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I am very confused by the statement made in Haag's, Local Quantum Physics: Fields, Particles, Algebras (page 46):

... the idea that to each particle there is a corresponding field and to each field a corresponding particle has also been misleading and served to veil essential aspects. The rôle of fields is to implement the principle of locality. The number and the nature of different basic fields needed in the theory is related to the charge structure, not the empirical spectrum of particles. In the presently favoured gauge theories the basic fields are the carriers of charges called colour and flavour but are not directly associated to observed particles like protons.

However, in my understanding of the SM, to each field (or linear combination of) we do assign a particle, even if we do not directly observe it. So, even though I agree that there is no field in the SM which corresponds to the proton, there is a field corresponding to the quarks, which we do not observe on their own. In fact, it seems to me that we define the notion of fundamental particle by the fact that there is a field associated to it.

Another aspect of this discussion may be that, once we have the full SM, the relationship between fields and the symmetry groups is what defines the particle. Quarks are described by the multiplet of fields that transform non-trivially into each other under the $SU(3)$ gauge symmetry.

Can somebody help me understand more clearly what Haag is referring to in this quote from his book? Thank you very much.

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    $\begingroup$ Whether or not you want to associate a particle to the quark field, Haag's points remains independently of that, doesn't it? For while QCD has many formulations believed to be equivalent, in the ultraviolet as a theory of quarks and gluons, in the infrared as a theory of mesons, nobody would have ventured a theory where there are both quark AND proton fields (this would be taking "for each particle a field" literal). $\endgroup$ Aug 4, 2020 at 15:19

4 Answers 4

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Good question. Some preliminary remarks.

  1. The map "one particle" $\leftrightarrow$ "one field" holds, at best, in the weakly coupled regime, where fields are (by construction, cf. ref.1) interpolating fields for one-particle states. In a strongly-coupled theory, a single field may (and usually does) create many different particles, and there are fields which may not create particles at all.

  2. The map is specially subtle in gauge theories, because the fields themselves are not physical (they are not observable). The states of the theory are (by definition) gauge-invariant; the fields are not.

  3. A given theory usually has an infinite number of different descriptions (in its simplest incarnation, due to the fact that one may integrate in/out auxiliary fields; in more subtle situations, due to the existence of non-trivial dualities, where apparently different QFTs describe in fact the exact same dynamics). Therefore, it is not correct to claim that to every particle there is one field: the particles are intrinsic to the system, the fields are user-dependent.

Recall the basic definitions:

  • A particle is a (special) vector in your Hilbert space $\mathcal H$. It is typically defined as one that is a common eigenvector to (the maximal torus of) some algebra of observables, usually containing Poincaré.

  • A field $\phi\in\mathrm{End}(\mathcal H)$ is said to create the particle $|a\rangle\in\mathcal H$ if and only if $\langle 0|\phi|a\rangle\neq0$, where $0$ denotes the vacuum state. The field may be Lagrangian (i.e., it is a variable you path-integrate over) or not (e.g., a composite).

In a weakly coupled theory, one can associate a field to every particle. In a strongly coupled one, every field typically creates all the particles of the theory (unless there is some selection rule, a la Wigner-Eckart). Indeed, unless there is a specific reason for the matrix element to vanish, the field will create everything.

Reference

  1. Weinberg S. - Quantum theory of fields, Vol.1.
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Even if there were some valid relaxed sense in which every field in every QFT has an associated particle, the important point is that fields are inputs (used to define the theory mathematically) and particles are outputs (phenomena that we derive from the theory). Particles are transient and not always sharply defined. As examples that challenge the particles-correspond-to-fields idea, we could point to the Schwinger model, or to conformal field theories, or even to topological QFTs.

More generically, we can often improve clarity by distinguishing between three parts of the scientific process:

  • The creative task of inventing a theory that has a chance of agreeing with experiment.

  • The clean-up task of expressing a given theory in the simplest possible way (something resembling "axioms") with the benefit of hindsight, so that the starting point for predictions/intuition/teaching is more clear.

  • The computational task of extracting a given theory's predictions, starting from those "axioms."

Haag's comment is presumably trying to emphasize the second part. All three parts are essential, and they all interact with each other (the lines are fuzzy), but different texts may emphasize different parts. Weinberg's QFT book spends considerable effort on the first part, using a particle-centric viewpoint to motivate the structure of QFT. Most QFT texts spend most of their pages on the third part, especially on perturbation theory.

The specific axiomatic system that Haag's book was advocating might not be the panacea of powerful theorems that some might have hoped it would be (although it still has value as a conceptual framework), but the core of his comment is still relevant today: in QFT, fields are mathematically more basic than particles, and we should not approach the study of QFT expecting to find any tidy correspondence between them.

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    $\begingroup$ A strong argument for treating fields as more basic than particles: In curved spacetime, the particle content of your system can depend on your reference frame, but the expectation values of fields do not. $\endgroup$
    – user1504
    Aug 4, 2020 at 20:03
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    $\begingroup$ Further to @user1504's point, in QFTCS you can't always define a concept of particle number. There might not even be a well-defined vacuum state, & if there is one it might not respect the same symmetries as the spacetime itself. $\endgroup$
    – J.G.
    Aug 4, 2020 at 21:32
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It's worth remembering when reading Haag's book that it was published in 1965, well before the reality of quarks was accepted. There was no Standard Model at the time; instead they had piecemeal understanding of gauge fields, muddled together with a lot of S-matrix thinking. Bjorken's light cone scaling arguments didn't come along until 1968 and the definitive deep inelastic scattering experiments that established the reality of partons weren't carried out until 1969.

So at the time Haag was writing, the observed particles were the baryon/meson zoo (plus electrons and neutrinos), and it was a point of debate whether there were any particles associated to the fields.

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    $\begingroup$ I may be mistaken, but you might think of Wightman and Streater's book; Haag's was published in the 90s. $\endgroup$ Aug 4, 2020 at 15:14
  • $\begingroup$ Yeah, the version I have is from 96. $\endgroup$ Aug 4, 2020 at 15:31
  • $\begingroup$ No, i was just confused. I thought the first edition was considerably older, but it was from 92. Let me submit instead that Haag's viewpoint was stuck in the middle 60s. That was when he was at his most active as a researcher. It's somewhat astonishing that someone could publish a text like this in 1992. $\endgroup$
    – user1504
    Aug 4, 2020 at 16:09
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    $\begingroup$ Perhaps the point Haag was making is that protons are non-perturbative objects. A proton decidedly is not a perturbative excitation of QCD fields like an electron is of QED fields. As you correctly point out, the fundamental, perturbative excitations of QCD are quarks and gluons. $\endgroup$ Aug 4, 2020 at 19:32
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As an experimental particle physicist my knowledge of fied theory is on a working level, i,e, how it is used in order to calculate interaction crossections and decays for particle physics.

The course I took in field theory was back in 1964 and the professor used the book of Bogolyubov, and after struggling with creation and annihilation operators for some months, I saw the light at a CERN school where Veltman gave lectures on how to calculate crossections.At the same time I was being introduced to a field theory of nuclear physics interactions, so since then it is clear to me the field theory is a calculational tool for quantum mechanics,dependent on the subject under study. Since then quantum field theory is applied to other branches of physics also.

For particle physics, the axiomatically assumed particles in the table of the standard model are assigned a field on every point in space time represented mathematically by the plane wave solution of the appropriate equation for each particle, (for example Dirac for electrons, the quantized Maxwell equations plane wave for the photons,etc). Thus it is the particles assumed in the theory that define the fields, not the fields that define the particles.To calculate crossections and decays Feynman diagrams are used and creation and annihilation operators define the function to be integrated. This works, fits the data and is the success of the standard model.

BUT the progress in space of free particles cannot be modeled by a single plane wave creation and annihilation , one has to use wave packets to get localized particles.

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  • $\begingroup$ Klein-Gordon is really more of a toy model. For example, it doesn't describe the Higgs field; you need a nonlinear model for that. Bosons with spin also don't admit a Klein-Gordon description. $\endgroup$
    – J.G.
    Aug 4, 2020 at 21:30
  • $\begingroup$ @J.G. I am just using the terms generally, the important thing is to use the appropriate QM free particle (no potential) equtation so as to have the approriate QM plane waves. edited $\endgroup$
    – anna v
    Aug 5, 2020 at 4:10

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