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  Did Dirac ever publish anything about his string trick?

+ 1 like - 0 dislike
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Paul Dirac often used scissors and strings to demonstrate the properties of spin 1/2 and of SU(2). The demonstration is variously called the "string trick" or the "belt trick" or the "scissor trick".

Did Dirac ever publish anything about it - in a paper or in one of his books on quantum theory?

Searching on Google scholar it seems that he didn't - but is this conclusion correct?

asked Feb 10 in Theoretical Physics by Johanna [ no revision ]

@Johanna , I am not sure if Dirac wrote about it; but you can find that in Roger Penrose's and Wolfgang Rindler's textbook called Spinors and space-time volume 1 on page 43.

2 Answers

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P. Dirac's string trick does not and may not demonstrate the properties of spin 1/2 of SU(2), as a matter of fact. First, SU(2) transformations are not real rotations, but recalculation rules from one still reference frame to another one. Second, his trick works for real rotations whereas a rotation to $2\pi$ in SU(2) does not change physical (observable) results of spin 1/2 in QM. I heard that he demonstrated his "analogy" for fun, but, probably, he reconned it as an "analogy" rather than as a "proof". There is nothing about it in his list of publications.

answered Feb 10 by Vladimir Kalitvianski (132 points) [ revision history ]
edited Feb 10 by Vladimir Kalitvianski

Did Dirac write about the trick in any of his books on quantum theory?

As usual, Vladimir's statements are all wrong or misleading. Why does he waste his talents, his life and his time in this way?

@Frank: Frank, frankly, point out where I am wrong or misleading, please. I wrote sincirely what I knew about the subject. For example, I have a full list of publications and the publications themselves in Russian (they were translated in Soviet times).

Dirac's belt trick is actually very relevant to the mathematics of spin 1/2 particles.  It demonstrates that the group of real rotations, $SO(3)$, is not simply connected-- it has a double cover by the  "space of belt configurations".   This double cover is precisely $SU(2)$.  From a mathematical perspective,  it is this double cover that is responsible for the spin 1/2 representations of $so(3)$.

@PhilTosteson: It is interesting. Can you explain me what it means for SO(3) to be "not simply connected", please? And what is "the space of belt configurations"? Thanks.

Bob.

I think that Phil means that \(\pi_1(SO(n,\mathbb{R})) = \mathbb{Z}_2 \forall n>2\) and that \(\mathrm{Spin}(n)\) is its double cover.

@IgorMol: And I think it is Phil Tosteson who must answer my question.

+ 0 like - 0 dislike

I don't think Dirac ever published his trick in his scientific papers and neither in his books, but only used it as a classroom illustration. When Martin Gardner was writing the Mathematical Games column in Scientific American, he contacted Dirac to learn more about his "Scissor Problem," and received the following (typical Dirac) concise and precise reply:

You can find this story, together with an elementary explanation of the trick, in Gardner's popular-level book, "Riddles of the sphinx, and other mathematical puzzle tales," Ch.23. This book is very entertaining and contains many other nice mathematical puzzles for the general public.

Another interesting source is "The Spinor Spanner" by Ethan D. Bolker published in the American Mathematical Monthly where, after talking about a typical neutron interferometer experiment that one read about in introductory quantum mechanics textbooks, one reads: "There is, however, an easy experiment with an analogous outcome. P. A. M. Dirac invented it to lessen, in lectures, the implausibility of the neutron's predicted behavior."

By the way, in the above mentioned notes, Bolker uses Dirac's trick as a motivation to introduce the student to notions of homotopy theory and covering spaces, and you can read it for free in JSTOR here. Also, an excellent discussion of spinors and homotopy in the rotation and Lorentz groups, using again the Dirac's scissor problem as an illustration, can be found in G. Naber's "The Geometry of Minkowski Spacetime," Appendix B. See Google Books here. Another source is M.H.A. Newman's "On a string problem of Dirac."

So, it seems to me that such a trick was, for Dirac, only a classroom explanation, and didn't make it to his Oeuvre, otherwise I would expect one of the above reviews to cite Dirac directly. This is interesting for Dirac fans, since many former students of Dirac claimed that his lectures were too dry, basically repeating the same phrases of his Principles of Quantum Mechanics

Finally, a similar gedankenexperiment that illustrates a similar point can be found in Sec. 41.5 of Misner, Thorne and Wheller's Gravitation. 

answered Mar 12 by Igor Mol (510 points) [ revision history ]
edited Mar 12 by Igor Mol

As I said previously, there is a big difference between "rotations" as recalculation rules between differently placed reference frames (the body stays intact) and physical rotations of the body seen from the same reference frame.

In case of "Dirac strings", it is not even "physical rotations", but twisting the body. This twisting has an irreversible effect on the deformed body which is especially evident in case of a fragile twisted body.

As I said previously, there is a big difference between "rotations" as recalculation rules betwenn differently placed reference frames (the body stays intact) and physical rotations of  the body seen from the same reference frame.

This difference is called passive versus active transformations (see Ryder's QFT).

In case of "Dirac strings", it is not even physical rotations, but twisting the body. This twisting has an irreversible effect on the deformed body which is especially evident in case of a fragile twisted body.

Of course one should consider an idealized elastic string, not a "fragile" one.

In such a case, what is interesting about the gedankenexperiment is that, if you rotate the scissor \(2\pi\) around its axis, it is impossible to untwist the strings by any motion keeping the scissor pointing at a fixed direction and the chair at rest, but if you rotate \(4\pi\), it is possible to untwist the strings, as shown here. To use your terms, after a \(4\pi\) rotation, the twisting becomes reversible.

In this sense, the trick concerns a rotation not only in the "physical," Euclidean space \(\mathbb{E}^3\) of the lab, but a rotation in the configuration space of the lab and the "state" of the strings. In the Bolker's paper which I quoted above, it is called the configuration space \(\Omega\) of the wrench, see section 4 how it is constructed.

@IgorMol: Thank you, Igor, for teaching me about passive and active "transformations", but know it for more than forty years.

Yes, it is possible to untwist the strings by a rotation to $-2\pi$ and keeping the scissor pointing at a fixed direction and the chair at rest.

The only thing that is essential for SO(3) representations is non commutativity of rotation generators (that can be "demonstrated classically"). And there is no classical analogue to spin 1/2 object - there is nothing that only has discrete projections $J_z$ and the corresponding fermi-statistics.

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