None, since particles of spin -1/2 don't exist.

It's not useful to confuse spin of a particle, which can take only non-negative values, with the value of the helicity of the particle, that can take values of either sign.

Virtual particles don't have well defined mass, or spin.

All this is standard material in all courses on quantum field theory, e.g. Article Notes from Sidney Coleman's Physics 253a

However, if addition of angular momentum isn't known, a review of quantum mechanics is a prerequisite, e.g. http://hitoshi.berkeley.edu/221A/index.html

The bound state of two particles of spin 1/2 can have spin 1 or spin 0.

The spin 0 state is antisymmetric under the exchange of the two particles; the spin 1 state is symmetric under the exchange.

The interactions between spin 1/2 that produce such bound states are then determined by imposing, for instance, invariance under rotations. It's a standard exercise to write down all such interaction terms.

Dear Richard,

In the BCS superconductivity model you have an example of what you are asking. There are pairs of electrons with opposite spins (+1/2 and -1/2) which interact through phonons of the lattice which act as virtual bosons particles. The spins plus and minus one half are the eigenvalues of the z component of the spin for the electron

Sz|+>=1/2 |+>

Sz|->=-1/2 |->

I suspect Mr. Petti didn't mean spin as the eigenvalue of the operator S, which is 1/2 for an electron, but the eigenvalue -1/2 of the operator Sz.

Another example would be the decay of positronium, where the virtual particle is an electron going either forward or backward in time.

Actually, as i understand it, no particle has well-defined properties until a measurement is made. Since a virtual particle is not subject to direct measurement, i guess Dr. Nicolis' statement about them is OK. However, it does seem to me that the virtual electron intermediating the positronium decay having a spin of 1/2 is consistent with what is known.

Just to clarify the positronium decay interaction for Mr. Petti, assuming it is appropriate to his question, the interpretation i learned back in the day has the electron emitting a gamma-ray photon and a virtual electron and disappearing. The virtual electron can either go forward or backward in time to intercept the positron. Both disappear as a second gamma-ray photon is emitted.

By the way the total spin of the photons of the decaying singlet state of positronium is zero.

Thank you Stam, Daniel and Francis; yes, I mean z-spin.

Your comments stimulate me to clarify my understanding of a virtual particle. My understanding is that a virtual particle has a momentum vector with positive or negative length, so the particle appears to move either on a future timelike, past timelike or spacelike trajectory. However, I believe no particle need move in a spacelike direction: could the virtual particle be a construction that masks a sequence of future and past-directed particles that link up to form a pathway that can communicate momentum, spin, and charge (electric, strong and weak) from one real particle to the other?

Is this interpretation correct? Is this the best way to interpret virtual particles and their internal lines in Feynman diagrams?

I am most interested in the case where the z-spin of two real particles are combined via a virtual particle so the z-spin of the reaction products is zero.

Daniel, does the phonon interaction you describe fit my description above?

Francis, does the positronium decay you describe fit my description above?

I am reaching to apply the geometry of dislocation networks in crystals and of spin/torsion in Einstein Cartan theory to conclude that a virtual particle that fits my criteria has the geometry of an edge dislocation that connects the ends of two oppositely-oriented screw dislocations.

a) In the continuum theory of dislocations, a screw dislocation is described by antisymmetric torsion (a 3-tensor), and edge dislocations are described by the torsion trace (a 1-tensor = vector). Dislocation lines cannot end inside a crystal; so when two screw dislocations of opposite helicity end, an edge dislocation that runs from one screw to the other is he simplest way to maintain continuity of dislocation lines.

b) In Einstein–Cartan theory, a spin 1/2 Dirac particle is described by antisymmetric torsion; it has the geometric structure of a screw dislocation. A virtual particle that behaves as I have described would seem to have the geometric structure of an edge dislocation that joins the ends of two screw dislocations; that is, the virtual particle would carry torsion trace.

Torsion is fundamental entity in affine differential geometry. My thinking is guided by concepts of classical differential geometry and classical spacetime physics, deferring quantum mechanical issues until later.

Thank you for your help.

--Richard

Dear Richard,

If you want to apply the concept of virtual particle to elastic defects as the dislocations, you have many different difficulties:

1. They are not particles and therefore it is very difficult to apply them a virtual particle. The virtual particles are thought for allowing the interaction between particles using the uncertainty principle of quantum mechanics. I cannot imagine it for these macroscopic objects.

2. They are always artefacts of a perturbation method which means that you need a part of the Hamiltonian much smaller to be considered.

3. In the case of BCS the boson exchanged (a singlet), between the two electrons for forming the Cooper pair, allows us to conseder it as virtual particle. I recommend you the book of Piers Coleman, in Chapter 14 you have a development of this interaction and how the boson acts as a virtual particle, besides you have the Nambu spinors which can be close to the spinor representation that you have asked

https://www.cambridge.org/core/books/introduction-to-manybody-physics/B7598FC1FCEE0285F5EC767E835854C8

Dear Francis,

It seems to me that the decay of the positronium is not a clear example of what was asking Richard. One of the difficulties is that the electron and the positron have spin 1/2 both, given for the positronium S=0 (singlet) or S=1(triplet) with Sz=-1, o, +1.

The singlet state is the para-prasitronium which decays in two (or in an even number) photons (511 keV each one), while the triplet state decays in three photons (or an odd number) called ortho-positronium.

But there are also othe decays taking into account the weak interaction and therefore the neutrinos where the virtual particles are the bosons W+,W- and Z. This makes much more difficult to analyze the spins configurations as it was asked in this thread.

Dear Daniel,

Yes, but there is still the case of the singlet state annihilating into two photons. I don't really follow your objection.

On the other hand, do you, Richard, mean the TOTAL spin is zero or none of the particles that result have spin? If the latter is the case, then my example is not appropriate, since although the total spin of the photons is zero, each has a spin of one but in opposite directions. I tend to agree with Daniel, however, in that i don't see the application to the problem you are addressing. But, again, i don't see a lot of things. :-(

On the other, other hand, it sounds to me like you are describing a (virtual?) phonon with torsion, but a phonon is a boson with spin zero. So, i must admit i'm a bit confused.

Daniel: "If you want to apply the concept of virtual particle to elastic defects..."

It is more like the reverse: I want to use dislocation theory to support a conjecture about some virtual particles.

I want to establish a credible analogy between a particular process in dislocation theory to show that some virtual particles (that enable two particles with opposite z-spins to cancel their z-spins, and that appear to have spacelike momentum) carry a field that is the abstract geometric embodiment of an edge dislocation.

The mechanism that enables two oppositely-oriented screw dislocations to end -- without violating the law that dislocation lines cannot end inside a crystal grain -- is to join the ends of the screw dislocations by an edge dislocation. The continuum geometric essence of an edge dislocation is the torsion trace.

Torsion already describes physical phenomena in two fields of physics at different length scales.

a) Torsion is the core of the continuum theory of dislocations. This has been an established part of rational mechanics since the mid-1950s. The key authors include Bilby and Kondo.

b) Torsion describes all intrinsic angular momentum (a.m.) in Einstein-Cartan theory (EC), which is a modest extension of GR that fixes the treatment of a.m. in spacetime physics. This includes quantum spin, especially in situations with very high spin density and very strong gravitational fields (the early universe).

c) I suspect that torsion must be included in a complete theory of turbulence. The essence of turbulence is transfer of orbital a.m. to length scales too small to be represented as orbital a.m. So turbulence must create classical intrinsic a.m., which is not included in current theories of turbulence. Feynman considered turbulence to be least-well-understood phenomenon in classical physics. This is a separate discussion.

Thank you for your thoughts.

For quarks, leptons and neutrinos

S_z=1/2 or -1/2. h-bar=1.

And the magnitude of the spin

S=sqrt[1/2(1+1/2)]=sqrt(3/4).

Dear Richard,

Many years ago I tried to employ the geometric concept of torsion (besides curvature) for explain the high-temperature superconductors (with perovskites). I attach you the paper. Obviously these objects are not particles and it is very complicated to stablish a Hamiltonian for applying the perturbation method where the virtual particles can play an important role (if any).

Dear Francis,

You wrote:

Yes, but there is still the case of the singlet state annihilating into two photons

It is true that the main properties of positronium are governed almost completely by QED – strong and weak interaction corrections are below the level of current interest due to the small value of the electron mass. But some features of positronium tend to complicate the analysis compared to, say, hydrogen or muonium, because positronium is composed of a particle and its antiparticle, it exhibits real and virtual annihilation into photons.

Francis, this process is quite complex and even if you reduce it to a para-positronium decay where is involved the virtual particle corresponds to the vacuum polarization modifying the photon propagator. Its ground state energy level has a correction due to vacuum polarization effects in the two-photon-annihilation channel which is not simple to see.

In summary, the positronium decay has many different cases to study and not only a singlet can be assumed and even in this case the concept of virtual particle is mixed with the vacuum polarization for particle-antiparticle annilation. I don't say that this is possible, what I said is that is mixed with many other different subtle concepts.

Dear Daniel and Francis,

The Feynman diagram for pair annihilation of an electron and a positron shows:

a) an electron line and a positron line entering from the bottom, each line ending at a vertex.

b) two photons leaving from the top, one from each vertex where a lepton line ends.

c) a horizontal line in the middle that connects to two vertices. I believe this line represents the bound positronium state; is that right?

This diagram appears in books about relativistic quantum mechanics and in the Wikipedia article on “particle–antiparticle annihilation”. https://en.wikipedia.org/wiki/Annihilation

Why is the positronium state included? Why doesn’t the diagram omit the horizontal line, and show the electron and positron lines ending at the same vertex, and the two photons emerging from the same vertex?

The horizontal like looks like it represents a virtual state that is not stable, that has spacelike momentum, and that carries z-spin between the electron and the positron. Is this correct?

Sorry for the delay; I have been exhausted by overwork for the past few months, and his is my first week with a little rest.

Thanks,

-Richard

Intrinsic spin (not the z-component of it) of a particle is always non-negative.

However, SU(2) isospin can be either positive or negative. For example a u-quark having isospin +1/2 may couple to a d-quark of isospin -1/2. At the interaction vertex a W boson is either emitted or absorbed, depending on your Feynman diagram. W boson is a isospin one particle.

If you want to know how to change z-component of intrinsic spin of a particle (say electron) then look at the interaction of magnetic field with electron spin. Magnetic fields can be manipulated to reverse z-component of intrinsic spin. See literature on Hahn-echo, e.g, "Spin echoes", E.L. Hahn, Physical Review. 80: 580–594 (1950).