Spin has a geometrical basis in how fields or particles behave when rotated. In quantum field theory, which is the basis for particle physics, particles are the quantum excitations of a field, and their spin reflects the behavior of the field under rotations. A field with no directional characteristics, such as the Higgs field, is just a numerical function of space-time, and has spin zero. The electromagnetic field is described by a vector potential, which gives it the directional characteristics of a vector, which we say has spin 1, and so do the photons which are its quanta. The gravitational field is a second rank tensor (two vector indices), and has spin 2, as would gravitons. For integer spin, this generalizes in the obvious way, with rank n tensor fields having spin n. All such fields are bosons. Fermion fields, such as the electron, change sign under a full rotation. The classical field describing electrons is a spinor field, which changes sign under a full rotation and requires two full rotations to return to its original form. Such a field is said to have spin 1/2, and so do its quantum excitations, which we call electrons. (Spinors are a kind of square root of a vector.)

Spin is an abstract concept which illustrates the magnetic properties (magnetic moment) of a particle...

Historically you must start from this point: For centuries a puzzle intrigued scientists: How is it possible to break a magnet into parts and still all parts have two poles?

The theory of modern magnetism explains this by using Spin. Please refer to the book "The new ambidextrous universe" by Martin Gardner if you want to pursue this route. But a message of warning: If you read the book you may find out how simple-minded were scientists were, are, and will be most of the times...

The spin of a particle is always greater or equal than zero. We can, however have the spin projection in a particular frame to be negative. For example, we can have an electron (spin 1/2) with spin up Sz=1/2 ez and another electron with spin down Sz=-1/2 ez. This is just the projection of the spin in a certain frame.

From a "classical point of view" spin would be a rotation of a body through its axis of symmetry, and therefore, spin up or down would just be reflecting the way it rotates. From the quantum point of view, spin is an intrinsic property of particles like charge (or color for quarks) and is related to the way particles behave through the spin-statistic theorem, which basicaly tell us that spin half particles satisfy the Fermi-Dirac statistic while integer spin particles satisfy the Bose-Einstein statistic.

Conserning anti-particles, these share the same spin what are different are the other quantum numbers: charge, color,...

For example, the positron (the anti-particle of the electron) have spin 1/2 but oposit charge. The projection of the spin in some frame can be up or down, just like the electron.

It is abstract concept with no classical analog like particle rotating about its own axis or so. Every elementary particle is thought to be born with an intrinsic angular momentum (don't ask how it looks like!) which is called the 'spin' of the particle.

Now the question is, how should we assign the 'spin' quantum number of a particle?

This is my understanding:

Every elementary particle has a field associated with it whose excitation or de-excitation cause the particle appear or disappear. This field transforms under the Lorentz Transformations in a certain way. The spin of the particle depend on how the corresponding field transforms under Lorentz Transformations. For example, the 'Higgs' field does not transform at all. Consequently, the Higgs particle is a 'scalar', i.e., it has spin 0 !

Dear Dipankar

I believe when you are speaking in the context of QFT, you have to consider Gauge invariance (local or global) too. This is how the Lagrangian approach to QFT works. Indeed it is the choice of your gauge which determines polarization states of your particle; this later concept is more clear in the Hamiltonian approach to QFT(Relativistic extensions of the Schrodinger equation) which is the basis of the necessary concepts and elements of QFT.

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Dear Marc

On more thing regarding the second question: First of all anti-particle is different from anti-matter. Anti-particle is a particle, i.e. it exists in the world where other particles are. Therefore it obeys the limits of the ordinary world, for instance positive rest mass.

secondly we do not have negative Spin anywhere, but the components of the Spin vector can be negative or positive.

The concept of particles need to have an intrinsic characteristic rather than mass, charge, ... was introduced by Wolfgang Pauli in 1925 as Dr. Serodio explained to justify the exclusion principle. Later the theoretical extensions of the concept was used wherever physicists needed to categorize particles, Isospin-Weak isospin-Weak hypercharge...

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Please read this if you are eager to enter studying Quantum mechanics which I do not suggest. Because I believe the book I introduced really answers lots of questions and avoids lots of time and energy waste:

Both electron and positron are spin-1/2 particles...I believe your misunderstanding comes from encountering phenomena such as pair production or annihilation....in these interactions due to the conservation of angular momentum in the CM(center of momentum) frame, for instance the z component of spin must be conserved.

In pair production since we have nothing at the beginning (Spin 0 along the z axis) therefore in the final state we must have Spin 0 along the z axis too. Since electron and positron are spin-1/2 particles, for their z components to cancel each other one must be opposite the other, i.e. one must be positive and the other must be negative.

Mohammad, Thanks for this explantion.However, I have come across that the pion has spin zero; so I assumed that since pions are mesons made up of a quark particle and quark anti-particle, then to have zero spin the quark has possibly +1/2 and the anti-quark has -1/2 spin. Perhaps you can explain if the spin of the quark components effect the total spin for the pion but obviously my explanation with anti-quark negative is not the answer.

Marc

Marc, I will give you an answer first. Please always be cautious to find out the presumptions behind these answers.

Suppose you have two vectors which both have the same size(a), but you don't know their directions(alignments). When these two vectors add up together, they can generate a vector which can have a size between 0(if they are counter-aligned) to 2a(if they are co-aligned)....

Now suppose the vectors we are talking about are the Spin of a quark and the Spin of an anti-quark; therefore we can still have 0 Spin as a result of their gathering.

In quantum mechanics when we say that a particle is spin-1/2, it means that the size of its Spin vector is 1/2...These are the components that help you determine the alignment of this vector....you cannot also determine all those components together....refer to the STERN-GERLACH experiment

If you are really eager to jump into this field and you don't have troubles with mathematics, I suggest reading the first few chapters of J. J. Sakurai Modern Quantum Mechanics(the first chapter explains the construction of quantum mechanical quantities and specially starts with Spin)....I believe this book has a sound and logically true reconstruction of quantum mechanics....

Before that I suggest a little bit of research on different interpretations of quantum mechanics, starting with wikipedia.

Please be very careful..

@Mohammad: So would zero spin mean the quark and anti-quark would have opposing spin direction, up and down?

Yes...partially true

+another thing in quantum mechanics, i.e. quantization. Spin is a discrete quantity meaning it is quantized. It cannot have any value. Pay attention not all quantities are discrete in quantum mechanics. We have also continuous quantities such as momentum, space,...

Example: If you have a spin-1 particle and a spin-1/2 particle which for instance form a bound state(a new particle). The total spin of the new particle would be: 1/2 or (1/2+1)3/2...these are half integer values which means the new particle is a fermion. Also you can see we are not saying that any value between 1/2 and 3/2 is possible.

In the case of mesons which are bound states of two spin-1/2 particles(quarks) clearly the final state would be 0 or 1. In your words: If they are up and down the result is 0 and if they are up-up or down-down the result is 1. But In a physicist word up and down refer to the sign of the z component of the spin-1/2. So be cautious. Here your words and a physicist words are different.

The reason is that when a physicist imagines an up state of a spin-1/2 particle, he understands that what he knows and can know about the spin vector is only and only its size and the value of its z component. he cannot know anything certain about the x and y components. in other words the other two components can have any value +1/2 or -1/2.

You see what I am saying: if you want to dig deeper, other concepts come through and you have to dig wider. Or you could have asked your main problem which has guided you here. I saw your questions: I feel you want to find out the logic of summation rules in quantum mechanics for discrete quantities such as Spin,color charge,....

I have given you the two rules here and in the last post which summetrize as:

For two discrete quantities with values A and B. The final state can be A-B , (A-B)+1 , (A-B)+2 , .... , A+B

These are in the framework of Copenhagen interpretation of quantum mechanics.

I forgot to add something in the final rule: all the values are absolute values.

So just to confirm; a pion always has zero spin in terms of its discrete energy magnitude or angular momentum but can have integer spin direction such as zero or one depending if the quark and anti-quark is up (positive 1/2) or down (negative 1/2).

No. A Pion always has spin_0...by the way you must understand that when we say Pion we are talking about 3 particles indeed...the neutral Pion(Pi_0), Pi_+ , and Pi_0 all of which are Spin_0 partcles.

The story for Pion:

I briefly mentioned before that particles have other intrinsic properties other than Spin which physicists correspond to them in analogy with Spin to categorize them.

Pion is one of the isotriplets of the {u,d,s} generation of quarks

One of the spin-1 partners of Pi is Rho which has the same quark content and therefore the same Isospin states.

Or, it can also be said such a way that we call, for example, Iu ¥bar{d}> quark state with spin-1 as "Rho meson" and spin-0 as "Pion".

The pretty different masses between them can be understood by the difference of the relative spin direction like magnetic dipole-dipole interaction of classical analogy.

So there is not essential difference between Pion and Rho meson.

Then, both total spin-0 and -1 state exist. We just call them differently due to their different masses and properties caused by the different masses.

Particle quantum spin can be shown classically by breaking the spinors into their vector components for a 3-d model of spin, however it can not be defined in that way with a point particle model common in QM. Here is an example.

http://www.academia.edu/879621/Symmetry_In_Pair_Production_and_The_Importance_of_Quantum_Spin_in_Particle_Formation

A work in progress for an undergraduate model to introduce concepts like Maxwell polarization of the vacuum, pair production and the Dirac equation.

Quote from the famous paper by Hans C. Ohanian "What is spin?" (Am. J. Phys. 54 (6), June 1986):

"According to the prevailing belief, the spin of the electron or of some other particle is a misterious 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, 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 electron."

See also this link:

http://physics-quest.org/Book_Chapter_Gordon_Decomposition.pdf

Spin is quantum number which is used to distinguish between particle and antiparticle.It has a big role in nuclear forces

Spin alone cannot be used to distinguish between particles and antiparticles.

Spin observable is purely quantum mechanics aspect and there is no corresponding classical aspect. This is an intrinsic property of all elementary particles. Addition of spin vectors is the same of the addition of angular momentum vector. This quantity effects on all dynamical properties of the system and limits the structure of many body systems such as atoms and nuclei. Experimentally, this quantity measures by the its interaction with magnetic field. The eigenvalues of this operator are +1/2 and -1/2 for spin up and spin down, respectively. Total spin of the systems including odd number of particles is half integer and fr the systems of even number of particles is integer. In shell model both for atoms and nuclei, spin-orbit coupling is an important contribution for energy eigenvalues, as well as taken into account of this effect is necessary.

Spin is an abstract mathematical concept in quantum mechanics, because of QM's underlying premise of particles being zero dimensional points. For a descriptive explanation of spin you will have more success by switching to the hidden variable theories. http://dx.doi.org/10.5539/apr.v5n5p107

Spin has a geometrical basis in how fields or particles behave when rotated. In quantum field theory, which is the basis for particle physics, particles are the quantum excitations of a field, and their spin reflects the behavior of the field under rotations. A field with no directional characteristics, such as the Higgs field, is just a numerical function of space-time, and has spin zero. The electromagnetic field is described by a vector potential, which gives it the directional characteristics of a vector, which we say has spin 1, and so do the photons which are its quanta. The gravitational field is a second rank tensor (two vector indices), and has spin 2, as would gravitons. For integer spin, this generalizes in the obvious way, with rank n tensor fields having spin n. All such fields are bosons. Fermion fields, such as the electron, change sign under a full rotation. The classical field describing electrons is a spinor field, which changes sign under a full rotation and requires two full rotations to return to its original form. Such a field is said to have spin 1/2, and so do its quantum excitations, which we call electrons. (Spinors are a kind of square root of a vector.)

Yes! No one knows what is this spin, but everyone believes it exists! Isn't that great science!

Thank you Scott Yost, said the same thing in response, but not as well as you put it.

Jim Baggott. 2012. Higgs. The invention and Discovery of the 'God Particle'. Oxford University Press

Hello! There is an interesting theory that proposes an classical analogous of spin [E. Entralgo and V. Kuryskin, Il Nuovo Cimento 103A, 561 (1990)], which is called Theory of a Structural Point-Particle. It was motivated by a quantization rule that requires a classical analogous of the quantum mechanical operators. In this theory, the classical spin is not an intrinsic mechanical (angular) momentum of the particle. The spin definition includes the intrinsic electric dipole moment and its rate of change in time. This classical spin does not interact with a gravitational field, only interacts with electric and magnetic fields. The reference above might be useful.

Cheers,

Alexis

What do you all think of the explanatory videos on the Looking Glass Universe - YouTube? I find the one on spin particularly helpful in visualizing, conceptualizing this difficult concept.