Talk:Spin (physics)/Archive 1

Archive 1

Archive 2

Why is spin so important?

A particle may exist in many different quantum states. However, it seems that spin is somehow more fundamental to various other properties of particles than other things. E.g., spin states add and give bosons or fermions which have dramatically different properties, but principle quantum numbers don't. Also, most fundamental particles and even larger quantum mechanical systems are grouped most readily by their spin states (spin 1/2, sprin 0, spin 1, spin 2 (graviton?)). Can someone explain this in the article? Ryanluck 17:11, 27 February 2006 (UTC)

Well, the spin quantum number of particles is more important because it is intrinsic, i.e. it is a property of an electron, whereas the principle quantum number depends on how the electron is moving relative to a nucleus. The same can be said of mass and charge. This fact is mentioned in the article already. -lethe ^{talk +} 17:21, 27 February 2006 (UTC)

Hmm, I see that this mention was added by you. Well I agree with your edits. -lethe ^{talk +} 17:32, 27 February 2006 (UTC)

Upon looking closer, I've decided that I actually do not agree with your edits. Specifically, the use of the word "analogous". I've reverted. -lethe ^{talk +} 18:00, 27 February 2006 (UTC)

spin's relevance

I was hoping to find a correlation/link between spin/rotation and e=mc2. The other correlation/link is whether sound waves are generated and measureable by particles and large scale structures. Does the mass of an object include these energy outputs?

Sound waves are not generated by particle spin. Sound is a phenomenon meaningless at much smaller scales than that of mid-sized groups (a few thousand) of particles. It involves the transmission of vibrations between particles. Unless you are dealing with densities on the order of those in a neutron star or higher, these generally be atoms or molecules (molecules being made up of 2 or more atoms, and atoms being made up of 4 or more elementary particles (3 quarks and an electron for basic hydrogen, more for more complex atoms (not counting gluons, etc.))). Particle spin is an intrinsic property of individual elementary particles, and as far as I know, the spin of one particle does not induce vibrations in its neighbors. 12.37.33.3 05:09, 18 March 2006 (UTC)

Exchange of identical particles

Mention must be made of the connection between spin and the sign change of wavefunctions upon an exchange of identical particles. References should include the Dirac chapter on permutation and the Feynman chapter on the Stern-Gerlach apparatus. —The preceding unsigned comment was added by 59.92.152.114 (talk) 17:12, 25 February 2007 (UTC).

SU(2) symmetry

Could someone please tell me what SU(2) symmetry is and its relation to spin? I've had an entire mathematical course on group theory and to them all SU(2) is is a group. I understand you can generate any eigenstate of spin with an SU(2) transformation (a rotation in spin space), but what does it mean to have SU(2) symmetry?--Loodog 15:17, 20 August 2007 (UTC)

An alternative calculation of spin

We have all noticed that spin is described as being a multiple of hbar/2. I thought that it would be better to set this value to a constant giving,

hdot = hbar/2 = 5.2728584118222738157569629987­554e-35 J.s

But now the equations for spin did not work with hdot, so I had to correct them.

Here are the corrected equations,

|sv| = sqrt(s(s + 2)) * hdot

and

Sz = ms.hdot

where,

sv is the quantized spin vector,

|sv| is the norm of the spin vector,

s is the spin quantum number, which can be any non negative integer,

Sz is the spin z projection,

ms is the secondary spin quantum number, ranging from -s to +s in steps of two integers

For spin 1 particles this gives:

|sv| = sqrt(3).hdot and Sz = -hdot, +hdot

For spin 2 particles this gives:

|sv| = sqrt(8).hdot and Sz = -hdot, 0, +hdot

Now that the spin equations have been corrected, the definitions for fermions and bosons are incorrect, and must be redefined as follows.

Fermions are particles that that have an odd integer spin.

Bosons are particles that have an even integer spin.

Would these redefinitions have any other effects on the Standard Model?

Can these redefinitions explain any currently unexplained phenomena?

Are there any experiments that could confirm or refute these claims?

I would like eveyone to have a good think about this, and give me your objections to it, or even data to support it.

Giving a name to $\hbar /2$ has absolutely no affect on anything physical, just the way we describe it. Thus there are no experiments that could confirm or refute your claims as you claim nothing different from the status quo. However, you have made a massive change to notational conventions that would require most of the world's physics books to be rewritten, and therefore I think it is a bad idea. Uberdude85 01:40, 21 November 2006 (UTC)

I had a radical thought I'd just like to put out there for Visionaries. What if gravity is just the combined effect(similar to centrifugal force) of the Quantum Spin of large masses of matter Dark or Common. And what if a series of Quantum Gyroscopes could be created to Nullify this effect.... —Preceding unsigned comment added by 99.252.195.32 (talk) 15:42, 2 January 2008 (UTC)

(In)accurate prediction of the electron g-factor

In the section "Spin and magnetic moments" there is a discussion of the prediction of the electron g-factor, and it's experimentally determined value is listed as "2.0023193043768(86)." The whole number portion and fracational portion of this value are then considered. It is stated that the whole number portion, two (2), arises from "the Dirac equation." It is further stated that there is a "correction of 0.00231456893..." which arises from its own and surrounding electromagnetic fields.

I might be confused but when I do the math that is implied in that section:

  2.0023193043768(86) − 2 - 0.00231456893... = 0.0000047354435(53)

  2.0023193043768(86)≠ 2.00231456893...

Where did this correction of "0.00231456893..." come from? To what number is the correction made? Maybe it would be better to state:

  the predicted value of the electron g-factor,
  the experimentally determined value of the electron g-factor,
  then expand upon or reference a discussion of any correction(s)

Cbobo01 (talk) 02:21, 18 January 2008 (UTC)

Applications

This section doesn't really explain what a direct application is, and what an indirect application is. — Werdna • talk 12:21, 7 July 2008 (UTC) Im interested in the logic of spin numbers between successive accumulated nucleons. And in particular the elements EE4Be8 with Spin=0, EO4Be9 with one additional (Spin 1/2) neutron and Spin=-3/2, and OO5B10 with an additional (Spin 1/2) proton and Spin=+3. The question is 1: whether there is any logical inference I can make about this accumulation process based on an indication of their given "Spin" values and 2: how were these values determined in the first place.WFPMWFPM (talk) 19:46, 10 September 2008 (UTC)

Half-integers?

This page says the spin is a "half-integer (0, 1/2, 1, 3/2, ...)". But the page about Half-integers says integers are no half-integers, i.e. only 1/2, 3/2, ... Which one is it? If both pages are right, at least the link is misleading. Or "half-integer" has two meanings that should be distinguished.

The listed numbers are correct: N/2. Integer spin means boson and half-odd-integer spin means fermion. I'm not sure about the meanings of half-integer. --MarSch 12:53, 3 November 2005 (UTC)

I think you have been confused by the slightly unclear phrasing; it says "integer or half-integer (0, 1/2, 1, 3/2, 2, etc.)". 0, 1 and 2 are integers; 1/2, 3/2, 5/2 are half integers. So both pages are correct. Uberdude85 01:40, 21 November 2006 (UTC)

The confusing concept of half-odd-integer spin is only historically justified. The quantum of angular momentum was believed to be h/2π until it was discovered that the spin angular momentum of the electron is h/4π. When measured in units of h/2π the spin angular momentum is either integer or half-integer, but when measured in units of h/4π the spin angular momentum is integer, and fermions have odd spin angular momentum and bosons have even spin angular momentum. Using h/4π as the unit is easier to understand. Bo Jacoby (talk) 06:52, 11 June 2009 (UTC).

Theoretical value missing

"One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor". It does not say which number this factor is!! The accuracy comes with just comparing the experimental value with the theoretical value...--190.190.87.136 (talk) 17:56, 30 July 2009 (UTC)

I just rewrote the section "Measurement of the spin along an arbitrary axis"

It was unnecessarily complicated in that it brought in spherical coordinates(!!) which have nothing to do with spin and vastly complicate calculations in this case. I also provided the general, normalized solution for spin-1/2 in an arbitrary direction.

I suspect that it is possible to write down a general solution for higher spin in arbitrary directions, given that an appropriate representation of SU(2) is provided. If someone thinks this is worth writing up, give me a ring at carl@brannenworks.com and I'll look around for a reference.

---

—Preceding unsigned comment added by 24.19.189.240 (talk) 07:32, 3 March 2008 (UTC)

"A normalized spinor for spin-1/2 in the $(u_{x},u_{y},u_{z})$ direction (which works for all spin states except spin down where it will give 0/0), is:

${\frac {1}{\sqrt {2+2u_{z}}}}{\begin{bmatrix}1+u_{z}\\u_{x}+iu_{y}\end{bmatrix}}.$

The above spinor is obtained in the usual way by diagonalizing the $\sigma _{u}$ matrix and finding the eigenstates corresponding to the eigenvalues."

My complaint here is that surely there should exist two spinors in this direction?144.173.228.8 (talk) 14:56, 23 November 2009 (UTC)

Under the Measurement of the spin along an arbitrary axis subsection the article currently reads

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let $u=(u_{x},u_{y},u_{z})$ be an arbitrary unit vector. Then the operator for spin in this direction is simply $\sigma _{u}=\hbar (u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z})/2$.

The operator $\sigma _{u}$ has eigenvalues of $\pm \hbar /2$, just like the usual Pauli spin matrices.

I'm not really into quantum physics so I may be totally wrong, but shouldn't this rather read something like this:

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices, simply by forming a new matrix $S_{u}={\hbar \over 2}\sigma _{u}$, where $\sigma _{u}=u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}$. Just like the usual Pauli spin matrices, the latter has eigenvalues of $\pm \hbar /2$, ...

Cool dude ragnar (talk) 14:41, 5 September 2008 (UTC)

Edit Cool dude ragnar (talk) 23:51, 5 September 2008 (UTC) — Preceding unsigned comment added by 5.2.200.163 (talk)

Merger proposal

Spin is a special case of angular momentum. The angular momentum article has a good section on quantum mechanical angular momentum. There is a lot of overlap between the two articles. This spin article does not emphasize that spin is angular momentum measured in units of h-bar. Bo Jacoby (talk) 16:33, 17 May 2010 (UTC).

Oppose. Spin is related to quantum mechanics (micro world). The Angular momentum article is mainly focused on classical mechanics (macro world, particles in the classical mechanical sense, rigid bodies, and deformable bodies). Even though there is mentioned of the quantum mechanics understanding of angular momentum, it does not mean the articles are about the same topic.

I would even suggest to separate the Angular momentum article into two articles:

1. Angular momentum (mechanics)
2. Angular momentum (quantum mechanics)

Perhaps then you can merge the spin (physics) article into the Angular momentum (quantum mechanics) article. sanpaz (talk) 17:48, 17 May 2010 (UTC)

Historically the quantum mechanical angular momentum seemed to differ from the classical mechanical angular momentum, but logically there is only one type of angular momentum, and the classical mechanical angular momentum is but a limiting case of the general quantum mechanical angular momentum. That's why I propose a merger. Bo Jacoby (talk) 22:19, 19 May 2010 (UTC).

Oppose. Quantum spin is not about angular momentum. In other words quantum spin is not about an electron rotating. If you disagree, show me a reference. Daniel.Cardenas (talk) 00:09, 20 May 2010 (UTC)

I disagree. Reference: The Feynman Lectures on Physics, vol 3, chapter 18: Angular Momentum. Spin is angular momentum, even if it is not about the particle rotating. Bo Jacoby (talk) 23:12, 31 May 2010 (UTC).

Oppose. 1) Unmanageable quantity of information. 2) "classical mechanical angular momentum is but a limiting case of the general quantum mechanical angular momentum", this is only true if "classical" refers to theories of the electron - my understanding is that "classical" is pre-electron (and friends). Certainly, electrons may behave in some experiments as if they were prespun BUT this does not mean they were spinning in the same way as we consider a ball that rotates. This is because when a ball rotates it is the aligned motion of a lattice, when an electron or a molecule has atomic spin it includes other traditionally non-kinetic factors. Furthermore, from what I have seen nobody - in literature- has been able to reconcile the two forms of spin. As these guys (http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html) write "The property called electron spin must be considered to be a quantum concept without detailed classical analogy." 3) Lastly, if the two are the same or different is a contemporary issue, by merging an unorthadox, fringe, side is taken in this debate at no benifit to the reader --Frozenport (talk) 10:12, 31 May 2010 (UTC) 4)"logically there is only one type of angular momentum" -> unless the word refers to two different things. --Frozenport (talk) 10:18, 31 May 2010 (UTC)

Note that a flow of circularily polarized spin 1 photons provides classical torque. So the spin implies classical angular momentum, although it is quantized. Bo Jacoby (talk) 23:12, 31 May 2010 (UTC).

Oppose. Each subject is vast in scope. To unify the two articles and cover the appropriate material would make the article far too long. The split of the material between the two articles is fine, although duplicate information could be deleted and linked to instead. I think Sanpaz's idea is worthy of further discussion. JHobbs103 (talk) 15:33, 2 June 2010 (UTC)

Oppose. Yes, modern physics defines "angular momentum" as "operator that generates of infinitesimal rotations", and then spin is a type of angular momentum. But there is a lot to say about angular momentum that is unrelated to spin (e.g. classical angular momentum, orbital angular momentum, etc.), and a lot to say about spin that is unrelated or marginally related to the general concept of angular momentum (e.g. spin-statistics theorem, spin of different nuclei and how that relates to NMR, etc.). --Steve (talk) 23:26, 9 June 2010 (UTC)

Oppose (Shinobeee, July 1 2010). In quantum mechanics spin and angular momentum are mathematically related but conceptually quite different. Spin is not (really) attributed to the rotational motion of a particles, it is more of an inherent magnetic property. Angular momentum, while covering a wide range of subjects, does cover instances that are directly attributed to motion such as the angular quantum numbers of the hydrogen atom, and the particle on a ring example. In classical mechanics, angular momentum also has its own variety of subjects. While all quantum mechanical and classic mechanical angular momentum are under the same umbrella, merging them all into 1 article might be too crowded for a Wikipedia article. Shinobeee 17:29, 1 July 2010 (UTC/GMT +9 hours) 08:29, 1 July 2010 (UTC)

massless particle's spin

I'm no expert, but a massless particle's spin "behaves" differently. It can only have it oriented along its momentum or against it. A photon can only have m=+1 or m=-1, not m=0, unlike a massive particle with spin 1. I think it should be mentioned with reference to helicity and chirality.--132.64.102.160 (talk) 15:19, 11 July 2010 (UTC)

Units more obvious in header

Could it be made easy to see in the intro what units s has? -Craig Pemberton 22:55, 7 August 2010 (UTC)

Changed it, is that OK? --Steve (talk) 02:42, 8 August 2010 (UTC)

Awesome! Thanks. -Craig Pemberton 05:03, 8 August 2010 (UTC)

photon spin multiplicity

Photons are in doublet state contrary to the rule 2s+1 for the spin multiplicity because s=1 for photons. Please clarify and update.

Galium111 (talk) 11:31, 9 October 2010 (UTC)

various issues (including spin in irreps of SL(2,C)

There are many points that could use addressing.

1. Spin arises in the context of irreducible representations of the Poincare covering group, or SL(2,C). You can't understand spin without equation (I.3.23) of

http://reslib.com/book/Local_quantum_physics__fields__particles__algebras#44

and the discussion that precedes it (or the equivalent, which can be found in Araki's text, also in reslib). You should also consider the special cases where α is a pure rotation and where α is a pure boost. In the case of a pure rotation, you'll find that the result is a combination of (commuting) orbital and spin rotations:

  |p>⊗|s> --> |Λ(α)p>⊗|s>    (orbital rotation)
  |p>⊗|s> --> |p>⊗D(α)|s>    (spin rotation)

Here |p> is an improper momentum eigenstate in Haag's ℋc, and |s> is any element of the little Hilbert space. In the case of a pure boost, you'll find that γ(α,p) = 1, so that pure boosts don't alter spin content.

2. The subject "Spin and Lorentz transformations" is equation (I.3.23) mentioned above.

3. The notion that spin is ineffable is mistaken. In the special case of α a rotation and p being at rest, we see that spin is simply the angular momentum of a state at rest. And the spin content is unaffected by a pure boost. This defines the nature of spin.

4. It would be nice to write down the spin operator in closed form using elements in the Lie algebra of the SL(2,C) representation. I personally don't know how to do that.

5. You might want to mention how the Pauli-Lubanski vector W fits in. Again, the context is irreducible (massive!) representations of SL(2,C). Here are two interesting points about W:

a. W⋅W = S⋅S. (I set mass = 1 for simplicity.)

b. For all p on-shell, there exists a 4-vector n(p) such that n(p)⋅W |p>⊗|s> = S₃ |p>⊗|s>.

6. You might want to correct the following misconception about spin, ie, (3-155) of:

 http://reslib.com/book/Quantum_field_theory__Claude_Itzykson_#164

This is a flat-out blunder. The energy-momentum tensor isn't even symmetric. With the correct energy-momentum tensor (eg, the expression obtained in Weinberg's gravity textbook), J is given by the

• first two terms* of the second equation. The intrinsic spin density

is a misconception.

Gregweeks (talk) 03:28, 28 November 2011 (UTC)

Spin need not be so technical.

I sympathize with posters that say that the page is too complicated. It could begin with content accessible to most readers and get more jargon-esque from there.

One source of confusion is the assertion -- erroneous, I would argue -- that the spin and magnetic moment are ineffable. The issue is contentious, so I'll simply state my version of the facts and be prepared to demonstrate them.

1. In the context of (massive) one-particle Hilbert spaces, spin is the angular momentum of a particle at rest and is not affected by a boost.

2. In the context of free field theory -- that of the Dirac field, let us say -- the expression for the angular momentum in terms of an angular momentum density (based on the energy-momentum tensor) is identical to its classical field counterpart.

Likewise, the expression for the magnetic moment in terms of an integral involving the current J is identical to its classical field counterpart.

The fact that g=2 can be explained by noting that energy-momentum flow T and charge flow J are not be locked together as they would be in a classical charged fluid.

Gregweeks (talk) 03:49, 28 November 2011 (UTC)

Ohanian's paper yields a missing viewpoing

[My apologies if my comments generate more heat than light. I'll try not to comment again.]

A comment on:

   This modern definition of angular momentum is not the same as the
   historical classical mechanics definition, L = r × p.

Yes and no.

YES in the context of the Hilbert space of a single particle, which is where a clear understanding of spin was born. Even in that context, though, there is a tidy distinction between the orbital and spin parts of the angular momentum. But there is no mention of space, so there is no r.

NO when you introduce quantum fields. The whole of the angular momentum, orbital and spin, is found in the integral

   ∫d³x r x T⁰⁻

where T⁰⁻ represents the momentum _density_.

Here is my reference:

---

What is spin? by Hans C. Ohanian

American Journal of Physics, June 1986, v 54, Issue 6, pp 500-505 Issue Date: June 1986

From the abstract:

According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ``internal—they are not associated with the internal structure of the electron, but rather with the structure of its wave field.

---

Alas, Ohanian's article treated the Dirac equation in its 1928 interpretation, rather than as a quantum field. But his arguments go through for the (free) quantum Dirac field as well. But I have no reference for this, I'm sorry to say.

Anyway, I don't see the Ohanian viewpoint in the Wikipedia article.

---

Another viewpoint missing from the Wikipedia article is relevant to the section "Composite particles". My reference is the paragraph that you find yourself in the middle of as you read this:

http://reslib.com/book/PCT__spin_and_statistics_and_all_that#32

In particular, a deuteron is an "elementary system", and the Wignerian analysis of its spin is the same as the Wignerian analysis of the spin of a (free) electron. The idea of an _additional_ analysis in terms of components is not needed, although it may be interesting.

Gregweeks (talk) 15:49, 28 November 2011 (UTC)

Bose

Take a look here http://ysfine.com/einstein/bose.html It's complementary, and quite interesting.

Yoron. 90.132.22.104 (talk) 05:51, 28 October 2011 (UTC)

Please also see Satyendra Nath Bose, His Life and Times, Selected Works (with commentary), Edited by Kameshwari C. Wali, World Scientific, 2009 — Preceding unsigned comment added by 122.183.243.98 (talk) 04:50, 22 December 2011 (UTC)

A couple of typos in the section subtitled 'Measuring spin along an arbitrary axis'?

An arbitrary unit vector is introduced here as (u_x, u_y, u_x), and this form is repeated a few lines below. Shouldn't the third component be u_z?

Roscalen (talk) 18:35, 15 January 2012 (UTC)

Definition is not clear in the Intro

1. That the spin is a kind of angular momentum that doesn't have a counterpart in classical mechanics is clear. That it is a vector quantity is also understood. But the intro doesn't clearly say what exactly is the spin quantum number. Those who have mastered the subject, please give a definition spanning 3 or 4 sentences which includes the physical significance of the term.
2. The use of the word "intrinsic" is confusing. What is intrinsic and extrinsic with regard to angular momentum?

Aravind V R (talk) 11:48, 30 October 2012 (UTC)

I think "intrinsic" is being used to mean both "not associated with motion through space" and "an unchanging part of the identity of the particle, like its mass or charge". I agree that "intrinsic" should be explained in the intro...

For "what is the spin quantum number", I'll try...

A particle's spin is characterized by two things: A spin quantum number and a spin state. The quantum number, which is a nonnegative half-integer (0 or 1/2 or 1 or 3/2 etc.), is a characteristic of each particle; for example, every electron is spin-1/2, while every gluon is spin-1. The particle's spin state is not fixed, but can be changed by magnetic fields and other external forces; yet the range of possible spin states depends on the spin quantum number. For example, a spin-0 particle has only one possible spin state, but a spin-1/2 particle has a two-dimensional space of possible spin states, and a spin-1 particle has a three-dimensional space of possible spin states, and so on.

Any good? --Steve (talk) 13:59, 30 October 2012 (UTC)

I'm concerned about the statement that spin is a vector in the introduction. I'm currently reading "The Story of Spin" by Shinichiro Tomononaga (or Sin-Itiro Tomonaga) who shared the Nobel Prize with Richard Feynman. Chapter 7 of that book is entitled "The quantity which is neither vector nor tensor" A vector is not actually something with a magnitude and direction. Rather, a vector is a quantity which transforms in a certain way under rotation -- basically, the velocity of a particle doesn't change in some sense no matter who is looking at it, though one observer may describe it as moving to the left while another as moving to the right. But spin does not behave this way (except for spin-1). I've modified the introduction to reflect this David s graff (talk) 15:17, 14 February 2013 (UTC)

rewrite of Fermion and Boson section

I replaced/ greatly modified the old "elementary particles" and "composite particles" sections. It seemed that a discussion of spin should not start off with the distinction of elementary particles. I replaced it with a section on Fermions and Bosons, which is much more central to the idea of spin. I've summarized some of the composite and elementary particles in this section.

I also renamed the spin direction chapter by adding scare quotes as spin "direction" to indicate that spin is not really a vector, the concept of it having a direction is only an approximation useful in some contexts. David s graff (talk) 16:16, 14 February 2013 (UTC)

Revert and clarify is spin a vector in the introduction

An anonyous user reverted a couple of sentences in the introduction. I have undone his revert with a detailed discussion of why on my talk page where he justified his edit. User_talk:David_s_graff#on_spin. In particular, I think that most of my sentence is completely in accord with other sentences in this article, and merely rephrases them or summarizes them as is appropriate in an introduction. I modified one part of my sentence which I think was poorly written and may have led to the confusion of the editor. I think that we all agree that in many ways spin is like a vector, but there is a difference in how spin 1/2 particles behave under a geometric rotation, and the question is how best to word this idea in a sentence or two in the introduction to an article. I think that the more detailed descriptions of this later in the article at Spin_(physics)#Spin_vector and in the spinor article are fine, and the question is how to quickly and accurately summarize this complex idea. David s graff (talk) 15:45, 18 April 2014 (UTC)

I have reverted the edit to remove the false information reintroduced. The only confusion here is due to the above user: he fallaciously identifies spin-1/2 particle states with the actual spin of such a particle. As I took pains to expalain, spin is a vector. Specifically, it is an axial vector, but I won't argue for that distinction. One can observe a meticulous development of the theory of angular momentum from its infinitesimal generators in, e.g., Weinberg Lectures on Quantum Mechanics, wherein spin is described as both "a three-vector" and, in accordance with its satisfaction of the tensor transformation law, "a tensor". Certainly, the world's reigning theorist won't be discounted as easily as "an anonymous user" who wasted plenty of his time attempting to kindly educate the above user as to his mistake. (One might also confer more immediately with Fitzpatrick here, one among many web resources that very clearly define spin as a vector.)

Weinberg and Fitzgerald call spin a vector because spin is a vector. Spin is not a spin state. Spin is an operator: an observable. There is no "definite spin" associated with any spin state due to the conjugate indeterminacy of its components (one can only at any given time affirmatively determine one of its three axial components); nonetheless, it can be instructive to investigate the behaviour of one of the spin operator components (or rather its expectation value) under the application of the rotation operator. One finds that the expectation value of the operator behaves 'exactly as does any given Euclidean three-vector under rotation (it is only under parity transformations that the aberrant nature of the axial vector is appreciable). The spin state (described by a spinor) is the only entity that behaves as described by the misinformative reversion.

A spinor represents a spin-1/2 state and is not the same as spin. The edit was, if anything, more confused than the initial mistake; to say that "a spin has the sign of its wavefunction reversed" is a thorough confusion of two very different concepts. Spins, to wit, do not have wavefunctions. A spin is not a quantum state; it is an observable of a quantum state, and that observable is a vector...strictly a vector. Finally, as described before, it is absolutely the case that a spin operator, when rotated 360 degrees, maintains its sign.

To recap: it is incorrect to say all of 1) A spin is not strictly a vector (when contrasting it with anything other than an axial vector"; 2) A spin is described by a spinor; 3) A spin has a quantum wavefunction; and 4) A spin reverses sign when rotated by 360 degrees.

I do not personally agree that the introduction of this article, which is on spin in general, needs any reference to the behaviour of spinors; the focus of this article is on the quantity spin itself and it suffices to treat the geometric properties of a spinor in its respective article. Nonetheless, I have no quarrel with overinformation, only with misinformation. Should an editor desire to reintroduce the behaviour of spin states into the introductory portion of the article, he should do so in a way that neither conflicts with reality nor perpetuates the very confusion to which the editor above has twice succumbed. In my opinion, any introductory discussion of the behaviour of spinors under rotation should be kept separate from the paragraph about the vectorial nature of spin in order to avoid a confusion of the concepts in the minds of the uninitiated. 72.179.38.56 (talk) 08:02, 22 April 2014 (UTC)

With a minor nitpick on the description of spin as a vector, I agree with what 72.179.38.56 is saying technically. Spin, like angular momentum, is described most accurately as a bivector (in any number of dimensions), though in three dimensions, a bivector is typically been represented by its Hodge dual, this being an axial vector. —Quondum 04:45, 23 April 2014 (UTC)

purely quantum phenomenon

I'd like to dispute the claim that spin is a purely quantum phenomenon. For example, the classical electromagnetic field has intrinsic angular momentum. It is only half integer spin that is purely quantum mechanical. -Lethe | Talk 01:59, 9 August 2005 (UTC)

It's not just that. If you model the electron as a classical spinning object, then, given the experimental upper limit on the electron radius, the surface of the electron would have to be moving faster than light. -- CYD 02:41, 9 August 2005 (UTC)

Sure, so intrinsic angular momentum isn't due to a rotating sphere. It's intrinsic, just like the charge. but what does QM have to do with it? -Lethe | Talk 03:57, 9 August 2005 (UTC)

from Baez on spr:

Spin angular momentum is the angular momentum intrinsic to a particle. Most people are unfamiliar with spinning point particles in the context of classical mechanics, so they think of spin as a purely quantum business. This isn't quite right: there's a perfectly nice notion of a spinning point particle in classical mechanics. However, nobody thought of it until after quantum mechanics came along.

I think that you could just write a classical Hamiltonian like

$H={\frac {p^{2}}{2m}}+\mathbf {S} \cdot \mathbf {B}$

or something, and do classical mechanics with it. no? -Lethe | Talk 09:46, 9 August 2005 (UTC)

Wait. Why would the speed of rotation have to be greater than that of light under relativity? I can see how in Newtonian mechanics a rotating body might need to have a rotational speed greater than that of light, but FTL isn't forbidden in Newtonian mechanics. Under relativistic mechanics FTL is forbidden, but mass increases with velocity (and thus momentum increases faster than predicted by Newtonian mechanics), so no object should have to have an FTL velocity of rotation to have a given amount of angular momentum. 12.37.33.3 04:49, 18 March 2006 (UTC)

Well, I don't like to say "mass increases with velocity". In modern usage, "mass" always refers to invariant mass. Nevertheless, you may be right; the formula l=Iω is a nonrelativistic formula, and with a relativistic version of this formula, I guess we may find that we can have a very high angular momentum without rω being greater than c. But what is the relativistic formula for angular momentum of a rigid body? I don't know. We have to be a little careful, because there is actually no such thing as a rigid body in relativistic mechanics, but still we should be able to come up with a relativistic version of that formula. I'm going to think about that. It's a good point. -lethe ^{talk +} 05:08, 18 March 2006 (UTC)

OK, relativistic mass increases with velocity. This, I believe, is because of Energy/Mass equivalence (in fact, I believe that all of relativity can be derived from adding En/Ma equivalence to Newtonian mechanics, though I don't have the math skill to prove it, my knowledge being more of the theory). As I understand it, kinetic energy has mass, so it "weighs down" a particle. To get to a certain speed, you need to add a certain amount of energy, to go beyond that speed, you need to overcome the inertia of the rest mass *plus* the inertia of the kinetic energy already on the object. The speed of light is the point where the KE of an object is infinitely greater than its rest mass (this is why it is impossible for a massive object to reach the speed of light. It would take infinite energy). Photons travel at exactly c because any positive, non-zero, real number is infinitely greater than 0 (the rest mass of a photon). Another way of looking at it might be to say that the speed of a particle is a function of what fraction of its mass is KE, with 50% KE giving a speed of ~.86c and 100% KE giving a speed of c. Anyways, since one of the determining factors of momentum is mass, and since angular momentum is a kind of momentum, a rotational speed greater than c should not be required to give ANY finite, positive, real angular momentum. 12.37.33.3 05:35, 18 March 2006 (UTC)

Well, I'm sick to death of conversations about relativistic mass versus invariant mass. Let me just tell you that no practicing physicist uses a concept of relativistic mass. Please read this. Mass does not increase with velocity. Nevertheless, it may be the case that angular momentum increases without bound as radial velocity of rotation approaches c, so your question might still be a good one. But please, I don't want to talk any more about what relativistic mass means. -lethe ^{talk +} 06:00, 18 March 2006 (UTC)

Sorry. I'm a recent high school graduate, not a proffesional physicist (and as my new username suggests, I'm likely to go into a totally different field). Was my error in not properly specifying invariant vs. relativistic mass, or in even treading on the topic of relativistic mass in the first place? If it was not being specific enough, is there a way of specifying relativistic mass in less than 6 syllables? If it was in treading on the topic at all I'll shut up. Linguofreak 06:27, 18 March 2006 (UTC)

You didn't make an error. Sorry if I was terse before, but talking about relativistic mass makes me edgy; I've been involved in countless battles about that matter. The point is, relativistic mass, while not wrong, is rather inconvenient. It's not invariant, meaning it's observer dependent, two different people see different relativistic masses if they're moving. This isn't so bad, the same holds true of energy and momentum, but we also need to have a word for invariant quantities. Rest mass is a bad word, since some things are never at rest (photons). And "relativistic mass" is already redundant, since it's the same as energy. So let's use "mass" for rest mass, and "energy" for relativistic mass. Relativistic mass isn't really a mass anyway, since it's only inertia in one direction. Transverse inertia (inertia in a direction perpendicular to the direction of motion) is always equal to rest mass.

The way to specify "relativistic mass" in fewer than 6 syllables is to say "energy". Energy = mγc². If you prefer units of mass, then say E/c². And m is always rest mass.

But this issue is purely linguistic. It doesn't affect the physics, and you raised a good point. Your point was that moment of inertia should increase with the rate of rotation (since mass does, and moment of inertia depends on mass). The better way of saying that would just be that the linear formula for angular momentum must be corrected at relativistic speeds. I think it's a good point, one I hadn't thought of, but it's probably true, though I've never studied relativistic tops, so I don't know the answer. I'm researching it now, and as soon as I find something, I'll post back. Anyway, thank you for raising an interesting question. -lethe ^{talk +} 06:59, 18 March 2006 (UTC)

Ok, yeah, "energy" works. I'll have to try and remember to use E or E/c². It's kinda weird though, because since I don't have any formal training in relativistic physics I'm used to calling it "energy" when it exhibits the classical charachteristics of energy, and mass when it exhibits the classical characteristics of mass (such as inertia).

I do have a couple more questions. Wouldn't a point mass elementary particle (as modern physics assumes), be a black hole? I think I've heard something about the ratio of charge to mass for such particles excluding them being black holes, but what about neutral particles? But a sphere of non-zero size doesn't really fit either because of the point you made earlier of there being no such thing as a truly rigid body. So neither assumption really seems to work.

No, a point particle need not be a black hole. For example, electrons are not black holes. I suppose you think that should happen because the classical energy density (which goes as 1/r²) becomes infinite as the radius goes to zero. This is true, but it's not the whole story. Mostly, that is a relic of the classical point particle theory. The problem persists in a different form in the quantum theory, where renormalization is the answer. We can't answer this question with a point particle anyway, because point particles can't rotate. We have to answer it with a body of finite size. As we both agree, rigid bodies are tricky. Our choices are to either figure out how to deal with rigid bodies (and there are ways), or else to try to deal with nonrigid bodies. I'm looking at both ways to answer the question. A third way is to just assume perfect rigidness, but this is unphysical at relativistic velocities (a good approximation at slow speeds though). -lethe ^{talk +} 23:05, 26 March 2006 (UTC)

A particle isn't a point. It's a distribution that temporarily becomes point-like when observed. The "radius" a particle is observed to have is the radius of the region that queried it; for example, the radius of a pixel on a detection screen. A particle can also have an unobserved radius if it's trapped in a potential well; for example, an electron in an atom.

As a distribution, a particle with spin has a vorticity, and thus an angular momentum. There are only two weird things about this: its magnitude is a non-commuting operator instead of a numeric value, and it's impossible to observe. The common practice of going beyond this, to stress tensors left un-symmetrized, azimuthal dimensions, and eventually a host of mythical partner-particles, is totally uncalled-for.

(Perhaps this is the fault of string theorists. If they promote the "particle is a point" meme, then they can advertise themselves by saying "We disagree. We say particles are strings." Of course, there isn't any kind of observation that makes a particle string-like!)

166.137.101.174 (talk) 19:34, 26 June 2014 (UTC)Collin237

While you're researching and calculating, could you tell me if relativity can be derived by adding Energy/mass equivalency to Newtonian mechanics, or am I missing something?

Linguofreak 18:29, 18 March 2006 (UTC)

No, relativity cannot be derived by adding mass energy equivalence to Newtonian mechanics. The finiteness of the speed of light is an experimental factor which has to be put in by hand. If you don't add that to your theory, you won't get relativity. However, it works nicely the other way; the equivalence of mass and energy can be derived from the finiteness of the speed of light (along with the conservation of momentum), so that is definitely the way you should try to understand it. -lethe ^{talk +} 22:53, 26 March 2006 (UTC)

I was going to write something else, but then I found in the article on Momentum what I was looking to show. Under relativistic mechanics $\mathbf {p} =\gamma m\mathbf {v}$. So when v=c, and gamma is infinite, shouldn't momentum be infinite? (forgot to sign... again...) Linguofreak 21:27, 26 March 2006 (UTC)

Right, this was your original point: momentum and energy become unbounded as the speed approaches c, which implies that for a sufficiently large velocity less than c, the particle can attain any amount of energy or momentum. It's natural to assume that the same phenomenon occurs with angular momentum. If that is the case, then the statement "a rotating body with the angular momentum of an electron must be rotating with a radial speed greater than c" should be incorrect; you should be able to attain any angular momentum before your radial velocity surpasses c. It sounds reasonable. The question is, when the rω goes to c, does the angular momentum L become infinite? To answer that, we need a formula for angular momentum. One such formula is L=mγrv, and this expression certainly does become infinite. However, that formula is only valid for a point particle moving in a straight line. For a rotating disk, the answer becomes much more subtle. I've already done a calculation that shows that a rigid rotating disk has an angular momentum that stays finite, but rigid bodies are quite tricky in relativity. I'll say more about this later. Finally, let me just say that I think that while there are certainly difficulties with the classical theories of point particles, I don't think it's as bad as Xerxes makes it seem below. -lethe ^{talk +} 22:49, 26 March 2006 (UTC)

Assorted answers: A classical point mass particle would be a black hole. Ignoring quantum mechanics at that scale is foolish, however. Relativity is a broad unification of scalar and vector concepts into 4-vectors with Minkowski metric. Mass-energy-momentum is just one such unification. Infinite momentum is the equation's way of telling you "This is impossible." -- Xerxes 21:37, 26 March 2006 (UTC)

Did you mean finiteness of the speed of light, or constancy in all reference frames?

Finiteness of lightspeed without constancy does not give relativistic theory. However, if you knew that light carried energy, and were to somehow observe that E and m were related by a finite, non-zero ratio without knowing anything about the constancy of the speed of light or its finiteness, you should be able to deduce both, along with all of relativity from this. The thing is that it's easier to measure the speed of light directly (and find that it is finite and the same in all reference frames) than to figure out mass/energy equivalency directly.

This point is actually worth emphasizing. It is the finiteness of the speed of light, not the invariance, which should be viewed as the fundamental concept. The invariance of the speed of light can be derived mathematically. However, the math doesn't tell you whether this invariant speed is finite or infinite, that's something you have to choose for yourself, based on physical experiment. If you choose infinite invariant speed, you get Newtonian mechanics, while if you choose finite invariant speed, you get relativity. This point is not well-known, it's not taught in most of the books, I suppose because Einstein himself chose to adopt invariance of the speed of light as a postulate. But that's not fundamental; it can be derived. To sum up: yes indeed, finiteness alone without invariance of the speed of light alone is enough to get you the entire theory of special relativity. (I guess I should add a proviso here. When I say "speed of light", I don't necessarily mean the speed at which light travels. I just mean some universal invariant speed. Whether or not electromagnetism actually travels at that speed is a question of the details of electromagnetics, independent of the geometry of spacetime. It would be better if that speed had a more independent name, like say "the universal invariant speed" or something. Who knows? Maybe one day we'll find out the photon has a nonzero mass!)

Now about your proposal to adopt the equivalence of mass and energy as fundamental instead. Despite what I said earlier, I think probably this can work. The thing is, when you use the invariance/finiteness of c, you can use a purely geometric argument about distances between points. Flash a light inside a rocket, that sort of thing. If you want to argue with energy and mass instead, then you have to move to a less geometric picture, using variables that are a bit more abstract (energy versus distance). Also, I think you'll have to use the conservation of energy and/or the conservation of momentum. The argument is probably much less obvious, but maybe it still works. We should try it.

First question about your rigid rotating disk: Have you tried it for cases where the rate of rotation is greater than c? I have a hunch that it may work at c, but not faster. But as you said, relativity doesn't like rigid bodies. What about if you took it to be a rotation in a non-rigid body? The whole thing about point particles was more of an aside, because I didn't think a point particle or a rigid sphere would work. But now that I think you've explained quantum waveforms to me in a way that I can grasp on the uncertainty talk page, they seem like a viable alternative. I think I saw that you had left another response to me there, so I'll go check it out. Linguofreak 03:08, 27 March 2006 (UTC)

Yes, for the rigid rotating disk, the angular momentum is finite for all finite rω, below c and beyond it. But like I've said, rigid objects at relativistic speeds are quite unphysical. At the very least, there would need to be an enormous amount of internal energy holding the atoms together, to counteract the centrifugal force, and this ought to contribute to the angular momentum, so I don't put too much stock in the calculation. Now, Born provided a way to approximate a sort of relativistic form of rigidity (it's not a true rigidity, but rather some constraint about internal tensions), but I don't really know how to use that method. So I also have this citation for an old paper by Regge where he does such a calculation for a rigid sphere, but I haven't tracked down that paper yet. The other options are, as you mention, to consider non-rigid objects. Those are more physical anyway, but of course also more complicated. For that approach, there are books about relativistic neutron stars and such. A nice astrophysics book ought to have something about the angular momentum of a rapidly rotating star. The extreme example of this is the rotating black hole. A rotating black hole is described by the Kerr metric, and there is a formula in the classic GR textbook by Misner, Wheeler and Thorne (known affectionately as "the phonebook") which says that the angular velocity looks like this:

$\omega ={\frac {a}{a^{2}+r^{2}}}$

with r the horizon radius given by

$r=m+{\sqrt {m^{2}-a^{2}}}$

and the angular momentum given by L=ma. This isn't too hard to solve for L, and you get L=2m²β, where β is the ratio of the radial velocity at the horizon radius to the speed of light. Clearly this expression is finite for all β, so once again, it looks like L will not diverge. However, it's still not entirely satisfactory to me, because well, there is no mass at the horizon of a black hole, or indeed anywhere outside the singularity. Therefore we cannot conclude that this solution actually entails stuff going faster than light.

To sum up: the rigid disk doesn't diverge if you ignore rigidity issues, and I haven't found the calculation that takes them into effect. I haven't found the formula for the fast neutron star either, and I'm not satisfied with the black hole case. Thus, I don't have an answer to your seemingly simple question yet. We'll see what we can do though. 144.92.241.204 05:00, 27 March 2006 (UTC)

When Uhlenbeck and Goudsmith had the idea of spin, their professor, Ehrenfest, asked Lorentz what did he think about it. Lorentz sent him some papers covered with formulas, showing that it is impossible to have an electron spinning with the predicted angular momentum. Now, does anyone here know relativity better than Lorentz? Omsharan 09:44, 25 June 2007 (UTC)Om Sharan

spin

i think it would be really good if someone actually explains what is spin. fact is i dont know what it is

Spin is the property shown by quantum objects which causes them to show the behavior we associate with the word 'spin'. There really is no simpler answer than that - it is what it is. What I can tell you that it is [i]not[/i] is the rotation of a particle around its own axis. It's merely named with such an analogy in mind - but it is just a name. 81.156.75.42 14:32, 2 April 2007 (UTC)

Spin is a basic property of the particles that comprise all matter, and its behavior is well-defined and verifiable in physical experiments. Something that important, dare I say that real, is not "just a name". Obviously it can't be explained in the discourse of classical physics. Perhaps not even in any discourse suitable for Wikipedia. But what is known about it, and conjectured about it, has been discovered by mere mortals, and researched by mere mortals, so it can most certainly be explained to mere mortals.

What is probably not possible is to separate verified knowledge from conjecture. Any simple explanation of quantum concepts must mix the two, while carefully remaining within the art of physical inquiry (which unfortunately seems to have been driven into hiding since Jimmy Carter left office). But Wikipedia is not meant to be a forum of inquiry; it's meant to be a sending-off point. So an explanation needs to be found and linked to, but not quoted from. It should have a title that admits orphaned links, such as "further reading".

166.137.101.154 (talk) 21:11, 26 June 2014 (UTC)Collin237

mass could be a product of an objects contained energy ie. speed of its quantum spin more energy....faster spin....more gravity —Preceding unsigned comment added by 99.252.195.32 (talk) 15:51, 2 January 2008 (UTC)

I agree with the first post of this subject that the definition of "spin" needs to be explained in ordinary language and that this should appear near the start of the article. Even explaining what "spin" is not may be of help 90.205.123.105 (talk) 16:49, 10 February 2011 (UTC)

Spin is a solely quantum-mechanical phenomenon

The statement in the introduction: " Spin is a solely quantum-mechanical phenomenon; it does not have a counterpart in classical mechanics (despite the term spin being reminiscent of classical phenomena such as a planet spinning on its axis).[2]" is false! In order to help motivate my calculus students I often show how calculus can be used in sports; this actually often works. In particular, I often describe the motion of a curve ball in baseball. The configuration space of a baseball is R^3xSO(3) and its associated phase space is T*(R^3xSO(3)) ( the cotangent bundle of R^3xSO(3)). SO(3) is too complicated to describe to Freshman. So, instead, I use a simpler, though adequate, model of a baseball as a point particle with extra spin angular momentum. Now the phase space is T*(R^3)xS^2. The dynamical equations can now be set up and numerically solved and graphed on a graphing calculator. One can then watch the ball curve! The phase space (S^2,v) (use the canonical area 2-form v obtained by thinking of S^2 as the standard unit sphere in R^3) is not the cotangent bundle of any configuration space. For any s in R+, T*(R^3)x(S^2, s*v) provides a classical phase space of a point particle of spin s. Using Geometric Quantization (see: http://www.amazon.com/Geometric-Quantization-Oxford-Mathematical-Monographs/dp/0198502702/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1219711387&sr=8-1 ), one can quantize this phase space for s satisfying appropriate integrality conditions and obtain the usual quantum theory of spinning particles. For all s one can deformation quantize (see: http://www.amazon.com/Deformation-Quantization-Mathematicians-Mathematiques-Irma-Lecture/dp/311017247X/ref=sr_1_1?ie=UTF8&s=books&qid=1219711918&sr=1-1 ) this phase space and get something that is probably also interesting! Thus, Pauli was wrong! There are reasonable classical limits of spinning point particles. You might also want to look at: http://www.amazon.com/Supersymmetry-Mathematicians-Introduction-Courant-Lecture/dp/0821835742/ref=sr_1_1?ie=UTF8&s=books&qid=1219712646&sr=1-1 The theory of Fermi fields requires a totally different mathematical structure than the usual quantum probability theory! David Edwards, dedwards@math.uga.edu David edwards (talk) 17:21, 17 May 2014 (UTC)

As a layman, I find the claim (that "Spin is a solely quantum-mechanical phenomenon") rather unsatisfactory. There are certainly classical analogues that at least look like they might stand in, for example the classical angular momentum attributable to the EM field of an electron. I even understood (but may misremember) that Dirac followed this line of reasoning. Do the sources make this actual claim? —Quondum 22:22, 17 May 2014 (UTC)

I totally agree. To this layman, the first paragraph of this article seems to be self-contradictory. The first sentence says that spin is an intrinsic form of angular momentum, and links to a page on angular momentum in classical physics. The second sentence then claims that spin has no counterpart in classical mechanics. I am clearly missing some understanding here, because I don't see how both of these sentences can be true. (I actually came to Wikipedia to try to firm up my understanding of exactly this - the question I wanted to find an answer to was: Is quantum mechanical spin a "quantum version" of angular momentum, or is the naming an analogy based on the fact that the maths behind angular momentum and quantum spin look similar despite being unrelated physical phenomena. I found it slightly comical that the first paragraph of the article appears to contain exactly the same confusion I have in my head!) 144.24.20.231 (talk) 15:14, 8 August 2014 (UTC)

Higher spins transferred from Pauli matrices

In this version of the Pauli matrix article, higher spin matrices were included by other editors, not me, so credit goes to whoever wrote them. It was agreed on the talk page there that the higher spin matrices should be moved out of that article and into other places. This article would be one obvious candidate to include explicit expression for the first few spins beyond 1/2, and the general matrix elements, so I placed the content in a new section Spin (physics)#Higher spins. M∧Ŝc²ħεИ_τlk 19:35, 13 January 2015 (UTC)

Spin direction - ambiguity?

Just to note that an obvious property seems not to have been mentioned - the direction of spin. Presumably a positive spin corresponds to anticlockwise rotation when the particle is viewed from above (with spin up) - in accordance with the right hand rule. Not sure how to work this fact into the article (if I have it correct). Bdushaw (talk) 21:33, 23 January 2015 (UTC)

I'm not quite sure what you're getting at. Spin is a property that is measured as in a direction relative to an axis (up/down for the z-axis, for example). Positive or negative would only be with respect to a reference direction. The right hand rule is used only as a convention to relate the direction of rotation (a bivector) to a direction (a pseudovector), but this is so standard as to not bear mentioning. —Quondum 02:05, 24 January 2015 (UTC)

Yes, my comment was confusing. I was getting at the sense of rotation. If a particle has spin up, what is its sense of rotation? Which way is it rotating? That's defined by the right hand rule, yes? It might be generally obvious or standard to most...but this is an encyclopedia that might be read by 8th graders... One way to define it is that spin up means the intrinsic angular momentum vector is up, which corresponds to rotation according to the right hand rule. My thought was that it would be worth being explicit about this point in the article. (I suppose my own issue is that we speak of "spin up" so often it becomes more like jargon and we forget about what it actually means.) Bdushaw (talk) 03:04, 24 January 2015 (UTC)

Perhaps a sentence like: "Like angular momentum, the sense of rotation from the spin of a particle is defined according to the right hand rule." I am not so sure this understanding could be taken for granted, but will go with the consensus. Bdushaw (talk) 03:10, 24 January 2015 (UTC)

Yes, you may have a point. I would have no objection to a brief mention like that. —Quondum 04:07, 24 January 2015 (UTC)

The interpretation of "clockwise/anticlockwise" is not applicable to QM spin, and neither is the "right-hand-thumb rule". Some sources may use it but it is misleading, a particle with spin does not "rotate" or "spin", those names are just historical. To confirm Quondum, it can be measured in any direction by projecting the (pesudo)vector-valued spin operator S along the direction of interest (defined by a spatial unit vector n). Mathematically it's just the dot product n·S. The component of the spin is either parallel to the direction ("up" or "positive"), or antiparallel ("down" or "negative"). Normally n is just the unit vector in the z-direction since the z-component of spin is the simplest case to calculate with. There is no "ambiguity" in direction, just whichever is most convenient. M∧Ŝc²ħεИ_τlk 09:32, 24 January 2015 (UTC)

I don't think we are in disagreement. But spin is angular momentum, even though, as you note, a point particle cannot actually rotate. But having angular momentum there is a "sense" of rotation (cf. precession of a Dirac particle - the direction of precession from an applied torque is in accord with the intrinsic "rotation"). Perhaps the better approach may be just to emphasize that spin is angular momentum, hence has the attributes and conventions of classical angular momentum, even though it is as you describe. The spin vector, once measured, corresponds to an angular momentum vector. But there is a convention regarding the direction of angular momentum and the sense of rotation, according to the right hand rule. It applies to spin, even though "sense of rotation" may lack a clear physical meaning. (To put it another way, a naive reader may be befuddled and ask "But which way is it spinning?") Bdushaw (talk) 10:44, 24 January 2015 (UTC)

(I don't mean to make a big problem about this; it just seems a slight omission)Bdushaw (talk) 11:24, 24 January 2015 (UTC)

You're not making this a problem at all, Bdushaw. One main theoretical motivation for saying spin is an angular momentum is that the spin operators satisfy exactly the same commutation relations as those of orbital angular momentum, while experimentally there is a magnetic moment associated with spin. But we should not indicate any rotation or spinning, and to not indicate does confuse a lot of people (me included) for reasons you describe. It's probably best just to leave it simple by stating the similarities and differences, e.g. spin is an angular momentum (inferred from commutation relations, magnetic moments) but cannot be visualized as rotation since particles are point-like, then the reader should not be able to ask "But which way is it spinning?". M∧Ŝc²ħεИ_τlk 15:12, 24 January 2015 (UTC)

One should be careful not to view QM spin too abstractly (mystically?). Due to its quantization, things get a little confusing, but just like the quantized energy of a photon is real energy, the QM spin of a particle is real angular momentum – you can in principle measure it directly by flipping the spins of a lot of He-3, for example. (To sidestep the issue of "a point can't rotate", refer to direction of angular momentum directly.) Also, in classical mechanics and QM both, if by convention we expressed angular momentum as a bivector, there would be no handedness anywhere (ignoring some subtle QM observations). However, since we represent angular momentum (and by inference spin) as a pseudovector, we have introduced a handedness by this convention. The right hand screw rule (or whatever you choose to call it) is purely a convention of relating a direction of rotation with the direction of an axial vector; we cannot avoid this fact about convention in either classical mechanics or QM. (We could potentially have avoided it in QM if spin was not related to a real angular momentum, with a real direction of rotation.) I see Bdushaw's point as essentially relating to this convention only.

One should perhaps also perhaps take care to distinguish between spin (a pseudovector) and spin quantum number (a (pseudo?)scalar). —Quondum 15:55, 24 January 2015 (UTC)

The screw rules are just mnemonical for orientations in 3d only (even then they are primarily used for cross products), and are misleading in the context of spin. Yes, handedness does occur in the definition of a psuedovector (how it transforms between coordinate systems), but that does not mean we need to mention it in this article. And no - the spin quantum number is not a pseudoscalar, but a scalar multiple of hbar. M∧Ŝc²ħεИ_τlk 14:38, 25 January 2015 (UTC)

Perhaps you're missing what I'm saying. Because the vectorial representation of angular momentum in three dimensions is a cross product, we are assuming the handedness of the cross product. If we used a bivector to represent angular momentum, the handedness disappears. This applies in both classical and QM settings. —Quondum 16:09, 25 January 2015 (UTC)

No, I follow what you are saying, that any angular momentum in all of physics can be represented as a pseudovector or an antisymmetric tensor (or equivalent), and that handedness is relevant for all pseudovectors. But the cross product does not come into the definition of spin. All my original post meant to say is that handedness (especially the screw rule mnemonics) must not conflate with literal spinning. M∧Ŝc²ħεИ_τlk 16:51, 25 January 2015 (UTC)

Agreed that one should not think of literal spinning (or even rotation): point particles don't do this. The name "spin" is rather unfortunate in this context. But one must still think in terms of angular momentum, however it arises. Bivectors such as the magnetic field do not have a geometric rotational direction associated with them, and we can think of angular momentum (and spin) in the same way. If one chooses a polarity of charge, one can associate a rotational direction (as in "which direction would I have to orbit this charge to produce this magnetic field polarity?"); angular momentum has a more "natural" polarity association though: there is a natural (given our prejudice of what direction of time is to 'the future') sense of positive mass, so through the angular momentum one can indirectly associate with the spin a geometric rotational direction ("which direction would I have to orbit this mass to produce this angular momentum polarity?"). —Quondum 17:35, 25 January 2015 (UTC)

Yes - the convention. The practical effect, for example, is that total angular momentum is the vector sum of spin and orbital angular momentum, J=S+L. It is a sum, rather than a difference, because S and L have the same convention for the sense of rotation. It may seem silly or obvious, but we all know how the basics can trip up beginners and experts alike. Still not sure precisely what words to use or where; I'll think about it. Bdushaw (talk) 08:44, 25 January 2015 (UTC)

Confusing statements

This article has a few statements that can lead to some confusion, especially for someone who is learning about QM for the first time.

1- "The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus)".

First of all, what is a trajectory in QM? Wouldn't it be better to say that angular momentum "arises" whenever the system has spherical symmetry?

2- The whole "Direction" section is a bit confusing. Forgive if I'm wrong, but doesn't the spinor correspond to the state of the spin system and is therefore an object living in an abstract mathematical state space (possibly infinitely dimensional)? The spin in itself, on the other hand, is an observable corresponding to a vector operator, living in a very real space, just like regular angular momentum. I mean, spin can be observed! One can measure its magnitude and its components (even all three of them at the same time, albeit not with an arbitrary degree of precision). That is why I find statements such as:

"For example, rotating a spin-1/2 particle by 360 degrees does not bring it back to the same quantum state, but to the state with the opposite quantum phase"

very confusing. It makes is sound as if the particle itself is being physically rotated, whereas it is it's state that is being rotated. What happens if the spinor describing the system is rotated by 360 degrees? It will get a minus sign, but will the expectation values of the spin operators change (I'm thinking they don't, but I need to study this again to be sure)?

Any thoughts on this? — Preceding unsigned comment added by 178.192.175.190 (talk) 10:29, 14 February 2016 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Spin (physics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Lacks inline citations, very important topic! Snailwalker | talk 00:34, 21 October 2006 (UTC)

Last edited at 00:34, 21 October 2006 (UTC). Substituted at 06:42, 30 April 2016 (UTC)

Nothing on nuclear spin

nuclear spin redirects here but its not mentioned (apart from just NMR as an application). - Perhaps the Application section could be explicitly divided into nuclear spin and electron spin ? - Rod57 (talk) 22:29, 20 September 2016 (UTC)

I agree that there should be separate sections on Electron spin and Nuclear spin. The Nuclear spin section could start with material copied or adapted from Nuclear magnetic resonance#Nuclear spin and magnets and from Nuclear magnetic moment. Dirac66 (talk) 00:16, 21 September 2016 (UTC)

Electron spin

Electron spin can be derived directly from the pair-production energy equation,

Eν = E- + E+

Where Eν is the minimum energy of a photon when pair product occurs, E- is the mass energy of the newly created electron, and E+ is the mass energy of the newly created positron. Here, for the moment ignore any residual kinetic energy while further noting the following: for a photon, Eν = hν0, where h is Planck’s constant and ν0 is the minimum frequency of the photon that creates the particle pair, and given that h = 2πħ, where ħ is Planck’s constant divided by 2π, and that 2πν0 = ω0, where ω0 is the angular frequency equivalent of the ν0,

hν0 = E- + E+ ⇒ 2πν0ħ = E- + E+ ⇒ ħω0 = E- + E+ ⇒ ħ = E-/ω0 + E+/ω0

Since the masses of electrons and positrons appear to be equal we can say that,

½ħ = E-/ω0 = E+/ω0 eq.1

The left side of the equation is electron spin. The other side tells that for the electron (or positron) the spin is equal to the mass energy divided by the angular frequency equivalent of the photon that created the particle(s). Angular frequency implies rotational motion so the electron is energy trapped in a spherical rotation. Also note that the spins of both particles is positive so the spins are in the same direction. Most likely spin direction is determined by which side of the nucleus the photon passes on relative to the nuclear magnetic moment.-- 7/25/2017 — Preceding unsigned comment added by WCherring (talk • contribs) 17:58, 25 July 2017 (UTC)

Spin observables

OK, the operators describing spin observables do form an operator valued vector space defined by $[J_{i},J_{j}]=i\epsilon _{ijk}\hbar J_{k}$. Why did you revert? And how about the part on SO(3) representations and Clebsch-Gordon coefficients?

• When one talks about an observable such as spin or energy, one is talking about the eigenvalue/eigenstate, not the corresponding operator(s). Spin does not transform like a vector, i.e. like the coordinates, under rotations. For example, if you have a spin-1/2 particle with a spin "up" in the x direction (i.e. it is an eigenstate of $\sigma _{x}$ with eigenvalue +1, if you rotate your coordinate system by 360 degrees, you do not get back the same state; rather, it picks up a π phase shift. Alternatively, if you rotate your measurement apparatus by 90 degrees to measure spin along the y direction, you do not measure zero, you get a 50% chance of spin up or down. It is only as the angular momentum becomes large that you asymptotically recover the classical behavior.
  • (Of course, strictly speaking, angular momentum is not a vector either; it is a pseudovector.)
  • You are correct that the operators ($\sigma _{1}$,$\sigma _{2}$,$\sigma _{3}$) for spin-1/2, or J's for higher spins, do form the basis of a 3-vector space. i.e. for the operator $\sigma =v_{1}\sigma _{1}+v_{2}\sigma _{2}+v_{3}\sigma _{3}$, the quantities $(v_{1},v_{2},v_{3})$ transform like a 3-vector (or maybe a pseudovector?) as the coordinate system is rotated. This is just a reflection of the fact that these operators are defined relative to the coordinate system, and not on the transformation properties of the underlying observable quantity.
  • (Also, please do not confuse 3-vectors with general abstract vector spaces, which of course include all quantum states.)

—Steven G. Johnson

I try to understand (I understand how to derive the phase-shift of π after a full rotation from theory) how the π phase-shift manifests itself in practice/experiment: suppose you do an interference experiment where the electron can go through a reference path/arm and through the Stern-Gerlach machine as another path, in order to measure the phase shift (constructive or destructive interference). now suppose the SG setup is detachable, and that we have a curtain, so that the physicist only sees the original setup, can he measure the current phase shift (0 or π), ask a volunteer to take the SG apparatus behind the curtain and secretly rotate it an even or odd number of times, and then return the SG apparatus to the physicist. Can the physicist (who does not know the number of rotations) insert the SG apparatus in the interference setup and determine if the volunteer's secret number of rotations was odd/even? — Preceding unsigned comment added by 213.211.139.49 (talk) 01:52, 10 September 2017 (UTC)

Spin value not exactly 1/2 ħ?

I was looking for the value of electron spin in SI units, which was fortunately mentioned in the introduction:

The SI unit of spin is the (N·m·s) or (kg·m2·s−1), just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same units of angular momentum, although it should be noted that this is not the full computation of this value.

however, I am completely confused by although it should be noted that this is not the full computation of this value. what is the author hinting at? I know that the magnetic moment is not exactly a Bohr magneton due to g-factor from QED, but I always thought the spin magnitude was exactly 1/2 ħ

Am I missing something, if so please refer me to a text that describes how the intrinsic angular momentum of the electron differs from 1/2ħ, or did the author probably think of the inaccurate Bohr magneton, and should it be removed? — Preceding unsigned comment added by 213.211.139.49 (talk) 05:17, 10 September 2017 (UTC)

Potential conflict in definition of the Newton

I'm new to this subject, and I noted that the definition of the Newton, on the Newton page, is N = kg * m * s^-2 but this page states that it would be N = kg * m^2 * s^-1

Is this a typo, or is there a second definition of the Newton?

Again, I'm new to the subject, so if there's an explanation, a pointer would be welcome. — Preceding unsigned comment added by Roger.lee (talk • contribs) 10:39, 3 October 2019 (UTC)

Are you referring to one of the sentences in the introduction, starting:

The SI unit of spin is the (N·m·s) or (kg·m²·s⁻¹)...

If so, this is defining the unit for spin rather than the unit for the Newton. If you insert the units for the Newton (from the Newton (unit) page) into (N·m·s) you end up with the units in the second brackets. Hope this helps! --  Newty  17:39, 3 October 2019 (UTC)

Spin is Instantaneous - Quantum Nonlocality

I added this to the intro... Spin between two particles - no matter how far apart - is instantaneous thus faster-than-light. Albert Einstein referred to this as "spooky-action-at-a-distance"/quantum nonlocality. 50.185.129.118 (talk) 14:35, 7 October 2022 (UTC)

What do you mean by "Spin between two particles"? Is it meant to be entangled particles? It seems a discussion about non locality would be better in its own section, not the intro. Mr snarf (talk) 09:01, 8 October 2022 (UTC)

points?

"even though they are believed to be point particles possessing no internal structure" Are they really believed to be infinitesmal points, or are they just modeled as points? — Omegatron 20:57, 30 September 2005 (UTC)

There is a radius called the classical electron radius. It is the radius that a charged sphere would have, such that the energy of the electromagnetic field would equal the known mass of the electron. Unfortunately, electrons are known to be much smaller. In most cases, the effective radius of an electron is about the de Broglie wavelength. But with higher and higher energy accelerators, the wavelength can be decreased. And even at the smallest wavelength from the highest energy accelerators, there is no suggestion that we are getting close. But otherwise, the theory doesn't give any radius. You might consider spin as the angular momentum of a sphere, in the limit as the radius goes to zero, while the angular momentum stays finite. Gah4 (talk) 06:55, 13 June 2023 (UTC)

A comment on the phrase "spin 1/2 particle"

This spin article, as well as many of the other QM articles, uses phrases like "in the case of spin 1/2 particles" where what is really meant is "in the case of two spin-state particles". The later seems more fundamental to me. Calling a particle "spin 1/2" is really just a matter of convention, though a sensible one, a reflection of the choice of having a quantum number increment by multiples of unity, as contrasted to, say, multiples of 2. It does not communicate the essential feature of the particle. More fundamental is the number of states, since a two-state quantum system is obviously different from, and arguably simpler than, a three-state quantum system. I thus propose that wherever an article contains the phrase "spin 1/2 particle", the first appearance is qualified by the parenthetical phrase "(that is, a particle with exactly two spin states)". —Preceding unsigned comment added by William.menke (talk • contribs) 02:43, 7 May 2010 (UTC)

I suppose, but I believe that "spin 1/2 particle" is the WP:COMMONNAME. Gah4 (talk) 07:08, 13 June 2023 (UTC)

What Spin Is

Spin exists because physics is invariant under Lorentz Transformations, instead of being invariant under Galilean transformations. Quantum mechanics doesn't create spin, it merely quantizes it.

There is a very close relationship between physical laws being invariant under groups of transformations, and conserved quantities. Translation invariance implies momentum conservation, time invariance implies energy conservation, rotational invariance implies angular momentum conservation, and so on.

Now Lorentz Transformations have this funny little property that Galilean Transformations don't have. Two velocity changes at angles to each other may be equivalent to a velocity change PLUS A ROTATION. Because of this, the total conserved quantity under Lorentz spacial rotations isn't exactly classical angular momentum, but it's classical angular momentum plus "a little extra." We call this extra quantity, which we identify with the intrinsic rotational properties of different types of fields under Lorentz Transformations, spin. When we quantize the fields, we get quanta that each carry a little bit of this spin, which depends on the type of geometric object we quantized. Scalars - spin 0, vectors - spin 1, rank 2 tensors - spin 2, and of course, for various flavors of spinors, which carry an half-odd-integral spin representation of the Poincare group, 1/2, 3/2, 5/2, etc...

So in a nutshell, spin is not a quantum mechanical property. Spin is a relativistic property. Spin getting quantized is a quantum mechanical property. Classical spin, and even classical spinors, are perfectly well defined concepts.

Note: A huge number of undergraduate texts get this completely wrong, and present spin as some magical incomprehensible quantum mechanical property. It becomes apparent later, when you employ Langrangian densities and Poincare invariance to directly calculate the equations of motion and conserved quantities for various flavors of fields, how spin is fabricated out of the odd way Lorentz rotations compose.

It would be nice if the article explained what spin actually is, in terms comprehensible to an ordinary person.Hermitian 09:27, 12 January 2006 (UTC)

I disagree with you: spin arises naturally even in nonrelativistic settings. The noncommutativity of boosts in relativistic mechanics is not really relevant. -lethe ^talk 10:50, 12 January 2006 (UTC)

It's not noncommutativity. It's the composition of two pure velocity changes yielding a velocity change plus a rotation. Velocity changes are just rotations in a plane which includes "t" as a coordinate, whereas spacial rotations do not. If there was no mixing between purely spacial rotations and velocity changes, then total angular momentum would be the usual classical thing. Spin arises naturally in the mechanics of fields when you calculate the conserved total angular momentum for Poincare invariance, and doesn't when you calculate it for Galilean invariance. All spin in the standard model is created in this manner.

I'm not sure what you mean by "spin arises naturally even in nonrelativistic settings." Spin is a consequence of Poincare invariance of physical laws. That doesn't mean that the particles in question are moving at relativistic speeds.

A user in an infinitesimally displaced frame will see different coordinates and field variables. If the laws of physics are invariant, the variation in the action will be zero, and we will get a conservation law, of the quantity canonically conjugate to what was perturbed.

When we do this for a field and infinitesimal spacial rotations under the Galilean group, the conserved total angular momentum is a single term, which we can quickly identify as classical angular momentum. When we do this for a field and infinitesimal spacial rotations under the Poincare group, the conserved total angular momentum is the sum of two terms, because of the mixing between pure spacial rotations and velocity changes. We can quickly identify one of the terms as the same classical angular momentum we saw in the Galilean scenario, and we call the new term "spin angular momentum." The total angular momentum is the sum of these two.

All quantum field theory based on the Lagrangian Formalism and Schwinger's Variational Principle derives spin through this mechanism, with the spins of the quanta corresponding to what variety of geometric object is being employed as a field variable.Hermitian 11:40, 12 January 2006 (UTC)

Spin is a consequence of Poincare invariance of physical laws. No, spin is a consequence of rotational invariance. The fact that Poincaré invariance also implies spin simply speaks to the fact that the rotation group is a subgroup of the Poincaré group. You have the same spin representations of the nonrelativistic Galileo group as you do of the Poincaré group, therefore it cannot be correct to classify spin as a relativistic phenomenon. -lethe ^{talk +} 23:50, 21 February 2006 (UTC)

You are both wrong. So Let's clear some things up. The Galilean group is generated by the rotation group SO(3,R) and Galilean boosts. In the Galilean group, the set of boosts K forms an Abelian subgroup and thus an invariant subgroup. So the Galilean group is a "semidirect" product of SO(3) rotations J with the Abelian group of boosts K which exist on a flat affine velocity space. So J^2 is a Casimir operator. It commutes with everything and is preserved under Galilean transformations In contrast, in the Lorentz group the set of boosts exists on the curved Lobachevski velocity space and does not form a subgroup, but a groupoid. So the Lorentz group is a "quasidirect" product of the rotation group (as a subgroup of the automorphism group of the velocity groupoid) with the weakly associative groupoid of velocities. Thus the Lorentz group is IN NO WAY a semidirect product like the Galilean group, hence the Lorentz group is a simple group (ie noncommutative and having no nontrivial invariant subgroups, such as the SO(3) group as you claim). NOW, why this often gets confused as being a result of mysterious quantum mechanics has to do with Lie algebras and irreducible representations and the complexification of your original representation space, which are mathematical concepts that are more advanced than you are when you first learn quantum mechanics. So instead of dealing with angular momentum, as a more direct illustration take the electric and magnetic fields. Under SO(3,R) rotations, the sixtors E and B transform separately, giving a representation of the rotation subgroup that forms a direct sum, and hence an irreducible representation. This irreducibility does not hold if we allow for complex numbers as the number field for the representation spaces. Thus we can form representations under SO(3,C) where F+=E+iB, F-=E-iB. Notice this looks a lot like the forms J+ and J- in quantum mechanics. So really, in the Lorentz group, we must do the same thing in order to get a compact representation. So we complexify and find that our full rotations under the Lorentz group are J+=J+iK, J-=J-iK. This DOES arise from the fact that Lorentz boosts are non-Abelian. Hence we arrive at the Dirac or Weyl representation of spinors for a relativistic theory, which reproduces our 4D Lorentz algebra. But notice J^2 is no longer a Casimir operator. Now our equivalent Casimir operator is formed from the Pauli-Lubanski 4-vector, W. In the timelike case for a massive spin 1/2 particle, no one cares about the boosts K and the commutator for W reduces SO(3), and the Casimir now looks like m^2 J^2, which is the product of our 2 Casimirs P^2 and W^2. However if we boost, our theory is no longer SO(3) invariant, and spin is no longer a good quantum number. When we boost we have to add a term that looks like the helicity operator, which couples on momentum and spin. So using the fact that SO(3)~SU(2), the Lorentz group turns into a helicity representation SU(2)L x SU(2)R, which are coupled by the mass of the particle. Each transform under different representations since it is a direct product. —Preceding unsigned comment added by 71.105.84.231 (talk • contribs)

You say it yourself: for a massive spin 1/2 particle, no one cares about the boosts K and the commutator for W reduces SO(3), and the Casimir now looks like m^2 J^2. It's not that no one cares about boosts, it's that Wigner taught us that the right way to find representations of a group is to look at the isotropy subgroup. The isotropy subgroup for a massive particle is the rotation group. The rotation group is a subgroup of any spacetime symmetry group of physics, thus any system must live in a rep of the rotation group, which we call spin. All the other stuff you mention is certainly relevant to the representation theory in general, but does not change this simple fact. -lethe ^{talk +} 20:41, 17 March 2006 (UTC)

This is all very well...but what is spin???

I agree with Hermitian's original comment - please try and explain this sort of in normal terms somewhere... I get that it can't be too basic, because it's a scientific article, but an analogy by a respected scientist for example would be good. Because the article talks about "velocity", "direction", "vector"...but all relative to what?! It says "spin does not relate to the mechanical principle of spin"...so what is it?
It also needs to be explained what 1/2 spin is, for example. I get what 1/2 mass is - when something's half as heavy as something else. But what is 1/2 spin - when something only "spins" half-way, or half as fast (compared to what???), or something else entirely? BigSteve (talk) 15:02, 8 July 2012 (UTC)

I see no reply to this important question after 10 years. It is not satisfying, but a fairly correct answer is that we cannot "know" what particle spin is. The reason is that macroscopic angular velocity doesn't really exist in the atomic size domain. A particle spin is a fixed and constant property of a particle that can be measured but does not really correspond to anything in our commonsense physics. Other familiar macroscopic quantities that similarly don't exist on the atomic scale include temperature and pressure. I hope that someone with advanced learning in physics can validate or correct what I've just written. David Spector (talk) 00:58, 15 June 2023 (UTC)

Forward referencing

It would be nice if this article didn't use the phrases, "which will be described in the remainder of this article", "A later section covers", "as we will see", etc. in describing information presented later in the article. I guess I'm saying the grammar could be improved.

I agree, but I disagree that the problem is grammatical. To make a forward reference, create a wikilink to the proper section, using '#' in the link to name the section. David Spector (talk) 01:00, 15 June 2023 (UTC)