Rotation of Spinors

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The electron, and other fermions with spin , is described in relativistic quantum mechanics by a spinor. A distinguishing feature of spinors is their behavior under rotation. Whereas a vector boson, with spin 1, will return to its initial state after a rotation by , a spinor requires two full rotations, with the angle advancing by to recover its initial state. A spinor is described by a complex phasor in addition to a helicity. This is represented in the graphic by rotation in a circle normal to its spin direction, with the complex phase color coded. A rotation in space by an angle is accompanied by a phase change of . Thus after rotation by , the spin direction of the particle is recovered but the phase changes by a factor . This can be observed experimentally in interference phenomena, most notably those done in neutron diffraction. In the course of rotation of by , the phasor traces out a Möbius band. This accords with the fact that a point on the surface of a Möbius band must go around twice in order to return to its initial location.

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In the terminology of group theory, the Lie group describing spinors provides a double covering for the 3-dimensionsal rotation group .

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Contributed by: S. M. Blinder ( {MonthName, Year, Day})
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: starting point

Snapshot 2: rotation by produces a phase difference of

Snapshot 3: rotation by is required to recover the initial state

Reference

R. Penrose and W. Rindler, Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry, Cambridge: Cambridge University Press, 1988.



Permanent Citation

S. M. Blinder "Rotation of Spinors"
http://demonstrations.wolfram.com/RotationOfSpinors/
Wolfram Demonstrations Project
Published:  Part[DateValue[, {MonthName, Year, Day}], 3] {MonthName, Year, Day}

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