# There are no particles, only fields!

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## Main Question or Discussion Point

It seems to be widely accepted on this forum that fields, not particles, are fundamental. In other words particles are made of fields. I have seen particles described in various ways such as being excitations of fields or eigenstates with known energy.

This creates a problem for high school students, particularly those who have to study the standard model. Should they be told that the electron and some of the other things they study are not really fundamental but are made of fields?

## Answers and Replies

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Should they be told that the electron and some of the other things they study are not really fundamental but are made of fields?
Yes, why not? I don't lie to my students if they ask, beacuse I see no point in doing so. I think that it is a really bad didactical method to make up things for them just so that they can feel like they understand something. Also that's the reason I really don't like pop-sci books...

A. Neumaier
2019 Award
Should they be told that the electron and some of the other things they study are not really fundamental but are made of fields?
''made of'' is not really the right terminology for the relationship between an electron and the electron field.

They can be told that electrons are discrete excitations of the electron field, photons are discrete excitations of the electromagnetic field, etc.. This can be understood by analogy to musical sounds, which are discrete excitations of the pressure field of the air.

This analogy has some merits since it naturally explains aspects of the uncertainty relation: Musical sounds have pitch and times of occurrence, but the more precise you specify the time the less meaningful is the specification of pitch, and conversely. This is completely analogous to the uncertainty of position and momentum for electrons.

However, the pressure field is a classical field and the electron is a quantum field, so that the analogy is not perfect.

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DarMM
Gold Member
This is a difficult issue, as it is difficult to say to what degree even fields are fundamental in Quantum Field Theory.

For example in Yang Mills theories the field operators act on a enlarged set of unphysical states, only quantities like Wilson loops have well defined expectation values on the physical hilbert space and even then only smeared versions of them.

One might say relativistic quantum systems have particle-like states and classical field-like states among the states in their Hilbert Spaces, but I'm unsure how one might say the field states are primary to the particle ones.

A. Neumaier
2019 Award
relativistic quantum systems have particle-like states and classical field-like states among the states in their Hilbert Spaces, but I'm unsure how one might say the field states are primary to the particle ones.
Multiparticle states need a Fock space for their description, hence are only asymptotically defined at times $t\to\pm\infty$, and hence at finite time always approximate (and reasonably defined only when the particles involved are approximately free). On the other hand, all states are states of the interacting quantum field theory, hence are field states.

This makes the fields primary and particles secondary. This can also be seen from the fact that relativistic quantum particle theory is almost nonexistent. On the other hand, particle physicists routinely use relativistic quantum field theory to analyze the properties of experiments involving particles.

For example in Yang Mills theories the field operators act on a enlarged set of unphysical states, only quantities like Wilson loops have well defined expectation values on the physical hilbert space and even then only smeared versions of them.
Even in classical physics, only fields $\phi(x)$ smeared with smooth test functions $f(x)$ according to $\phi(f)=\int f(x)\phi(x)dx$ are observable. This continues to hold in the quantum case for the field expectations. Doesn't this remains valid even in Yang-Mills theories, except that the space of test functions $f$ is constrained to those satisfying the continuity equation $\nabla\cdot f=0$?

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DarMM
Gold Member
Multiparticle states need a Fock space for their description, hence are only asymptotically defined at times $t\to\pm\infty$, and hence at finite time always approximate (and reasonably defined only when the particles involved are approximately free). On the other hand, all states are states of the interacting quantum field theory, hence are field states.

This makes the fields primary and particles secondary. This can also be seen from the fact that relativistic quantum particle theory is almost nonexistent. On the other hand, particle physicists routinely use relativistic quantum field theory to analyze the properties of experiments involving particles.
I agree particles are not a primary concept, especially for multiparticle states as for $\psi, \phi \in \mathcal{H}^{(1)}$ then $\psi \otimes \phi$ does not lie in the Hilbert Space.

My point is more that in Yang Mills for example there is the space of states $\mathcal{H}$, acting on which is a set of operators $\mathcal{O}$. This space of operators may have smeared fields as its basis or smeared Wilson loops and either can be expressed in terms of the other, though Wilson loops are hard to compute with as primary objects. One could equally say the states are Wilson loop states.

Even in classical physics, only fields $\phi(x)$ smeared with smooth test functions $f(x)$ according to $\phi(f)=\int f(x)\phi(x)dx$ are observable. This continues to hold in the quantum case for the field expectations. Doesn't this remains valid even in Yang-Mills theories, except that the space of test functions $f$ is constrained to those satisfying the continuity equation $\nabla\cdot f=0$?
I don't think so, $\int{A_{\mu}(x)f^{\mu}(x)dx}$ will map you outside the physical Hilbert Space and hence are not observables in general. Wilson loops however would be observables (Migdal and Nakanishi discuss this). In a sense one might say that fundamentally pure Yang-Mills is a theory of the noncommutative statistics of Lie Holonomies.

Or perhaps, the noncommutative statistics of Lie Fiber Bundles, with neither the connections nor holonomies being more primary, but equal descriptions.

(I'm not fully sure if this is right!)

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A. Neumaier
2019 Award
I don't think so, $\int{A_{\mu}(x)f^{\mu}(x)dx}$ will map you outside the physical Hilbert Space and hence are not observables in general.
But isn't $A(f)$ gauge invariant when $\nabla\cdot f=0$? (I am not an expert on gauge theories but at least it seems to me so for QED.) And shouldn't the physical observable space be the space of gauge invariant expressions in the fields?

<Moderator's note: broken tags fixed>

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... relativistic quantum particle theory is almost nonexistent.
Yes, particle-based approaches are not in the mainstream. But they exist, nevertheless. There is a 2-volume book explaining the particle-centric view on QFT:

Stefanovich E. Relativistic quantum theory of particles. Vol. 1 Quantum electrodynamics. Saarbrücken: Lambert Academic; 2015.

Stefanovich E. Relativistic quantum theory of particles. Vol. 2 A non-traditional perspective on space, time, particles, fields, and action-at-a-distance. Saarbrücken: Lambert Academic; 2015.

This content is also available online:

Stefanovich, E. V. Relativistic Quantum Dynamics: A non-traditional perspective on space, time, particles, fields, and action-at-a-distance. arXiv:physics/0504062

Eugene.

A. Neumaier
2019 Award
Yes, particle-based approaches are not in the mainstream. But they exist, nevertheless. There is a 2-volume book explaining the particle-centric view on QFT:
I know your books. They feature an approach in which superluminal effects are present. Full relativistic causality is lost in the transition from a covariant to a Hamiltonian picture. Not attractive....

I know your books. They feature an approach in which superluminal effects are present. Full relativistic causality is lost in the transition from a covariant to a Hamiltonian picture. Not attractive....
Yes, there are instantaneous interactions in this approach.
No, there are no violations of causality.
These two statements do not contradict each other. This is explained in section 17.3 of the arXiv version.

Eugene.

atyy
Yes, why not? I don't lie to my students if they ask, beacuse I see no point in doing so. I think that it is a really bad didactical method to make up things for them just so that they can feel like they understand something. Also that's the reason I really don't like pop-sci books...
I think there is a good case to be made for lying, as long as we are honest about it. Honest lying :)

Within the Copenhagen interpretation, I don't know whether the electron field, for example, is real at all.

Furthermore, even the electron field may not be fundamental, since string theory may be more fundamental.

A. Neumaier