Generalized gauge invariance and the integer quantum Hall effect

Generalizing a simple gauge-invariance argument, we introduce a k -space vector potential A k which allows us to obtain explicitly the identification of the Hall conductance with the quantized phase winding number of the wave function around the Brillouin zone. We also demonstrate, based on these winding number considerations alone, that a weak periodic potential which splits each Landau band into non-degenerate subbands results in Hall conductances which sum to unity, and satisfy a Diophantine equation.

Meanings and Criteria in Ethics

In Recent years noncognitivist ethical theories have been supported by an argument which has come to be widely accepted among moral philosophers.1 According to this argument, an ethical term like ‘good’ has both a commending function and a describing function, but between these functions there is the important difference that the commending function alone is invariant while the describing function varies greatly. For many and different things may be called good—hammers, sunsets, paintings, missionaries, cannibals—but despite these differences in the descriptive criteria for applying the word to objects, ‘good’ retains a common meaning in all these uses, for in each case the word is being used to commend. The conclusion drawn, then, is that because of its being the sole invariant or common feature of every use of ‘good’, the commendatory function rather than the varying descriptive function must be the primary meaning of ‘good’. I shall refer to this as the commendatory invariance argument