Exploring Real Fields and Spinors: Understanding Movement in Tangent Space

[Spinnor]

Suppose we have a real field, S(x,y,z,t), that satisfies E^2 = P^2 + m^2. Here the tangent space could be R^1? Say we can expand the tangent space and let it be R^3 but make the restriction that "movement" of the field in the tangent space was restricted to some orbit about the origin of the tangent space, and further, the orbit is contained in some arbitrarily oriented plane that contained the origin of the tangent space. If so then consider how we might graph this field, S, at a point in spacetime (X,t). As the movement is restricted to a plane let the length of a vector represent the magnitude of the fields displacement (the distance from the origin of the tangent space) and let the direction of the vector represent the normal to the orbit plane. An additional unit vector in the orbit plane gives the angular position in the orbit plane. This representation of the "state" of the field S at a point in spacetime smells of spinors, is this the case? If not spinor like can things be tweaked to make it so?

I refer to a paper by W. T. Payne:

http://adsabs.harvard.edu/abs/1952AmJPh..20..253P

In the paper by Payne a spinor is represented as an ax with the end of its handle at the origin. With this we get a direction, a magnitude, and an angle.

Thanks for any help.

 

[pellis]

Regarding your question of whether or not spinors are involved in your construction, and specifically in relation to Baker's (and others') Axe image of a spinor: I'm not an expert, but the sense I get from Penrose (e.g. Road to Reality) is that if a spinor is involved, then additional to the axe you need also to recognise that a 2*Pi rotation of the Axe involves the orientation-entanglement relation, which results in the sign of the spinor/Axe being inverted. The same point is made in MTW, referring to Penrose's original description of a spinor (Penrose & Rindler), where he describes a spinor as corresponding to a laser beam pointed at the moon, and projecting an arrowhead that can rotate its direction(cf blade of axe rotating direction), but again he reminds us that the rotating arrowhead embodies the orientation-entanglement relation (i.e. the sign inversion on 2*Pi rotation of the arrowhead).

Like me, you seem to be considering the geometric sense of a spinor; but bear in mind the quote from Sir Michael Atiyah who, like Roger Penrose, is also a real expert on the subject:

“No-one fully understands spinors. Their algebra is formally understood but their geometrical significance is mysterious. In some sense they describe the square root of geometry and, just as understanding of the concept of the square root of -1 took centuries, the same might be true of spinors.”

Quoted in G. Farmelo’s biography of Dirac “The Strangest Man” (2009)

Hope this helps (a little bit)

P

 

[Spinnor]

...


Like me, you seem to be considering the geometric sense of a spinor; but bear in mind the quote from Sir Michael Atiyah who, like Roger Penrose, is also a real expert on the subject:

“No-one fully understands spinors. Their algebra is formally understood but their geometrical significance is mysterious. In some sense they describe the square root of geometry and, just as understanding of the concept of the square root of -1 took centuries, the same might be true of spinors.”

Quoted in G. Farmelo’s biography of Dirac “The Strangest Man” (2009)

Hope this helps (a little bit)

P

I like the quote, thank you! I will have to reread the paper by Payne,

http://adsabs.harvard.edu/abs/1952AmJPh..20..253P

I'm not sure Payne brings up the entanglement relation but it seems like it must be there.

Thanks again!

 

[pellis]

The orientation-entanglement relation (OER) isn't well referenced explicitly, having only 45 hits on Google at present (at least a couple of which are my own enquiries), but a mention in MTW is copied in http://www.math.utah.edu/~palais/mtw.pdf .

As a property that switches sign between 2n*Pi and 4n*Pi rotations in 3-space (n integer) the OER does seem to represent the fundamental character of spinors (as I understand it – someone please correct me if I’m wrong). And Penrose gives another convincing way to demonstrate it, in a simple physical way while illustrating the quaternion rules, using a book and a belt (a long strip of paper works better), in the chapter on hypercomplex numbers in Road to Reality. (And Atiyah mentions, in "Paul Dirac: The Man and His Work" by A. Pais, M. Jacob, D. I. Olive, and M. F. Atiyah, that quaternions are all you need to know about spinors in 3D and 4D.)

The authors MTW, referenced above, stated (in 1973), referring to 2*Pi and 4*Pi versions of objects in 3D:

"Whether there is also a detectable difference in physics (contact potential between a metallic object and its metallic surroundings, for example) for two inequivalent versions of an object is not known [Aharanov and Susskind (1967)]."

I guess they wrote this before it became known experimantally that rotating a neutron by 2*Pi does invert the sign of the wave function, which seems to confirm the effect as having some physical basis. (I can’t quote a definite reference, but there was an excellent Scientific American article by Bernstein and Philips from July 1981, with a title, (from memory) something like "Vector Bundles in Quantum Theory" that mentions neutron interferometry, which I think confirms this.)

However, I've long wondered:

1) Is the OER a fundamental character of spinors, or just one type of situation that can be represented by spinors? (There are other physical systems displaying the 4*Pi identity, such as the "Balinese candle dance" beloved of Feynman, the "quaternion handshake" visible on YouTube; also pairs of cones and, relatedly, pairs of coins, rotating around each other; but it’s never quite made clear whether these illustrations are meant simply as analogies, to make the idea of a 4*Pi identity plausible (different systems having the same homotopic character?), or, alternatively, whether they are supposed to be direct illustrations of what might be fundamentally the same effect.)

2) Does the OER mean that particles with spin 1/2 are in some physical sense linked to their environment, and if so, how are they so linked? For example, could the exchange of virtual particles provide a sufficient linkage with the environment? Or is there some other form of entanglement? In http://www.iop.org/EJ/article/1742-6596/68/1/012036/jpconf7_68_012036.pdf?request-id=c9b0629e-25fa-4973-9b6a-3c70c52f4413 there's a suggestion, apparently by the author, Hristu Culetu [sic], that it's related to relativistic frame-dragging.

3) Or is this topological/homotopic property purely abstract, requiring no physical linkage?

Even after reading http://math.ucr.edu/home/baez/spin_stat.html, I'm still unsure.

 

[ytuab]

The comments in this thread are interesting, so I was awaken.

Spin is a very strange thing because of the sign inversion of 2*Pi rotaion and the faster-than-light spinning speed.

As far as I know, if "spin" really exists, it is very difficult to express spin as a real entity.
The 4*pi rotaion experiment (See this) shows the spinning neutron changes as $$e^{i\phi /2}$$ beautifully by the rotaion.

The fermion model (for example, the model using the ball and twisting string) which I saw before can not explain this change of $$e^{i\phi /2}$$ , I think.

So it is very difficult to express "spin" as a real thing.

To begin with, what is the spin ?
Spin was defined by the spectrum data and the idea of the atomic structure.
Only the Stern-Gerlach experiment could not define the "spin" (See this thread.)

I think spin has the strong mathematical property.
For example, the helium atom has spin-up and spin-down electrons(due to the spin-statistics) , it seems to generate no magnetic field.
But to be precise, in all areas except in the part at just the same distance from the two electrons, the magnetic fields are theoretically created, because the two electrons are apart by the repulsive Coulomb force.
(So as the electrons moves, they lose energy by emitting the (electro)magnetic waves.)

I think QFT also has the mathematical property because it introduces the new idea of the second quantization + Lagrangian without previous notice.

 

[pellis]

Spin is indeed strange, and I don't think anyone has a simple physical intuition that accounts for all its properties. As far as I'm aware, the idea of spin as a simple mechanical motion with F-T-L speed (in the thread you link to) was dropped very early on. Spin just emerges from the Dirac equation as an unexpected consequence of QM+relativity. I've never been able to get beyond that point.