The induced Chern-Simons term at finite temperature

It is argued that the derivative expansion is a suitable method to deal with finite temperature field theory, if it is restricted to spatial derivatives only. Using this method, a simple and direct calculation is presented for the radiatively induced Chern-Simons--like piece of the effective action of (2+1)-dimensional fermions at finite temperature coupled to external gauge fields. The gauge fields are not assumed to be subjected to special constraints, and in particular, they are not required to be stationary nor Abelian.Comment: 5 pages, REVTEX, no figure

Representation of Complex Probabilities

Let a ``complex probability'' be a normalizable complex distribution $P(x)$ defined on $\R^D$. A real and positive probability distribution $p(z)$, defined on the complex plane \C^D, is said to be a positive representation of $P(x)$ if $\langle Q(x)\rangle_P = \langle Q(z)\rangle_p$, where $Q(x)$ is any polynomial in $\R^D$ and $Q(z)$ its analytical extension on \C^D. In this paper it is shown that every complex probability admits a real representation and a constructive method is given. Among other results, explicit positive representations, in any number of dimensions, are given for any complex distribution of the form Gaussian times polynomial, for any complex distributions with support at one point and for any periodic Gaussian times polynomial.Comment: REVTeX, 15 pages, no figures, uuencode

Project 150: High School is Tough Enough

https://digitalscholarship.unlv.edu/educ_sys_202/1117/thumbnail.jp