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I am not a physicist, but my understanding is that the Standard Model of particle physics is not the foundational model but rather is emergent from Quantum Field Theory (QFT), and that QFT is within modern physics treated as the foundational model of reality (not withstanding speculative and empirically not-yet-supported ideas such as string theory).

My question is:

I know that spin is treated as a fundamental property of a particle within particle physics. But is spin also a fundamental property within QFT, or is it some emergent property of a quantum field?

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    $\begingroup$ I don't know what this question is trying to ask. Theoretical particle physics is mostly QFT, so I don't know what distinction between "particle physics" and "QFT" is drawn here nor what exactly an "emergent property of a quantum field" would be ("emergent" is a nice buzzword but without more explanation it just doesn't mean anything in a general context) as opposed to a "fundamental property of a quantum field". $\endgroup$
    – ACuriousMind
    Commented May 18 at 10:44
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    $\begingroup$ Are you asking if we could have constructed a quantum field theory that did not have the property of spin? If so then the answer is yes, but of course this theory would not describe the real world. $\endgroup$ Commented May 18 at 11:18
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    $\begingroup$ @ACuriousMind i think the OP refers to fundamental properties referring to properties that a field has just by quantizing it, and emergent properties as properties that the field acquires due to a particular choice of physical model. $\endgroup$
    – LolloBoldo
    Commented May 18 at 12:46
  • $\begingroup$ I'm honestly a little shocked that this question isn't clear to people. By fundamental property I simply mean, a variable that is defined directly in the model, while a property is emergent if it is a pattern in the fundamental variables (e.g. a wave in a field is an emergent feature of the field which is the fundamental object). I'm not sure why this is so confusing. $\endgroup$
    – user56834
    Commented May 19 at 4:53

3 Answers 3

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The Standard Model of Particle Physics is a quantum field theory, which describes 6 quark fields, 6 lepton fields (3 electrically charged + 3 neutral), and 4 gauge fields which mediate their interactions, as well as the Higgs field which gives a number of the aforementioned fields their observed masses (not all by the same mechanism).

The spin of a field refers to the intrinsic angular momentum possessed by quantized excitations (i.e. particles) of that field. It is not emergent, but rather part of the definition of the field; the quark and lepton fields have spin-1/2, the gauge fields have spin-1, and the Higgs field has spin-0.

When you write down a generic QFT, you need to decide how many fields to include, what their spins are, what symmetries they have, how they interact, etc. These ingredients don't emerge from the framework of QFT (except insofar as not all choices are compatible with one another), but need to be put in by hand.

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  • $\begingroup$ So each field in the standard model has a spin which is axiomatically associated to it, and which enters into the schroedinger equation for the model in such a way that it equals the intrinsic angular momentum of the particles? $\endgroup$
    – user56834
    Commented May 19 at 4:57
  • $\begingroup$ By the way, regarding "The Standard Model of Particle Physics is a quantum field theory": the reason I thought otherwise was because I thought based on some popular discussions that the standard model assumed the existence of particles rather than fields axiomatically, and that there was an underlying QFT. $\endgroup$
    – user56834
    Commented May 19 at 5:01
  • $\begingroup$ @user56834 When we write down a QFT, we need to decide how each field transforms under the action of whatever symmetry groups we decide to impose. For a relativistic QFT, that includes the Lorentz group. Without diving into technical detail, each possible choice corresponds to a choice of how the angular momentum operators act on those fields, and in particular to the intrinsic angular momentum carried by their excitations. This is what we call spin. $\endgroup$
    – J. Murray
    Commented May 19 at 19:10
  • $\begingroup$ @user56834 With regard to your second comment, that's an understandable misconception given that "particle" is in the name. See e.g. the wiki article for a lengthier explanation. $\endgroup$
    – J. Murray
    Commented May 19 at 19:17
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Murray's answer is not true. His description is valid only for QFTs which possess the Lorentz invariance.

A totally generic QFT must not possess an a priori Lorentz symmetry.

The standard model is a very very specific type of QFT, and possesses a Poincare (Lorentz + translations) symmetry, as well as other global symmetries and also possesses gauge symmetry.

The spin in the standard model, or any other QFT, emerges when your QFT has a Lorentz symmetry: the "charge" associated to the Lorentz symmetry is in fact the spin.

As stated above, the QFT framework never requires such symmetry: it does not require any symmetry at all.

Relativistic QFTs instead require the Lorentz invariance and hence possess a spin charge. However, relativistic QFTs are only a subset of all the QFTs.

I want to add that, as stated in the comments, our universe seems to possess such a symmetry, so all the QFTs who describe our reality should have spin, but not all QFTs describe our world, so spin is a property only of a subset of them.

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    $\begingroup$ Spin is a consequence of relativistic physics; the correlation between spin and statistics is a consequence of QFT physics. $\endgroup$ Commented May 18 at 12:46
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    $\begingroup$ Btw: Murray did not claim that spin was a consequence of QFT. He explicitly stated that spin and symmetries did not emerge from QFT itself. $\endgroup$ Commented May 18 at 12:48
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    $\begingroup$ You are misinterpreting my answer. My point is that if you write down a QFT for a field, then whether it is spin-1/2 or spin-0 (or more generally, how it transforms under rotations) is something you decide, not something which emerges magically from the framework. I can do conventional QED or scalar QED; nothing about the framework of QFT requires the electron to have spin-1/2. That fact is something which the physicist installs to match their observations. $\endgroup$
    – J. Murray
    Commented May 18 at 14:37
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    $\begingroup$ "You also don't need to put it by hand, but spin emerges as a consequence of a symmetry, is not imposed by hand." This is misleading, because you are the one who decides how the field transforms under that symmetry. The concept of spin exists in any Lorentz-invariant theory, but what spin a particular field possesses is a choice that you make. $\endgroup$
    – J. Murray
    Commented May 18 at 14:42
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    $\begingroup$ @LolloBoldo You are hyper-focusing on a single phrase of my answer and ignoring the context in which it appears. The color of a shirt is not a fundamental property of the atoms of which the shirt is composed; it is an emergent property of the aggregate material, which depends on how those atoms bond together and interact with light. The question is whether spin is a similarly emergent quantity in QFT, and my answer was no - the spin of a field is a fundamental part of its definition, not a feature which emerges on large scales due to subtleties of the underlying dynamics. $\endgroup$
    – J. Murray
    Commented May 18 at 16:39
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Quantum field theory is a mathematical formalism in terms of which one can model theories about the physical world. It is a very powerful formalism, allowing us to formulate the complicated fundamental dynamics in terms of the standard model. There are some fundamental aspects about the physical world that are already built into quantum field theory, such as special relativity, which has previously been established as being scientifically correct. So all theories formulated in terms of quantum field theory already satisfies this requirement.

Spin comes from physical observations. For instance, electric fields are vector fields because they produce a force on a charged particle pointing in a specific direction. The vectorial nature represents internal degrees of freedom that the field carries along with it at every point. Electrons also carry such internal degrees of freedom according to which they can be separated in a Stern-Gerlach experiment.

In quantum field theory these internal degrees of freedom can be represented in terms of the spin of the field. The spin is distinguished by the symmetry properties of the field (and is related to special relativity), which can be determined by experimental means.

So in the end, we use quantum field theory to provide the necessary formulation of the fields that we observe in the physical world together with all their properties including their spin. Then of course we go and test this theory that has been formulated in terms of quantum field theory. In the case of the standard model we saw that it works extremely well.

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