Path Integrals and the Scale Anomaly

The theme of this thesis is scaling, units, and dimensional analysis, along with symmetry, and the failure of these concepts at the quantum level, along with implications of such a failure. Such a failure can be quantified by a quantity called the quantum anomaly. The term anomaly indicates deviation from expected behavior, where expected behavior is of course classical behavior, and deviant behavior is quantum behavior. In the introduction, we will give a short, simple, self-contained example explaining what is renormalization. This example will get to the heart of what we mean by the anomaly - the complete destruction of a system's symmetries due to quantum effects.

The applications of the anomaly are enormous, spanning several branches of physics, from atomic to condensed matter to particle to gravitational physics. For example, just within particle physics, the chiral anomaly is responsible for the decay rate of the $\Pi 0$ meson to two photons. The scale anomaly is responsible for the Yang-Mills mass gap in pure QCD and the formation of glueballs, and its calculation is an intermediate step in lattice QCD to calculate the QCD phase diagram - indeed, the anomaly is responsible for $\Lambda QCD$ itself, through dimensional transmutation.

However, one of the most exciting applications of anomalies has been realized only in this decade in the study of ultracold gases, where the measurement of various manifestations of the anomaly has only now become experimentally accessible to atomic physicists. In 2008, a set of universal thermodynamic relations known as the Tan relations was published in a series of 3 back-to-back-to-back papers. In (2+1) dimensions, the Tan contact is merely the anomaly.

In this thesis, we develop a novel framework for calculating anomalies using the path-integral and Fujikawa's determinant. In particular, we derive 4 results: the anomaly for a (3+1) relativistic Bose gas, the Tan-pressure relation for a (2+1) nonrelativistic Bose gas, a new derivation of the virial theorem, and the relationship between the Fujikawa determinant and the quantum effective potential using the background field method. Some unpublished results will also be discussed, and as how this all began, wildly speculative ideas end the thesis.