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…if the real nature of electron spin is something like this, it is truly “classically indescribable,”, is it not?

 

Sin-itiro Tomonaga

[The Story of Spin (Translated by Takeshi Oka, Univ. of Chicago Press, Chicago, 1997) p. 62.]


What is spin?

The term spin has two meanings in physics. The first meaning is the ordinary one, namely rotation about a local axis. For example, a spinning top has angular momentum equal to the moment of inertia multiplied by the angular frequency. This rotational angular momentum can be (but usually isn't) called spin angular momentum. The top can also have angular momentum due to translational motion. For example, if the top is tied to a string and spun around in a circular orbit, the angular momentum would be the linear momentum (mass times velocity) multiplied by the length of the string. This type of angular momentum can be called orbital angular momentum. In this example the spin angular momentum may also be considered as a type of orbital angular momentum since the rotational motion can be interpreted as each piece of the top orbiting around the axis.

 In quantum mechanics, the spin angular momentum has not (yet) been explicitly related to rotational motion.  For a classical wave, the spin angular momentum is associated with motion of the medium whereas orbital angular momentum is associated with propagation of the wave.

The second meaning of spin relates to the transformation of variable components under rotations. In this context the spin is p/2 radians (90 degrees) divided by the angle between two independent states. For example, a scalar field (e.g. temperature) is described at each point in space by a single number independent of orientation. Since there is only one independent state at each point,  scalar fields have 'spin zero'. A vector field (e.g. position vector) is described at each point in space by three independent components with an angular separation of p/2 radians between any pair of independent components (e.g. coordinate axes). Therefore a vector field has 'spin one'.

Waves always have two independent states p radians (180 degrees) apart. This situation is called 'spin one-half'. For example, a scalar wave propagating in one dimension consists of a superposition of forward and backward propagating waves. The forward and backward directions are of course p radians apart. [This is fundamentally different from positive and negative amplitudes which are related by a multiplicative factor and therefore are not independent states. Forward and backward waves are not related by a simple multiplicative factor. The mathematical functions have different arguments.] Generalization of this wave solution to three dimensions is mathematically complicated, requiring the use of complex matrices called 'bispinors' (or Dirac spinors). Bispinors have the interesting property that they change sign upon 360 degree rotation. (This property is customarily considered as defining spin one-half systems, but such definition is misleading since this property does not apply to physical observables.)

The four components of a one-dimensional bispinor represent the forward positive, forward negative, backward positive, and backward negative components of the wave. Extension to three dimensional vector waves requires eight free parameters which can be regarded as forward and backward wave amplitudes (2 parameters) , rotation of polarization axes (3), rotation of velocity axes (3) . (Although only two rotation angles are required to specify a direction, a third rotation angle is necessary to also specify the plane of changes of direction).  A change in the orientation of the velocity axis or in the ratio between forward and backward waves is called a 'Lorentz boost'.

Elementary particles with half-integer spin are called 'Fermions'. Elementary particles with integer spin are called 'Bosons'.

Bispinors were first proposed in 1928 by Paul Dirac in order to model the electron.  This author has not found any other physics literature mentioning  that bispinors have a simple classical interpretation as solutions of ordinary classical wave equations.

For more detail go to Chapter 3 of The Classical Theory of Matter Waves.

 


Created: 27 February 2006;  Last updated: 08 January 2007

Copyright © 2006-2007  Robert A. Close