Widdekind

i personally perceive, at present, that one cannot construe "spin" as resulting, from "global" coherent spinning, of an electron's entire wave-function... that would have to be represented, by an azimuthal [math]e^{\imath \phi}[/math] type of term, which would have already appeared, in the Schrodinger solutions for H atoms

instead, the Schrodinger solutions can be construed, as "stable super-positions" of position eigenstates, [math]\Psi(x) = \Psi (x_1) \delta(x-x_1) + \Psi(x_2) \delta(x-x_2) \cdots[/math]

and the electron, partially present at each point, is, at each of those points, spinning

in analogy, if the electron wave-function is likened to a flock of birds, swarming around the proton, like crows around some type tree...

then each individual crow, hovering in place, is itself spinning around...

so that the overall spin of the entire flock is positive and pointing "up"...

but w/o any coherent rotation, of the flock as a whole, about the tree

if the electron wave-function collapsed, to some specific place in space, then it there would become fully present at that place, and would there be seen spinning

http://wiki.physics.fsu.edu/wiki/index.php/Klein-Gordon_equation

The above website says the KG equation admits "spontaneous transitions from positive-energy particle solutions, to negative-energy anti-particle solutions"...

Q1: could someone succinctly show, such a "spontaneous transition" sort of solution ?

Q2: i personally perceive, a parallel, between the opposite spins, of particles (left-handed) and antiparticles (right-handed)... and the opposite signs, of their frequencies, from the KG equation, which signs cause the phase-factors of particles to spin left-handed [math]e^{- \imath \omega t}[/math], and the phase-factors of antiparticles to spin right-handed [math]e^{+ \imath \omega t}[/math], around on the complex-plane... is it correct, to interpret the positive and negative energies, of particles and antiparticles, as actually merely positive (LH) and negative (RH) phase-factor frequencies ?

Q3: the "density" of antiparticles is conserved, but equals -1 (whereas particles normalize to +1)... so the KGE seemingly states, that antiparticles are "absent missing (but whole) holes" in the Dirac sea... is that not essentially the same, as what the Dirac equation demonstrates ? What does the Dirac equation actually demonstrate, that the KGE does not ?

from post 333 ?

 

The really important paper which attempts to describe this is

 

http://www.cybsoc.org/electron.pdf

 

Not only does it outline why there are problems with pointlike systems in the math, but it uses math to explain why particles like an electron are detected pointlike.

 

Not sure what makes you think this is correct.

is this correct :

[math]e_R^- + \bar{\nu}_R \longrightarrow W^- \longrightarrow \pi^-[/math] ?

and, can i ask, how to balance the so-called "hyper-charge", in the above equation ? For instance, is not the (net) hyper-charge, on the RHS, = 0 ? (And, is not the hyper-charge, of [math]\bar{\nu}_R = +1[/math] ?

Approximately stated,

photons carry no EM charge...

so, photons cannot become sources, for the creation, of new photons

Thus, photons are emitted by EM'ly charged particles... and then stream away thru space, essentially in fixed, constant, unchanging number...

and, thereby, (virtual) photons flight-paths trace out the "field lines" of EM fields...

so accounting for

[math]F \propto \frac{number \; field \; lines}{area} \propto \frac{number \; virtual \; photons}{area} \propto \frac{constant}{4 \pi r^2} = \frac{q}{4 \pi \epsilon_0 r^2}[/math]

But, gluons carry "color" charge...

and thus gluons can create new gluons...

so, as gluons propagate away from some "color" charged source, i.e. some quark...

those gluons will "breed" new gluons...

each of which will then also "breed" new gluons...

each of which new gluons will then "breed" further new gluons...

so on so forth...

And so, would not the number of gluons grow, exponentially, with distance from said quark source ?

And if so, then would not the Strong force, and Strong interaction potential, be dominated, by some term of the vague form

[math]F \propto e^{+\frac{r}{\alpha}}[/math]

where i emphasize the "+" to distinguish the same from the famous Yukawa potential, prominent in nuclear physics

does the PXP require, that two fermions, exist in orthogonal states, [math]< \Psi_1 | \Psi_2 > = 0[/math] ?

and, would that requirement only apply, to stationary states, i.e. time-independent solutions ? (In some hypothetical collision, the wave-functions, of two free electrons (say), could conceivably "crash" into each other, resulting, if ephemerally, in a positive overlap integral, between both particles ?

can you conceptualize, a charged W boson, as some sort of "crash zone", between matter and anti-matter ?

[math]e^- + \bar{\nu} \longrightarrow W^-[/math]

could you correctly claim, that the matter electron collided with the antimatter anti-neutrino, into a "crash zone complex", embodied by the charged W boson ?

If so, seemingly W bosons represent what occurs, when antimatter and matter encounter and collide and combine...

--------------------

can the following occur:

[math]e^- + e^+ \longrightarrow Z^0[/math]

?

Symmetry would suggest:

[math]e^- + \bar{\nu} \longrightarrow W^-[/math]

[math]e^+ + \nu \longrightarrow W^+[/math]

[math]e^- + e^+ \longrightarrow W^0[/math]

But according to the Electro-Weak theory, [math]W^0 \rightarrow Z^0[/math] (Weinberg mixing angle, etc.)...

so could and electron & positron "crash" into a [math]Z^0[/math] boson ?

imagine a 3D flat space, embedded in a 4D flat space

imagine a 3D object, in the 3D "hyper-3-plane" space

imagine a linear vector through the 4D flat space, from the center of the 3D object in the 3D "hyper-plane"…

hyper-dimensionally "out", to higher hyper-dimensional altitude, in the 4D space

i.e. the center of the 3D object = (0,0,0 | 0)

and the 4D linear vector = (0,0,0 | 1)

with its "tail" at the center of the object (0,0,0 | 0)

Now, please ponder spinning the 3D object, about the 4D linear vector, threading through its center, "out" to higher hyper altitude

in analogy, a 2D object, in a 2D flat-land, rotating about a 3D linear vector, from its center, "out" to higher 3D altitude…

would be rotating around, through the 2D perpendicular to the 3D vertical vector…

subject to the constraint, that all perpendicular distances were kept constant…

leaving (2-1)=1 degree-of-freedom to spin around in, which 1 DoF = azimuthal angle

so, if the 2D flat-land object were rotating with constant frequency, then it would be spinning around through azimuthal angle, periodically:

[math]\theta = 2 \pi f t[/math]

so, by analogy, a 3D object, in 3D space-land, spinning around a 4D axis threading through its center…

would be spinning through the 3D "xyz" perpendicular to the 4th hyper-dimension "w"…

subject to the constraint, of keeping constant all distances from that hyper axis…

so leaving (3-1)=2 DoF to spin around in, which 2 DoF = surface of a sphere of [math]4 \pi[/math] steradians…

i.e. a 3D object, spinning periodically about a 4D axis, would be seen "from the side" in 3D space-land, to be periodically spinning around, through the [math]4 \pi[/math] steradians of the surface of the sphere, i.e. of solid angle, centered at its center

i.e. from 2D to 3D:

[math]\theta(t) = 2 \pi f t[/math]

[math]\longrightarrow[/math]

[math]d\Omega(t) = 4 \pi f t[/math]

Question: is the above a correct extrapolation, from 2D "flatland", to 3D "spaceland" ? Would a 3D object in spaceland, spinning about a 4D axis threading through its center, be seen, to be spinning and twirling and tumbling around, through the 2 DoF of polar-angle and azimuthal-angle, coordinatizing the [math]4 \pi[/math] steradians of solid angle, about its center ?

A photon of wave-length = Planck length, has a mass-energy equivalent to the Planck-mass / energy...

so, even at the smallest something can be, its energy density still doesn't dominate gravitationally...

if that were not true... then wouldn't high-energy phenomena have already "twisted and contorted the fabric of space-time into donuts" and ripped out "holes to hyper-space" (for want of worthier words) ?

is another way of saying "Gravity is weak", to say instead, "the fabric of space-time is strong" (and pretty impervious to (individual) particles) ? Can you construe, the "Hierarchy Problem" of particle physics, i.e. that the Planck mass-energy scale is ~17 orders of magnitude larger than typical particle mass-energies... can you construe the same, as essentially "safeguarding" the fabric of space-time, from becoming overly curved ?

on page 2, the cybsoc article argues that internal EM stresses may exist, w/in the electron, trying to propel apart its charge distribution...

but, such "self-interaction" is not part of the successful Schrodinger solution, for Hydrogen atoms...

at the radius of the 1S orbital, Classical equations would predict an "auto-interaction" energy, of order

[math]\frac{e^2}{4 \pi \epsilon_0} \frac{1}{a_0} = \alpha^2 m_e c^2 \approx 30eV[/math]

so, if "self-interaction" from one region of the electron's wave-function, to another, were important; then such would already have been obvious, as a dominant term, in the famous Hydrogen wave-functions...

evidently, the electron wave-function does not interfere w/ itself EM'ly

if matter (electron) and anti-matter (anti-neutrino) can interconvert, to charged pions, via charged W bosons

[math]e^- + \bar{\nu} \leftrightarrow W^- \leftrightarrow \pi^-[/math]

(does anybody disagree w/ the above reactions?)

and if matter (electron) and anti-matter (positron) can interconvert, to neutral pions

[math]e^+ + e^- + \gamma \leftrightarrow \pi^0[/math]

(does anybody disagree w/ the above reactions?)

then why couldn't the neutral W boson be involved?

W bosons seem to be the pathways by which Fermions can interconvert, to-and-from Leptons & Hadrons. Is that true ?

electrons have two-component wave-functions, [math]\alpha \Psi_u (x) |u>[/math] + [math]\beta \Psi_d (x) |d>[/math]

at any place in space (xyz), you need four numbers to describe an electron's probability amplitude (R+I for spin up, R+I for spin down)

so, some electron, excited but bound to a proton, could conceivably exist with the following wave-function:

[math]\alpha \Psi_{1S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

and another, bound to the same proton, existing with

[math]\alpha \Psi_{2S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

if i understand correctly, the overlap integral is

[math]< \Psi_1 | \Psi_2 > = |\beta|^2[/math]

so the two electrons' wave-functions are not orthogonal...

Q: does PXP prevent two electrons from partially occupying the same (e.g. 2P-with-spin-down) state ?

Q: Can two electrons exist, with exactly the same (say) spin-down wave-functions (but w/ differing spin-up components) ?

electrons have two-component wave-functions, [math]\alpha \Psi_u (x) |u>[/math] + [math]\beta \Psi_d (x) |d>[/math]

at any place in space (xyz), you need four numbers to describe an electron's probability amplitude (R+I for spin up, R+I for spin down)

so, some electron, excited but bound to a proton, could conceivably exist with the following wave-function:

[math]\alpha \Psi_{1S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

and another with

[math]\alpha \Psi_{2S} (x) |u>[/math] + [math]\beta \Psi_{2P} (x) |d>[/math]

if i understand correctly, the overlap integral is

[math]< \Psi_1 | \Psi_2 > = |\beta|^2[/math]

so the two electrons' wave-functions are not orthogonal...

does PXP prevent two electrons from partially occupying the same (e.g. 2P-with-spin-down) state ?

@Enthalpy

most conduction electrons are available, for thermal conduction in metals, if i understand correctly...

TC is proportional to KT x Del(KT) = Cs x Cs x d(KT)/dx = speed for flux density x (dKT/dt) carried by electrons down gradient

i may have mis-stated something somewhere, about thermal conduction involving only electrons near the Fermi energy... but the above relations require no such sort of stuff... all electrons are involved... if they are more involved, when their metal is subject to strong EM forces, vs. passive meandering thru the metal in heat conduction, then such seems sensical

the combo, of an electron + neutrino, suggests the intermediary to be the W^- boson, i.e. for the reaction to involve the Weak force, not the Strong force

still... under certain circumstances, an electron can collide with a neutrino, and they both "break up" into two new Fermions, i.e. quarks (which, after the collision and "break ups", then begin emitting gluons, but not before, if i understand correctly)...

that seems sensible, and potentially important... electrons can "break up", losing some of their electrical charge, to a neutrino, in intensely energetic interactions... then, afterwards, subsequently, the "broken" particles begin emitting gluons, in "trying to knit themselves back together", which is my understanding of "color confinement"... the electron ----> d-quark, !v ----> !u-quark, so some of the electron's charge is "chipped off" and "lodges" in the neutrino... then, in their strange broken configuration, they then afterwards (only) begin giving off gluons...

only particles possessing partial electrical charges, emit gluons, and interact Strong-ly, yes ?

so i guess that the electron and anti-neutrino interact Weak-ly... re-configure into quarks... and only then begin generating "glue"

no spontaneous spin / chiral transitions occur ?

so, then, the energy of RH = LH = invariant w.r.t. spin-polarization-to-momentum-direction ? cp. photons, E = hv either forward-or-backward spinning ? simplistically stated, particles "don't care" whether they spin one-way or another... only Weak-interactions "care" about chirality ?

if i understand correctly, then there are no natural LH <----> RH transitions, at all... no RH electron can spin-flip, into a normal LH electron, via any known reaction... RH electrons can be produced, or be destroyed, by interactions... but no electron can "flip over on their own", or even "be flipped over" w/o also being destroyed, as an electron, in intense (Weak) interactions, which reconfigure them into quarks...

still, something seems amiss... in the reaction(s)

[math]e_R^- + \bar{nu}_R \leftrightarrow \pi^-[/math]

what is the hyper-charge, on the RHS ?? is the hyper-charge not (-1) on the LHS ?? so it cannot be (0) on the RHS ??

in general, GR metric matrices (can) have (up to) ten independent components…

so…

[math]ds^2 = \begin{array}{r} \left( dt dx dy dz \right) \end{array} g_{\mu \nu} \begin{bmatrix} dt \\ dx \\ dy \\ dz \end{bmatrix}[/math]

and we would want to embed the 4D manifold, into a higher dimensional hyperspace, of some unknown number of dimensions, which was flat, such that

[math]ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2 + \cdots + dx_N^2[/math]

you'd have to imagine mapping each manifold-enscribing coordinate (t,x,y,z), into an (unknown number) of "hyper-space coordinates":

[math]\begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix} \longrightarrow[/math][math] \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ \cdots \\ x_N \end{bmatrix}[/math]

so that, when, on the manifold, you began at point (t,x,y,z), and sought a small step to another nearby point (t+dt,x+dx,y+dy,z+dz), you were actually taking a straight step, through a higher-D hyperspace:

[math]dx_1 = \frac{\partial x_1}{\partial t} dt + \frac{\partial x_1}{\partial x} dx + \frac{\partial x_1}{\partial y} dy + \frac{\partial x_1}{\partial z} dz[/math]

[math]dx_2 = \frac{\partial x_2}{\partial t} dt + \frac{\partial x_2}{\partial x} dx + \frac{\partial x_2}{\partial y} dy + \frac{\partial x_2}{\partial z} dz[/math]

etc.

so, how many dimensions would the higher-D hyper-space require ??

on the RHS:

[math] = \left( \frac{\partial x_1}{\partial t} dt + \frac{\partial x_1}{\partial x} dx + \frac{\partial x_1}{\partial y} dy + \frac{\partial x_1}{\partial z} dz \right)^2 + \left(\frac{\partial x_2}{\partial t} dt + \frac{\partial x_2}{\partial x} dx + \frac{\partial x_2}{\partial y} dy + \frac{\partial x_2}{\partial z} dz\right)^2 + \cdots[/math]

you would wind up, with all of the same sorts of manifold-four-coordinate-differential pairs, e.g. [math]dt^2, dt dx[/math], on the RHS, as on the LHS...

but on the RHS, those differentials would be multiplied, by partial derivatives of the unknown hyper-coordinate mapping functions...

whereas on the LHS, those same differentials would be multiplied, by the 10 independent components of the metric matrix...

so, to have as much mathematical flexibility, on the RHS, as on the LHS, you'd have to pick 10 independent coordinate mapping functions, i.e. 10 independent coordinates..

i.e. hyper-space has 10D

essentially, you're re-casting all of the curvature information, w/in the metric matrix' components, into "dis-entangled" linearly independent orthogonal coordinate components, of a higher-D hyper-space

is this correct ??

simple analogy:

a 1D line, can require 3D, to curve through... you can construct 1D linear line-land spaces, which curve more complexly, than solely simply thru 2D on a plane page of paper

extension:

if space-time itself has more dimensions, e.g. 5, then hyper-space would have to have even more dimensions, to accommodate the complex curvatures conceivable, w/in the fabric of space-time, e.g. [math]5!^+ = 15[/math].

what about the exact inverse of the hyper-physics site:

[math]\bar{\nu}_R + e_R^- \rightarrow \pi^-[/math]

?

are you then stating, that right-handed fermions, cannot flip over to left-handed... w/o interacting, w/ another fermion... only collisions can spin-flip, spin-reverse, RH'd fermions ?

if photons, in super-conductors, acquire an effective mass (by "bouncing between pairs of electrons")…

then can such photons start to spin sideways, in the (S=1, M=0) third polarization state, w/ the projection of their spin, onto their path of propagation, equalling zero ?

(1) regarding "sloshing" as a (non-)technical term...

there are available visualizers, for finite potential wells' wave-functions in 1D...

if you choose appropriate parameters, then you can produce potential wells w/ three (3) discrete bound states

if you then choose your wave-function, to be a super-position of those three static stationary eigenstates (best by choosing coefficients from Weak-interaction mixing matrices, e.g. Cabibbo-Kobayashi-Maskawa)...

then you see, that the wave-function super-position states "sloshes" side-to-side, with its biggest peak bouncing back-and-forth from well-wall to well-wall

i'm suggesting, that the Higgs force-field is what binds mass-energy w/in the fabric of space-time... w/o the Higgs force-field, quanta would drift off the "brane" of space-time, out in to the hyper-dimensional "bulk" beyond...

the Higgs force-field is a 1D quantum well, whose "walls" are the "inner" and "outer" (hyper-)surfaces of the space-time fabric...

quanta embedded w/in the fabric of space-time occupy discrete bound states w/in the well...

and their finite energies, w/in the well, are their rest-mass energies...

e.g. an electron is the lowest-lying bound-state, for electrons, w/in the 1D Higgs potential "across" the fabric of space-time, through the thin "thickness" hyper-dimension "w"

muons are the first excited state of the same, and taons the second excited state

the Weak-force can be construed, as electro-magnetism, through the thin "thickness" hyper-dimension "w"...

when wave-functions interact intensely, so as to try to occupy the same place in space "xyz", then they start to "stack" through the thin "thickness" hyper-dimension "w"...

and their hyper-charges make them begin blasting each other, w/ hyper-high energy, hyper-dimension directed, photons = Weak bosons

when a neutrino emerges from Weak interactions, it exists in a super-position, of neutrino mass states [math]\nu_e = \alpha \nu_1 + \beta \nu_2 + \gamma \nu_3[/math]... the neutrino mass states would be the actual eigenstates of them in the Higgs potential...

But, known neutrinos emerge from Weak interactions, in the "electron" or "muon" or "taon" super-positions, of those eigenstates, w/ coefficients tabulated in the CKM mixing matrix...

those coefficients make the neutrinos "slosh" through the thin "thickness" hyper-dimension "w", bouncing back and forth from wall-to-wall of the 1D ("w") well of the Higgs potential

since, in their intensely energetic interaction, at "point blank range", through the "w" dimension, the neutrinos were blasted by 90GeV Weak bosons, so you'd expect, that conservation of momentum, applied to the hyper-dimension "w", would require conservation of hyper-directed-momentum...

so you'd expect the neutrinos to oscillate "out" and then "in" and then "out" etc. as they recoiled, thru the hyper-dimension, in conservation of hyper-momentum

that picture potentially explains "neutrino oscillations"...

in a Weak-force intensely intimate & energetic interaction, the neutrinos are basically "blasted back to the Higgs'-well-wall" and the "outer" or "inner" hyper-surface of space-time... after which they start "sloshing" side-to-hyper-dimensional-side, through "w", a little like a train car tipping from side-to-side in a rapidly running-away train on tracks

(2) oops on the Strong-Force

the Strong-Force affects the E/W force, and gluons carry E/W charge to-and-from quarks...

but the SF is not directly any kind of E/W force... instead, it is the force of "color confinement", interpreted as confining quarks into collections, of omni-directional spatial charge

(2A) Mesons

[math]\pi^+ = u + \bar{d}[/math]

the up-quark is a 2D "flat-land" particle, positively-electrically-charged in 2 standard-spatial dimensions "xy"...

that naturally defines an orthogonal 3^rd normal direction, perpendicular to the "xy" plane, i.e. "z"

that naturally would define the up-quark's direction of spin

[math]\vec{S}_u = \pm \frac{\hbar}{2} \hat{z}[/math]

the anti-down anti-quark is a 1D "line-land" particle, positively-electrically-charged in 1 standard-spatial dimension "z"

which would naturally define the anti-down-anti-quark's direction of spin

[math]\vec{S}_{\bar{d}} = \pm \frac{\hbar}{2} \hat{z}[/math]

the SF between them both, is a "twisting torquing" force, somewhat similar to magnetic dipoles in magnetic fields...

since (most) mesons are spinless scalars, so evidently the SF makes quarks spin-anti-align

[math]V \propto \vec{S}_q \circ \vec{S}_{\bar{q}}[/math]

[math]-\vec{F} \propto \nabla V[/math]

the interaction-potential is minimized, when their spins point anti-parallel

(2B) baryons

a proton is composed of two 2D ups and a 1D down quark...

their axial charges net-sum to (q_xq_yq_z) = (+++) if-and-only-if they all spin in mutually-orthogonal directions [math]\pm \hat{x} \pm \hat{y} \pm \hat{z}[/math]

u₁ = (++0)

u₂ = (+0+)

d = (-00)

------------

p = (+++)

so, the SF between a bunch of quarks (or between a bunch of antiquarks) is different, from the SF between opposite-matter-kinds of quarks

[math]V \propto \left( \vec{S}_{q_1} \circ \vec{S}_{q_2} \right)^2[/math]

[math]V_{p^+} \propto \left( \vec{S}_{u_1} \circ \vec{S}_{u_2} \right)^2+\left( \vec{S}_{u_1} \circ \vec{S}_{d} \right)^2+\left( \vec{S}_{d} \circ \vec{S}_{u_2} \right)^2[/math]

the interaction-potential is minimized, when their spins point parallel

(2C) distance term

presumably, the SF interaction also involves inter-particle distance; all now-known natural forces decrease w/ distance as [math]\propto d^{-2}[/math]

oops

"the interaction-potential is minimized, when their spins point parallel" [math]\longrightarrow[/math]

"the interaction-potential is minimized, when their spins point orthogonal"

requiring quarks to be mutually orthogonal w/in nucleons, cogently accounts for the current conundrum, of nucleon spin, not deriving directly, from constituent quarks:

http://en.wikipedia.org/wiki/Proton_spin_crisis

this hyper-space hypothesis, predicts that the three quarks' spins are all mutually orthogonal, effectively forming a LH or RH cartesian coordinate basis... so that they all form large angles, to their nucleon's net spin

the previous posts seem thoroughly thoughtful

i'm simply stating, that orthogonal wave-functions "avoid" PXP, i.e. their orthogonality (integrating the product of their phases) permits the particles to co-exist (?)

something somewhat similar, to a time-share on a vacation condominium, one electron is "there" whilst the other is "somewhere else", crudely quasi-classically considering the meaning of their mathematical "orthogonality"

--------------------

[math]\Psi_1^*(r) \Psi_2(r) = (a' - \imath b') \times (a + \imath b) = (a a' + b b') + \imath (a' b - b' a)[/math]

overall, the only way of getting zero, is for one of the "phasors" to be zero, so that where one wave-function is, the other is not...

orthogonality seems to be a sophisticated sense of the same

Where would the energy come from in an electron emitting a pion?

KE ? (the same place the pion's energy is deposited, during decay)

i think i managed to muddle matter-antimatter X right-handed/wrong-handed (for want of worthier words)

in the decay [math]e^-_R \rightarrow e^-_L[/math], the hyper-charge changes by (plus) one unit, whilst the (regular) charge is conserved...

to balance hyper-charge, without affecting charge, requires an electrically-neutrally, hyper-charged, particle...

potentially a neutral pion...

but antiquarks are intrinsically opposite-handed from quarks...

so q + !q = 0 net charge...

but only -1 hyper-charge, if one of the quarks is "wrong" handed, and the other "right"...

which means both must be spinning in the same sense

e.g. down-right (Y_W = -2/3) + anti-down-right (Y_W = -1 x 1/3)...

or up-left (Y_W = 1/3) + anti-up-left (Y_W = -1 x 4/3)...

BUT if both quarks are spinning in the same sense...

then the electrically-neutral pion, would have a net spin of one, yes??

the hyper-physics site states, that the decay

[math]\pi^- \longrightarrow e^-_R + \bar{\nu}_R[/math]

can occur...

but the net total hyper-charge on the RHS = (-2) + (-1 x -1) = -1

so 'tis the same on the other (LHS) side also...

but the only way to have a negative pion w/ net negative-one hyper-charge, is

down-right (-2/3) + antidown-right (-1 x 1/3)

if both quarks are bound into a common pion particle, and so are traveling together, then their linear momenta point parallel...

so if they're spinning in the same sense also, then their spins are pointing parallel...

and the pion would have to have a strong spin

the decay does occur, commonly enough to note

but Wikipedia purports that pions are spinless scalars

(?!)

Electro-Weak-Strong unification ?

Only one force, into which quanta couple with charge. Force-carrying Bosons can propagate, either through the standard spatial dimensions "xyz", or through the thin "thickness" hyper-dimension "w"; and, the Bosons can be charge-less, or charged:

[math]\bordermatrix{ \; & xyz & w \cr 0 & \gamma & Z^0 \cr q & g & W^{\pm} \cr}[/math]

Charge is a four-vector quantity:

[math]\tilde{q}=\begin{array}{r} \left( q_x q_y q_z q_w \right) \end{array}[/math]

and the four fundamental forms of Fermions are charged, in either zero (neutrinos), one (down-quarks), two (up-quarks), or three (electrons) standard spatial dimensions "xyz"; all have hyper-charge into the hyper-dimension "w"; all are massive, spin-one (S=1), having hyper-spin half (S_w = 1/2):

[math]\tilde{\nu}=\begin{array}{r} \left( 0 0 0 - \right) \end{array}[/math]

[math]\tilde{d}=\begin{array}{r} \left( - 0 0 - \right) \end{array}[/math]

[math]\tilde{u}=\begin{array}{r} \left( + + 0 + \right) \end{array}[/math]

[math]\tilde{e}=\begin{array}{r} \left( - - - - \right) \end{array}[/math]

Particles can rotate around through the standard spatial dimensions, during their interactions. So, on time average, [math]q_x \leftrightarrow q_y \leftrightarrow q_z[/math], and qualitatively, only the number of spatial-dimensional chargings (0-3) is of qualitative importance, and defines their Classical (scalar) charge

[math]q_C = \frac{1}{3}\sum_{xyz}q_i[/math]

Charged particles cannot accommodate both types of charge (+/-), loaded within the same particle wave-function, i.e. particle axial charges are either all positive (or neutral), or all negative (or neutral), i.e. particle charge four-vectors are positive-semi-definite, or negative-semi-definite.

To lowest order, this universe consists of equal numbers of protons (uud) and electrons (e), and so is both neutrally charged, and also neutrally hyper-charged. Demanding charge (and hyper-charge) neutrality imposes a pair of possibilities upon matter (protons + electrons, or anti-protons + anti-electrons). Whichever one occurred in the Big Bang, humans would come to call the cosmos' constituents "matter", and the other "anti-matter". So, this hyper-space hypothesis, and the Anthropic Principle, can account for so-called matter/antimatter asymmetry.

Force-carrying Bosons also can carry charge, in either zero (photons, Z⁰), one-two (gluons), or three (W^+/-) standard spatial dimensions "xyz":

[math]\tilde{\gamma}, \tilde{Z^0}=\begin{array}{r} \left( 0 0 0 0 \right) \end{array}[/math]

[math]\tilde{g}^+=\begin{array}{r} \left( + 0 0 0 \right) \end{array}[/math]

[math]\tilde{g}^{++}=\begin{array}{r} \left( + + 0 0 \right) \end{array}[/math]

[math]\tilde{W}^-=\begin{array}{r} \left( - - - 0 \right) \end{array}[/math]

When partially-charged particles interact, they either give away none of their spatial-chargings ("hold"), or give away all of their spatial-chargings ("fold / flush"). They cannot give away only some of their spatial-chargings (or else protons could decay into positive pions). Meanwhile, they can receive any missing spatial-charging(s) ("fill"). This hyper-space hypothesis could account, for positive pion decay, [math]\pi^+ \rightarrow \bar{e} + \nu[/math], without invoking the triply-charged W⁺ Boson, but instead solely single or doubly-charged g⁺, g⁺⁺ Gluons:

[math]\bar{e}=\begin{array}{r} \left( + + + + \right) \end{array} \; \bar{\nu}=\begin{array}{r} \left( 0 0 0 + \right) \end{array}[/math]

[math]\nwarrow \; \; g^{+,++} \; \; \nearrow[/math]

[math]\nearrow \cdots \cdots \cdots \nwarrow[/math]

[math]u=\begin{array}{r} \left( + + 0 + \right) \end{array} \; \bar{d}=\begin{array}{r} \left( 0 0 + + \right) \end{array}[/math]

Unlike Fermions, force-carrying Bosons can accommodate both types of charge (+/-) within one wave-function; force-carrying Bosons can be charged-and-anticharged:

[math]\tilde{g}'=\begin{array}{r} \left( \pm 0 0 0 \right) \end{array}[/math]

[math]\tilde{g}''=\begin{array}{r} \left( \pm \pm 0 0 \right) \end{array}[/math]

[math]\tilde{W}=\begin{array}{r} \left( \pm \pm \pm 0 \right) \end{array}[/math]

The last Boson, a hypothetical "double W", has been observed, as the source of supposed Higgs Bosons, [math]W^+ + W^- \rightarrow H^0[/math]. The Strong-force, inside nucleons, is mediated by "double Gluons", carrying a charge and an anti-charge, by which emitting quarks can rotate, inside their nucleon:

[math]u=\begin{array}{r} \left( + 0 + + \right) \end{array} \; u=\begin{array}{r} \left( + + 0 + \right) \end{array}[/math]

[math]\nwarrow\tilde{g}=\begin{array}{r}\left(0 + - 0\right)\end{array}\nearrow[/math]

[math]\nearrow \cdots \cdots \cdots \cdots \cdots \nwarrow[/math]

[math]u=\begin{array}{r} \left( + + 0 + \right) \end{array} \; u=\begin{array}{r} \left( + 0 + + \right) \end{array}[/math]

Strong-force "color charge", with its three possible values, and (nearly) nine possible "color / anticolor charged" Gluons, can be explained, instead, as directional chargings of quarks, and double-directional charged / anticharged Gluons:

[math]R G B \leftrightarrow x y z[/math]

[math]\tilde{g}_{R\bar{R}} \leftrightarrow \begin{array}{r} \left( \pm 0 0 0 \right)\end{array}[/math]

[math]\tilde{g}_{R\bar{Y}} \leftrightarrow \begin{array}{r} \left( + - 0 0 \right)\end{array}[/math]

[math]\tilde{g}_{R\bar{B}} \leftrightarrow \begin{array}{r} \left( + 0 - 0 \right)\end{array}[/math]

[math]\tilde{g}_{Y\bar{R}} \leftrightarrow \begin{array}{r} \left( - + 0 0 \right)\end{array}[/math]

[math]\tilde{g}_{Y\bar{Y}} \leftrightarrow \begin{array}{r} \left( 0 \pm 0 0 \right)\end{array}[/math]

[math]\tilde{g}_{Y\bar{B}} \leftrightarrow \begin{array}{r} \left( 0 + - 0 \right)\end{array}[/math]

[math]\tilde{g}_{B\bar{R}} \leftrightarrow \begin{array}{r} \left( - 0 + 0 \right)\end{array}[/math]

[math]\tilde{g}_{B\bar{Y}} \leftrightarrow \begin{array}{r} \left( 0 - + 0 \right)\end{array}[/math]

[math]\tilde{g}_{B\bar{B}} \leftrightarrow \begin{array}{r} \left( 0 0 \pm 0 \right)\end{array}[/math]

[math]\tilde{g}_{i j}^{k} = \delta_i^k - \delta_j^k[/math]

to try tensor notation.

why do we (nearly) never notice the 4^th spatial D ?

why Weak alpha (~1/4) is 40x EM alpha ?

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/hadron.html

The symmetry which suppresses the electron pathway is that of angular momentum, as described by Griffiths. Since the negative pion has spin zero, the electron and antineutrino must be emitted with opposite spins to preserve net zero spin. But the antineutrino is always right-handed, so this implies that the electron must be emitted with spin in the direction of its linear momentum (i.e., also right-handed). But if the electron were massless, it would (like the neutrino) only exist as a left-handed particle, and the electron pathway would be completely prohibited. So the suppression of the electron pathway is attributed to the fact that the electron's small mass greatly favors the left-handed symmetry, thus inhibiting the decay.

 

[math]\pi^- \longrightarrow \mu^- + \bar{\nu_{\mu}}[/math]

[math]\gg [/math]

[math] \longrightarrow e^- + \bar{\nu_e}[/math]

however, when the decay does occur, the emitted electron is [math]e^-_R[/math]

http://en.wikipedia.org/wiki/Pion#Neutral_pion_decays

[math]\pi^0 \longrightarrow \gamma + e^+ + e^-[/math] (1%)

so, if the reaction can occur one way, then it can occur the other way...

leptons (electrons + neutrinos) can convert, to hadrons (quarks)...

(which seems a strong step towards Strong / Weak unification)

according to vector calculus, the photon's momentum density is [math]\propto \vec{E} \times \vec{B}[/math]... e.g. [math]\vec{A} = \hat{\phi} \frac{A_{\phi}{r} e^{\imath \left( k z - \omega t \right) }[/math]

so any angular momentum would require [math]\vec{r} \times \left( \vec{E} \times \vec{B} \right)[/math]... which would require photons' fields to be (partially) longitudinal...

if the electrons repelled, then they would propagate apart, and not continue to exist, in stationary energy-constant states

in stationary states, the wave-functions are unchanging (except for an energy-dependent phase factor)

 

But EM radiation is not the result of an electrostatic field. It's oscillating E and B fields. Circular polarization occurs when the E and B are 90º out of phase. The field rotates about the propagation axis.

 

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/polclas.html

not what i suggested... perhaps a photon carries a magnetic dipole field, associated w/ its spin, along w/ that oscillating field (B = B_oscillate + B_dipole)

what about other bosons ? What about charged Weak-bosons... w/ charge & spin, they plausibly possess magnetic moments (?)