Gauge Invariance in Counting and Combinatorics

Abstract

There is perhaps no part of mathematics that is more intimately connected with everyday experiences than probability theory and statistics. The element of chance dominates the physical world. Probability is the heart of physics, in particular – quantum physics. At the probability theory lies combinatorics. We make an observation in the combination of n objects taken r objects at a time. We find a sort of combinatorial gauge invariance hidden there in that the combination of n objects taken r at a time or n-r at a time is the same. It has been argued that the Pauli Exclusion Principle is not a principle or cause at all; rather it is an effect of the combinatorics which essentially delivers the Fermi Dirac Statistics. We explore the consequent applications in quantum mechanics and field theory with particle statistics.