Field Transformations and Spin
摘要
As stated in Chap. 1 , this chapter is structured with the goal of ensuring that “readers gain confidence in their understanding of the Bloch sphere”. Many books attempt to explain the essence of spin based on spinor fields, but these typically assume a higher level of prior knowledge, making them rather difficult and intimidating for beginners. In this book, I have made every effort to avoid abrupt definitions and introductions, instead building explanations using only the material covered so far. We begin with the discussion of “how the component representation of a field changes when a rotational transformation is applied to a three-dimensional coordinate system”, which leads to the introduction of the concept of spin in fields (Sect. 7.1). Next, we analyze how the “two-component wavefunction in the non-relativistic limit of relativistic electron theory”, introduced in Sect. 6.3 , transforms under rotations of the three-dimensional coordinate system. This naturally leads to the introduction of spinor fields (Sect. 7.2). We then carefully explain what it means to “identify the spin of a field with the three-dimensional orientation illustrated on the Bloch sphere”. Finally, through a clear understanding of transverse spin states, we reaffirm the meaning of “controlling spin orientation via an applied magnetic field” (Sect. 7.3).