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Initial post: Nov 29, 2012 11:10:42 AM PST
Last edited by the author on Nov 29, 2012 10:16:14 PM PST
D. Colasante says:
About this time every year, I treat myself to a review of condensed, highlighted and annotated versions of some favorite threads. Sure, there's a lot of "flack" here, but that's easy enough to edit. In just a few years, I've accumulated a valued collection of truly informative posts including both insightful questions and well-crafted answers. Thanks to everyone who made the effort!

I've enjoyed frequent discussions of relativity. Occasionally, the concept of 4-velocity and 4-momentum arise, which I will quickly summarize.

Every particle has a "world line", which describes its path through 4D spacetime. A constant progression, for example, appears in a typical spacetime diagram as a straight line. This is equally true for conventional velocity (e.g. meters/sec.) and for a particle "at rest", which has a vertical world line also known as its timeline. Typically, space is assigned a horizontal axis and time a vertical one. Because they have different units, a proportionality constant is employed.

It is foundational to relativity that there is a universal speed limit, c. As such, a dimensionless* fraction (between 0 and 1) corresponds to every conventional velocity. Dimensionless expressions would be especially useful in communicating with intelligent aliens who never heard of a "meter" or a "second". c is the proportionality constant note above.

A consequence of dimensionless velocities is that both momentum (p = mv) and energy (e.g. KE = ½mv²) are found to have the same units (mass). These are combined to express a "4-momentum"[1] (a.k.a. "momenergy"[2]), where the energy component (E) is ascribed to the time axis and the momentum components (p) remain along the spatial axes. This allows the expression of an invariant** mass (mi) where mi² = E² - p², from which (when p = 0) the more famous equation E = mc² is readily derived (recall, c = 1).

Given 4-momentum, I must assert the existence of a 4-ANGULAR momentum. Such a notion exists, indeed even "rotational invariance" [3]. But here's the problem. While angular momentum components about the spatial axes are recognized, the analogous component, which MUST exist about the temporal axis (my "chronaxial spin"), is disgracefully avoided. To wit:

"However, there is another type of angular momentum, called spin angular momentum (more often shortened to spin)... Almost all elementary particles have spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: Spin is an intrinsic property of a particle, unrelated to any sort of motion in space. All elementary particles have a characteristic spin, for example electrons always have 'spin½'..."[3,4]

Of course, quantum spin is "unrelated to any sort of motion in space". We're talking about spaceTIME! What a ridiculous cop out! Or, am I misreading this?

*Dividing every velocity by c, yields each as a fraction of "lightspeed" having no units.
**Relativistically "invariant" means viewed the same by all inertial (constant velocity) observers. Speed c is invariant (c = c'). So are the intervals (d) separating events in space (x) and time (t) by an analogous equation d² = x² - t².
1] http://en.wikipedia.org/wiki/Four-momentum
2] http://physics.bu.edu/~duffy/ns547_spri ... energy.pdf
3] http://en.wikipedia.org/wiki/Angular_momentum
4] http://en.wikipedia.org/wiki/Angular_momentum_operator

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Posted on Nov 29, 2012 11:40:46 AM PST
The Weasel says:
"Yours is superior"

In reply to an earlier post on Nov 29, 2012 12:45:07 PM PST
Last edited by the author on Nov 29, 2012 4:20:12 PM PST
D. Colasante says:
Thanks W,

A word to the wise, in case I haven't mentioned it recently. I am *permanently* banned from the Physics Forums for "continued promotion of personal theories".

Posted on Nov 29, 2012 10:42:59 PM PST
Last edited by the author on Nov 29, 2012 10:49:05 PM PST
D. Colasante says:
Short Version:

If we accept that the 4-velocity and 4-momentum of spacetime each have three independent spatial components and one temporal component, I can imagine no way to preclude analogous 4-angular velocity and 4-angular momentum, each having identical components.

Instead, I find for relativistic mechanics:

"In the language of four-vectors and tensors the angular momentum of a particle in relativistic mechanics is expressed as an antisymmetric tensor of second order"

and for quantum mechanics:

"total angular momentum ... combines both the spin and orbital angular momentum"
where orbital angular momentum is about the three spatial axes and *spin* is about a non-spatial axis.
"Spin is an intrinsic property of a particle, fundamentally different from orbital angular momentum...unrelated to any sort of motion in space."

That a temporal spin component is neglected is inexplicable.
That instead, a new "intrinsic" axis is construed seems patently absurd.

In reply to an earlier post on Nov 30, 2012 6:01:56 AM PST
If there is an angular momentum 4-vector composed of three orbital angular momentum components and one spin angular momentum component, then there must be transformations of this 4-vector in which the transformed components are functions of all 4 of the components of the untransformed vector. This would imply that orbital angular momentum in one reference frame would be dependent on spin in another, and vice versa. Is there any evidence for this?

In reply to an earlier post on Nov 30, 2012 9:44:52 AM PST
Last edited by the author on Nov 30, 2012 9:50:41 AM PST
D. Colasante says:
af >> Is there any evidence for [4-angular momentum]?

I'm too ignorant to tell. My point has always been that if translation is possible along four coordinates then spin must also be possible about four coordinates. I think the question must be, "Can that ever not be the case?"

As mentioned in my OP, I did find "rotational invariance" [1,2,3] which to me, suggests a relationship between spatial and temporal spin analogous to the known invariants: interval, 4-velocity and 4-momentum. These are said to exhibit a hyperbolic geometry (i.e. containing a minus sign between squared spatial and temporal components). I'm clearly out of my element, but I found a minus sign here (I will indicate subscripts as {i,j} ):

"In the language of four-vectors and tensors the angular momentum of a particle in relativistic mechanics is expressed as an antisymmetric tensor of second order:
L{i,j} = x{i}p{j} − x{j}p{i}" [1]

I also found this:

"Mathematically, the structure of rotations in the universe is not SO(3), the group of three-dimensional rotations in classical mechanics. Instead, it is SU(2), which is identical to SO(3) for small rotations, but where a 360° rotation is mathematically distinguished from a rotation of 0°. (A rotation of 720° is, however, the same as a rotation of 0°.)"[2]

and I found this:

"In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger's equation. That is:
[R, E − H] = 0 for any rotation R.
Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have [R, H] = 0." [3]

Unless we equate chronaxial spin (about a temporal axis) and intrinsic spin (a.k.a "quantum spin" and "spin½" for fermions), we are faced with a dilemma. We have a temporal coordinate upon which translation occurs but for which no rotation is recognized. And correspondingly, there exists some "intrinsic" coordinate axis about which spin occurs but for which no translation is recognized. Can you blame me for seeing a connection?

1] (see Angular momentum in relativistic mechanics) http://en.wikipedia.org/wiki/Angular_momentum
2] (see SU(2), SO(3), and 360° rotations) http://en.wikipedia.org/wiki/Angular_momentum_operator
3] (see Application to quantum mechanics) http://en.wikipedia.org/wiki/Rotational_invariance

In reply to an earlier post on Nov 30, 2012 10:07:07 AM PST
Total angular momentum is the sum of orbital and spin angular momentum. Total angular momentum is conserved, whereas orbital and spin angular momentum are not separately conserved and angular momentum can be transferred back and forth between them. But I don't find an angular momentum 4-vector which unites them.

The SU(2) group applies to spinors, such as Dirac spinors, which are solutions of Dirac's equation (electrons and positrons). As you point out, it takes 720 degrees of rotation to reach the same state again.

What do you think rotation about a temporal axis would be physically? I've never run across that, and I don't know why it would be necessary.

In reply to an earlier post on Dec 1, 2012 10:29:10 PM PST
Last edited by the author on Dec 2, 2012 8:09:42 AM PST
D. Colasante says:
af >> Total angular momentum is the sum of orbital and spin angular momentum... But I don't find an angular momentum 4-vector which unites them.

Nor could I find "4-angular momentum" explicitly in my review. Nevertheless, the concept is unavoidable.

Given: orbital angular momentum (about a spatial axis) of a particle translationally "at rest",
Given: the relativity of space and time in various inertial reference frames, (x' at an angle to x)[1],
Then: other observers, who see the particle moving along the spatial axis, must also find a temporal component to the particle's spin axis and thus, its spin (axis' at an angle to axis).

If I did that correctly, the existence of what I call "chronaxial spin" is a moot point. All the more surprising not to find it explicitly stated.

af >> What do you think rotation about a temporal axis would be physically? ... I don't know why it would be necessary.

Excellent question! The same might have been asked originally about linear 4-momentum. But as you have eloquently recounted [2], in the special case where conventional (spatial) momentum is zero, Einstein gave us E = mc² for a particle's rest mass. Quite a find!

In view of that, how could we NOT follow Einstein's example and pursue 4-angular momentum in the same way? As orbital angular momentum goes to zero, intrinsic spin remains, associated with rest mass. But there's a bonus. When the magnetic field of a fermion with orbital spin goes to zero what remains is electric charge. Charge is a manifestation of intrinsic spin. I see electric charge as "rest magnetism". And because time is unidirectional, there are exactly two kinds of electric charge, one for each of its absolute spin directions. (Orbital spin has no absolute direction, varying with perspective.)

af >> Dirac's equation (electrons and positrons)...it takes 720 degrees of rotation to reach the same state again.

I believe that confirms that intrinsic spin is chronaxial spin. With time as an axis perpendicular to a spatial surface, as in Minkowski space [3], the angle encompassed by a circle drawn on that surface (i.e. a single orbit) rises as a geometric progression with the dimension of that surface (angle = 90° * 2^n, where n is the surface dimension).

For example, an orbit in:
a 1D surface encompasses 180°
a 2D surface encompasses 360°
a 3D surface encompasses 720°

1] (see first diagram) http://en.wikipedia.org/wiki/Minkowski_diagram
2] (starting at 2/4/12) http://www.amazon.com/forum/science/ref=cm_cd_pg_pg14?_encoding=UTF8&cdForum=FxZ58KVEERYS5E&cdPage=14&cdSort=oldest&cdThread=Tx1G6K0HR4I53QD
3] (see first diagram) http://en.wikipedia.org/wiki/Minkowski_space

edit: added electric charge = "rest magnetism"

In reply to an earlier post on Dec 2, 2012 2:17:00 PM PST
[Deleted by the author on Dec 2, 2012 2:17:27 PM PST]

Posted on Dec 3, 2012 11:30:18 AM PST
D. Colasante says:
Consistent with the notion that electric charge is "rest magnetism" (seen when orbital angular momentum is zero), in the same way that there is a residual "rest mass" (or energy) seen when linear momentum is zero, I found the following:

"In particular, a phenomenon that appears purely electric to one observer may be purely magnetic to another, or more generally the relative contributions of electricity and magnetism are dependent on the frame of reference. Thus, special relativity "mixes" electricity and magnetism into a single, inseparable phenomenon called electromagnetism (analogous to how relativity "mixes" space and time into spacetime)."[1]

"...what one observer perceives as an electric field, another observer in a different frame of reference perceives as a mixture of electric and magnetic fields."[2]

"...charge is a relativistic invariant. This means that any particle that has charge Q, no matter how fast it goes, always has charge Q. This property has been experimentally verified..."[3]

If we accept charge as a (chronaxial) spin phenomenon, then parity reflection entails charge reversal (as it does for any spin) and the astounding finding of weak parity violation* reverts to a very rare occurrence rather than the current impression of complete breaking.[4] That is, C reflection and P reflection cease to be considered separate and are instead understood to be united into one CP reflection.

*The same applies to its complement, charge symmetry violation.
1] http://en.wikipedia.org/wiki/Magnetism
2] http://en.wikipedia.org/wiki/Electric_field#mw-head
3] http://en.wikipedia.org/wiki/Electric_charge
4] http://en.wikipedia.org/wiki/Parity_%28physics%29

In reply to an earlier post on Dec 3, 2012 3:45:02 PM PST
Last edited by the author on Dec 3, 2012 3:45:48 PM PST
There is one problem with positing spin as the temporal component of an angular momentum 4-vector: the spin of a particle has an enormous significance for its behavior in aggregates (spin-statistics theorem). So if you put spin into a 4-vector, you have to accept that it would change in accordance with Lorentz transformations. But if the spin changes, that would change the behavior of the particle in aggregates. So in a different frame of reference would electrons no longer obey the Pauli exclusion principle? Would bosons no longer undergo Bose-Einstein condensation? I think this would pose grave difficulties.

In reply to an earlier post on Dec 4, 2012 10:55:03 PM PST
D. Colasante says:
af >> positing spin as the temporal component of an angular momentum 4-vector
I don't see a way around a temporal component to angular momentum. If quantum spin is different than this component, it represents some 5th spin axis! I think the model is simpler without that. The uncertainty principle raises some challenge in positing even a spatial 3-axis for spin [1], but I don't' see how that's any more a problem for angular momentum than for linear momentum.

af >> spin...would change in accordance with Lorentz transformations.
Though relativistic mass changes according to Lorentz transforms the rest mass that it's based upon doesn't. It's always there, verifiable by an observer co-moving with the particle. I see rest magnetism (charge) the same way. It's always there, in every magnetic field observable. I relate charge to the temporal spin component (chronaxial spin), which persists even when orbital spin is zero.

af >> So in a different frame of reference would electrons no longer obey the Pauli exclusion principle? Would bosons no longer undergo Bose-Einstein condensation?
The Pauli exclusion principle is safe! While I would claim that two electrons sharing an orbital have the same charge (thus the same chronaxial spin direction), that does not stop them from having different phases. Remember the Pauli exclusion principle is a rule not a mechanism.
If you picture the two electrons as arrows, each spinning around a temporal axis then +½, and -½ may easily refer to their having opposite orientations on a clock face in their plane of rotation (e.g. 12 o'clock and 6 o'clock). Both spin the same direction but they would be 360° out of phase (since this is a 3-plane).

Bose-Einstein condensates, to my recollection are numerous particles appearing to share the same quantum state. I don't know much about it but I can offer the idea of arrows spinning in phase, like synchronized swimmers.

A spinning arrow models an unpaired force, the potential of forming a force pair.

[1] (see Spin Vector) http://en.wikipedia.org/wiki/Spin_(physics)#mw-head

In reply to an earlier post on Dec 5, 2012 7:02:40 AM PST
OK, I know this is an appeal to authority, not really scientific, BUT.......

If spin really is the temporal component of an angular momentum 4-vector, why isn't this in the physics books? I would conclude by its absence that there is some problem with this concept of which I'm unaware.

In reply to an earlier post on Dec 5, 2012 9:40:23 AM PST
Last edited by the author on Dec 5, 2012 9:43:05 AM PST
D. Colasante says:
af >> why isn't this in the physics books?...there is some problem with this concept

The "problem" is that the temporal axis is so non-intuitive. Physics already acknowledges the extraordinary 720° rotational requirement (which a temporal axis provides). But it's goes further than that. You can easily imagine spin around any axis along which you can aim in space but you can't aim along a temporal axis any more than a Flatlander inhabiting an XY-surface can aim along the Z-axis.

That is, UNLESS the Flatlander is a point particle. Point particles see every available spin axis equally well. They aim up and out of a plane containing them as easily as they aim within it. This scale asymmetry is a tangible* basis for things being different in the quantum realm. We are Flatlanders inhabiting a 3-surface [1], from which point particles readily view a temporal axis that we cannot.

*compared to Heisenberg's Uncertainty Principle
1] (see first diagram) http://en.wikipedia.org/wiki/Minkowski_space

In reply to an earlier post on Dec 5, 2012 9:48:40 AM PST
Well there are many other non-intuitive things that ARE in the physics books, so that can't be the whole answer.

In reply to an earlier post on Dec 5, 2012 2:58:00 PM PST
Last edited by the author on Dec 5, 2012 3:05:05 PM PST
D. Colasante says:
af >> physics books

As these books have presented to us *linear* 4-velocity and 4-momentum, I should think the burden must be on the authors to explicitly state any exception relating to *angular* 4-velocity and 4-momentum.

Based upon your many, well-informed and helpful posts, in addition to your extensive and thorough book reviews, I must conclude that you are indeed very well served by your reliance on books. However, sometime *after* you have been awarded an advanced degree (which seems inevitable), I hope you will consider the following:

A professional physicist may possess a Phelps-like ability to swim in the deep end of physics, yet a rank amateur may still doggie paddle right past him in the race for new discovery, if he refuses to release his white knuckle grip on the side of the pool! Ultimately, books were never meant to serve as life jackets. They're a platform made for DIVING OFF!

Come on in. The water's fine!

In reply to an earlier post on Dec 5, 2012 3:01:22 PM PST
I be skeptikal.

In reply to an earlier post on Dec 5, 2012 3:09:56 PM PST
Last edited by the author on Dec 5, 2012 3:48:31 PM PST
D. Colasante says:
And you're not skeptical about "weak symmetry breaking" that happens to be complete, though entirely unanticipated by books?!

edit: Meanwhile, the highly-anticipated (in literature) supersymmetry is sinking like a brick.[1]

1] http://www.scientificamerican.com/article.cfm?id=supersymmetry-fails-test-forcing-physics-seek-new-idea&WT.mc_id=SA_WR_20121205

In reply to an earlier post on Dec 5, 2012 3:24:44 PM PST
I'm not familiar with weak symmetry breaking. Do you mean electroweak symmetry breaking?

Thanks for the link. I've done very little reading or studying about supersymmetry. Just goes to show that no matter how promising a theory is, it requires experimental confirmation before it can be accepted.

In reply to an earlier post on Dec 5, 2012 3:45:42 PM PST
D. Colasante says:
Sorry. I was referring to parity violation with respect to weak interactions [1], as I had noted in my post of 12/3/12.

1] http://en.wikipedia.org/wiki/Parity_%28physics%29

In reply to an earlier post on Dec 5, 2012 3:50:26 PM PST
How does that tie in to the question of the angular momentum 4-vector?

In reply to an earlier post on Dec 5, 2012 8:19:50 PM PST
Last edited by the author on Dec 5, 2012 10:52:24 PM PST
D. Colasante says:
Before 1956, it was assumed that all the physical laws held under charge, parity or time reversal. (For our purposes, "parity reversal" means reversing one of the three spatial dimensions as occurs looking in an ordinary mirror, which reverses front-to-back.) Two scientists, Chen Ning Yang and Tsung-Dao Lee, found satisfactory evidence for this in the literature for all but the weak force (a.k.a. weak interaction), responsible for particle transformations that often result in beta (electron) emissions. It seemed to them that incidental evidence suggested the opposite, that there was not P-symmetry for the weak force. They proposed several experiments in which this might be verified.

You might think some of *my* ideas sound pretty unconventional. But what Yang and Lee were thinking was much more bizarre. They suggested that if you are a weak interaction, when you look into a mirror, instead of seeing your face, you see the back of your head (i.e. no front-to-back reversal)! These guys were clearly thinking "outside the book", so-to-speak.

They found a heroic Chien-Shiung Wu, who at the last minute, cancelled her long-awaited trip to China in order to attempt a difficult experiment testing their claim (involving beta decay from nuclear spin-synchronized, cobalt 60 isotope). Their claim was found correct and was quickly verified. While nuclear spin direction (and the associated magnetic field) reverses on parity reflection, the direction of beta emissions does NOT. The very next year, Yang and Lee shared a Nobel prize (compare that to Einstein's 15-year wait for his photoelectric effect Nobel prize)! It's a travesty that Wu was not included.

As it turns out, reversing charge fixes the anomaly.* That is, if the cobalt 60 protons were negative, there wouldn't be a parity violation. Now make no mistake, I believe all three scientists involved clearly deserved a Nobel prize for their discovery. But I don't think they discovered a parity violation. What Yang, Lee, and Wu discovered was...(this is important, so please forgive my emphasis)...

*** ELECTRIC CHARGE IS A SPIN PHENOMENON! ***

There is no parity violation. They simply didn't realize that a parity reflection must entail a charge reversal (because spins reverse on parity reflection). Why didn't they realize this? Because charge arises from chronaxial spin, not conventional spatial spin.

Now ask yourself, which is easier to believe?
A static electric field arises from spin.** -OR- A weak interaction sees the back of its head in a mirror.***

*Weak interaction separately violates both parity reversal and charge reversal but not their combined reversal (a "CP-reflection"). My point is that a parity reflection must, a priori, be considered to entail a charge reflection.
**Hint. You already believe a magnetic field arises from spin.
***Hint. It is abundantly clear to all concerned, including Yang and Lee, that gravity, electromagnetic and strong force all see their faces in a mirror.

In reply to an earlier post on Dec 6, 2012 7:35:05 AM PST
So Nobel-Prize winning scientists didn't realize something that only you noticed.

Don't get me wrong, it is not impossible, just highly improbable.

If electric charge is a spin phenomenon, then how can a spin zero particle like a pi meson have electric charge?

In reply to an earlier post on Dec 6, 2012 9:56:26 AM PST
D. Colasante says:
af >> So Nobel-Prize winning scientists didn't realize something that only you noticed...highly improbable.
They weren't "Nobel-Prize winning scientists" when they came up with the *outrageous* idea of parity violation (which is still a completely inexplicable phenomenon). They were just young scientists who let go of the safe edge of the physics pool...and swam.

In the most objective sense, what Yang, Lee and Wu discovered was...
EITHER: There is weak parity violation.
OR: Charge is reversed by parity reflection. (I merely add chronaxial spin as the mechanism.)

Here's what's crazy:
Yang and Lee (and everyone else) agree that gravity, EM and strong are reversed on parity reflection.
No one (but I) suggest that charge is reversed on parity reflection. Charge reversal is thought to require a separate "C-mirror".
Yet QFT describes an electric charge field in terms of virtual photons (EM entities)! Does it take a Nobel laureate to say that light (even virtual light) reflects from a mirror? Thus:

*** Parity reflection MUST entail charge reversal! ***

There is NO weak parity violation. A P-mirror automatically entails a C-mirror.
When you look in a mirror, you're not seeing arpard, you're seeing anti-arpard! As Feynman and Hawking advise, "Don't shake hands."

af >>If electric charge is a spin phenomenon, then how can a spin zero particle...have electric charge?
Composite particles involve more than one spin. (As you know, mesons each contain two quarks.) Similarly, some iron is magnetic and some is not. In that case, the orbital electron spins are summed. Composite chronaxial spins also sum, as is the case for so-called, "intrinsic" spin.

In reply to an earlier post on Dec 6, 2012 10:40:28 AM PST
I don't understand your last answer. Agree that a positively charged pi meson contains an up and an antidown quark with electrical charges +2/3 and +1/3, respectively, and spins 1/2 and -1/2. A negatively charged pi meson contains an antiup and a down. A neutral pi meson can be either up antiup or down antidown. But I don't understand how you think these charges could be generated from these spins.

And if a quark with spin 1/2 can have a charge of +2/3 or -1/3, how does the same spin in an electron produce a charge of -1?
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Discussion in:  Science forum
Participants:  12
Total posts:  171
Initial post:  Nov 29, 2012
Latest post:  Dec 15, 2012

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