No abstract

The Dirac belt trick is often employed in physics classrooms to show that a 2π rotation is not topologically equivalent to the absence of rotation whereas a 4π rotation is, mirroring a key property of quaternions and their isomorphic cousins, spinors. The belt trick can leave the student wondering if a real understanding of quaternions and spinors has been achieved, or if the trick is just an amusing analogy. The goal of this paper is to demystify the belt trick and to show that it suggests an underlying four-dimensional parameter space for rotations that is simply connected. An investigation into the geometry of this four-dimensional space leads directly to the system of quaternions, and to an interpretation of three-dimensional vectors as the generators of rotations in this larger four-dimensional world. The paper also shows why quaternions are the natural extension of complex numbers to four dimensions. The level of the paper is suitable for undergraduate students of physics.

No abstract

No abstract

We apply the method of zeta functions, together with the n * µ -prescription for the temporal gauge, to evaluate the thermodynamic pressure in QCD at finite temperature T . Working in the imaginary-time formalism and employing a special version of the unified-gauge prescription, we show that the pure-gauge contribution to the pressure at two loops is given by P gauge 2 = −(g 2 /144)N c N g T 4 , where N c and N g denote the number of colours and gluons, respectively. This result agrees with the value in the Feynman gauge.