they?

In a paper by Bain (2011), particles are left with little ontological value because of the Reeh-Schlieder theorem, the Unruh effect and Haag's theorem. The author claims (and here I am copying his conclusion):

First, the existence of local number operators requires the absolute temporal metric of a classical spacetime. This structure allows NQFTs (non-relativistic QFTs) to avoid the consequences of the Reeh-Schlieder theorem. In particular, it prevents the non-relativistic vacuum state from being separating for any local algebra of operators, and this allows for the possibility of local number operators.
Second, the existence of a unique total number operator requires the absolute temporal metric of a classical spacetime. An absolute temporal metric guarantees the existence of a unique global time function for non-interacting NQFTs, and hence a unique way to define an inner-product (or its equivalent) on the space of single particle states. This ultimately leads to a uniquely defined total number operator via a Fock space construction, thus avoiding the implications of the Unruh Effect.
Finally, an absolute temporal metric allows interacting NQFTs to avoid polarizing the vacuum, and this immunizes such theories against the consequences of Haag’s theorem. In particular, interacting NQFTs exist that are unitarily equivalent to non-interacting NQFTs, and hence the former can appropriate the Fock space structure of the latter, and, in particular, the total number operators defined in the latter.

My questions:
1. So what are we to conclude? There are no particles? Or that we need absolute time (a heresy in much of the Physics community)?
2. If we accept an absolute time, thought, will we be able to integrate and explain the observation of the Unruh effect? Does the accelerated detector click at all in a background with absolute time? Or may the observation of the Unruh effect be used to falsify the notion of absolute time?
(I already have some hints to answers to the second question, coming from the two papers below by Dundar and Nikolic, but I would like to cast the question wide-open for now in order to gather several opinions)

Link to Bain (2011):
https://www.sciencedirect.com/science/article/abs/pii/S1355219810000493?via%3Dihub

Link to Unruh in a Shape Dynamics context, Dundar 2017:
https://arxiv.org/abs/1706.05890

Link to Nikolic 2022 on defining objective particles in Unruh-like effects:
https://arxiv.org/abs/0904.3412

 

[lindberg]

Hello, @Demystifier
Here is the separate post I have talked about to properly address the issue of particles in QFT.
So, if you don't mind, several questions (or specifications), since you are one of the rare proponents of absolute time (from what I know) and among those rare proponents, one of the few, who have actually considered this matter.
Maybe my questions will seem naive, please consider it is a topic I am still discovering.
1. If I get it correctly, while the Minkowski vacuum has the definite value zero for the Minkowski number operator, the particle number is indefinite for the Rindler number operator, since one has a superposition of Rindler quanta states. You advocate for the incorrectness of the Rindler approach. But in this case, an accelerating detector should detect zero particles, right? Unless... it is a squeezed state (?), which will attribute a non-zero number of particles due to the uncertainty between time and energy. Do not hesitate to correct me if I misunderstand things here.
2. In a different paper, you talk about the distinction between a photon and a phonon. Do we actually need the distinction between a particle and a quasi-particle here? Can't we stick to "real particles", using the above explanation?
Thanks again for your time and patience.

 

[Demystifier]

Hello, @Demystifier

1. If I get it correctly, while the Minkowski vacuum has the definite value zero for the Minkowski number operator, the particle number is indefinite for the Rindler number operator, since one has a superposition of Rindler quanta states. You advocate for the incorrectness of the Rindler approach. But in this case, an accelerating detector should detect zero particles, right? Unless... it is a squeezed state (?), which will attribute a non-zero number of particles due to the uncertainty between time and energy. Do not hesitate to correct me if I misunderstand things here.

I guess you mean this paper: https://arxiv.org/abs/gr-qc/0103108
I'm not saying that detector will not detect particles, but I'm saying that those particles are not well modeled by Rindler particles. The concept of Rindler particles highly depends on the existence of horizon, while particles detected by the accelerated detector should not depend on it.

2. In a different paper, you talk about the distinction between a photon and a phonon. Do we actually need the distinction between a particle and a quasi-particle here? Can't we stick to "real particles", using the above explanation?

I think that the distinction between particles and quasi-particles is not really necessary, because both can be viewed as collective excitations of certain fields. We usually think of particles as excitations of fundamental fields, and of quasi-particles as excitations of effective fields. But in the modern view the so called "fundamental" fields are not really fundamental; they are effective fields too, emerging from some even more fundamental but currently unknown theory.

 

[PeterDonis]

So what are we to conclude?

That some physicists do not like the way QFT is standardly done. Or, if you like, that "giving rigorous foundations to QFT" is still an open research problem.

 

[PeterDonis]

The author claims...

You're leaving out a key reason why the author wanted to establish these claims; as he states in the Introduction:

"This suggests that the intuitions that underlie the Received View's treatment of particles are non-relativistic, and to the extent that such intuitions are inappropriate in the relativistic context, they should be abandoned when it comes to interpreting RQFTs."

In other words, the lack in relativistic QFT of a "particle concept" that matches our intuitions is a problem with our intuitions, not with RQFTs. We don't need to fix RQFTs. We need to fix our intuitions. That is what Bain is arguing.

 

[vanhees71]

I'd also say that the notion of "particles" is different in Newtonian and relativistic quantum theory from a physical point of view, and the (not yet fully established) mathematically rigorous formulations of relativistic quantum field theories reflect this physics.

E.g., from our "classical intuition" a particle is some localized "small" object. In non-relativistic quantum theory it is at least an "arbitrarily precisely localizable" object. Formally that becomes clear that there is the formulation in terms of wave mechanics with a well-defined position operator, and the wave function $\psi(t,\vec{x})$ has the meaning that $P(t,\vec{x})=|\psi(t,\vec{x})|^2$ is the probability distribution of the position (at time $t$). Nothing in the formalism of quantum theory nor any physical plausibility constraints forbids to make $\psi(t,\vec{x})$ to be sharply peaked around an arbitrarily small region around an arbitrarily given position. It must only be square integrable, so that you can normalize the total probability for the particle to be "somewhere" to be 1. Also there's no problem with the standard description of the time evolution in terms of the Schrödinger equation with a given Hamiltonian, and all this can be extended to systems of many particles, that are interacting. The quantum-field theoretical description of such many-body systems ("2nd quantization") is just equivalent to this "1st-quantization formulation" but makes it easier to keep track of Bose (Fermi) (anti-)symmetrization of the product-bases states in terms of annihilation and creation field operators leading to the notion of Fock spaces, valid for the description of free as well as interacting particles.

Now the first attempt to generalize QT to a special relativistic description of course was to just use a wave function of this kind in the "1st-quantization formalism", but already the free-particle case is problematic, because even in that case you cannot fulfill Einstein causality, i.e., if you start with a well-localized "wave packet" and use the free Klein-Gordon equation to calculate how this state looks like at later times, it provides non-zero probability to be found in regions, where it can only have propagated with a speed faster than light (in vacuo). So already for free particles the usual interpretation of the (naive) position operator taken over from non-relativistic quantum theory leads to a dynamics violating fundamental principles like causality in the relativistic case.

Nowadays, being used to "particles" at relativistic energies, we know what's wrong with this all-too-naive picture of a particle: As soon as you have interactions of particles which involve relatistic energies and momenta, there's the possibility to create new particles and/or destroy the before present particles. If you want to confine a particle to a small region in space according to the Heisenberg uncertainty relation for position and momentum it's not unlikely that it has a large momentum and energy, and since confining it needs matter around it, it may interact with this "container particles" and being destroyed and other particles are created.

So the solution for free particles is to use quantum field theory and define local observables, which commute at space-like separated arguments (microcausality condition), and to be compatible with relativity you construct them by looking for unitary representations of the Poincare group. In this way you come to quantities like the energy, momentum, and angular momentum densities of quantized field quantizations, which fulfill the microcausality condition. Interestingly that only works when you introduce for each particle also its corresponding antiparticle (with the possibility to make the "strictly neutral" such that the particles and antiparticles are identical, like for the em. field and "photons" as their Fock states, which in some sense are somewhat "particle like").

In the case of interacting particles, there are many formal questions open, which includes Haag's theorem, according to which in a strict sense the "interaction picture" which is formally used to define interacting particles using the Fock space of the free particles and treat the interactions as perturbation. Famously this doesn't work so naively, and you encounter divergences and need to "renormalize" them to get useful finite results (e.g., to calculate cross sections for scattering processes as observed with the help of particle accelerators). In any case, there is no "naive" particle interpretation of relativistic QT.

Interestingly there's also no "naive" particle description in classical relativistic physics. One famous "no-go theorem" says that there's no relativistic theory of interacting point particles (which in non-relativistic physics is the first thing you encounter in your introductory mechanics lecture). Another famous problem is the problem to introduce a consistent theory for a point particle and the electromagnetic field: A particle accelerating in some somehow created electromagnetic field leads to the radiation of electromagnetic waves ("bremsstrahlung" or "synchrotron radiation" etc.) and to take the corresponding loss of energy and momentum of this particle into account leads also to divergences and the famous "radiation-reaction problem". So "naive" point particles make a lot of trouble already in classical relativistic physics, and the much more natural way are also here field theories (like magnetohydrodynamics as a macroscopic theory of a charged fluid in interaction with the electromagnetic field).

 

[lindberg]

Thanks a lot for such a detailed response @vanhees71

 

[lindberg]

I guess you mean this paper: https://arxiv.org/abs/gr-qc/0103108
I'm not saying that detector will not detect particles, but I'm saying that those particles are not well modeled by Rindler particles. The concept of Rindler particles highly depends on the existence of horizon, while particles detected by the accelerated detector should not depend on it.

I agree with you, although I know much less about it. There is a nice paper by Arageorgis, Earman and Ruetsche (2013) making similar claims: https://philpapers.org/rec/ARAFNA-2
This being said, if the detector does indeed detect any particles, these should be well modeled by the Minkowski vacuum. To quote Earman again, the Unruh-DeWitt detector can be said to detect "the noise or fluctuations in the Minkowski vacuum".
Then (and feel free to correct me here) this detector should act similarly irrespective of the relative or absolute nature of time and of the character of motion (inertial or accelerating or rotating).
What I do not understand then is... why do we need acceleration in this case? If we throw away the Rindler quanta, we can use the detector for fluctuations in Minkowski vacuum in both inertial and accelerated movements alike. No?

 

[Demystifier]

What I do not understand then is... why do we need acceleration in this case? If we throw away the Rindler quanta, we can use the detector for fluctuations in Minkowski vacuum in both inertial and accelerated movements alike. No?

The point is that the effect involves an average over a long time. Without acceleration the average is mathematically of the form $\int_{-\infty}^{\infty} dt\, e^{i\omega t}$, which is zero. Acceleration transforms it into a different type of an integral, which is non-zero.

 

[lindberg]

The point is that the effect involves an average over a long time. Without acceleration the average is mathematically of the form $\int_{-\infty}^{\infty} dt\, e^{i\omega t}$, which is zero. Acceleration transforms it into a different type of an integral, which is non-zero.

Ah yes, I see. Now I understand why we need acceleration! Thank you.
But then again, this should predict the same number of particles whether we do it for RQFT or NQFT, right?
What I mean is that an experiment would not be able to distinguish between the two.

 

[Demystifier]

But then again, this should predict the same number of particles whether we do it for RQFT or NQFT, right?

I don't know, I never seen a calculation of this with NRQFT.

 

[vanhees71]

That's subtle. Usually in relativistic QFT particle numbers are not observables but often "net-particle numbers" (i.e., #of particles - #of antiparticles) are! It depends on whether the corresponding densities fulfill the microcausality condition or not.

 

[A. Neumaier]

what are we to conclude? There are no particles?

You should conclude that the particle concept is observer dependent and valid only in spacetime regions where interactions are negligible, or when applied to sufficiently massive particles (so that a semiclassical description is adequate).