Tuesday, May 8, 2012

Spinors and Applications

Spinors are a type of mathematical quantity that expand on the notion of regular vectors and vector rotations. In the case of regular vectors in a space, it is a basic property that rotating a vector about any axis 360º will bring it back to its starting point. In other words, if we apply any 360º rotation r to a vector v, then

rv = v,

which can be simplified to r = 1, expressing the fact that the application of an arbitrary 360º rotation r will leave a vector unchanged. However, the key property of a spinor, or a space of spinors, is that a 360º rotation does not transform a spinor into itself, but rather its negative. Only a rotation of 720º, corresponding to two full rotations, will correspond to leaving a vector unchanged. Therefore, if s is a 360º rotation in a spinor space, then

s = -1, and
s2 = 1, i.e. the application of s twice returns any spinor to its original state.

To clarify, the mathematical objects that are actually spinors are elements, not rotations (strictly speaking), of a certain space. The above spinor rotation s expresses the way through which spinors are transformed into their respective negatives, and is not itself, in this context, a spinor. However, in reality, the set of possible rotations on a space often is itself a space, and the example provided below falls under this category.

One geometrically accessible example of a spinor is a quaternion. As discussed elsewhere, quaternions are an extension of the complex numbers to four dimensions, with four basis elements 1, i, j, and k.

The confusion arising from whether spinors are elements or rotations is related to the dual interpretation of quaternions: either as elements of a four dimensional space, or as rotations of three-dimensional space. However, the full "meaning" of i, j, and k, are not captured by the three-dimensional rotation analogy. As the orientation of the three-dimensional space is shifted through, for example, i2, returning to its former position, the space "remembers" that there has only been one rotation, and the orientation now is the negative of what it was before. In this sense, there is an added complexity to the ambient three-dimensional space-it has a way of "knowing" how many rotations have been performed.

The mathematical properties of spinors find a physical application in the area of quantum physics. It happens that the rotational properties of some particles are identical with those of spinors, in that a rotation of 360° produces not the same quantum state, but a "conjugate"-many properties are the same, but some are opposite to what they originally where. In the same way as spinors, they are restored to their former state through only a 720° rotation.

The family of particles that exhibit the properties of spinors are those with spin 1/2. One example is the electron. The electron, of course being very common, is then important to modeling particle interaction phenomena, and the knowledge of its properties is thus equally important. Thus spinors find an application in the study of subatomic particles.

Sources: http://en.wikipedia.org/wiki/Spinor, The Road to Reality by Roger Penrose, Chapter 11

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