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Showing posts from February, 2022

Zeta function regularization, heat kernels, and one-loop determinants

In the path integral formulation of quantum field theory, a free theory is one whose Euclidean action can be expressed as $$S[\phi] = \frac{1}{2} \int_{\mathcal{M}} \boldsymbol{\epsilon} \phi L \phi, \tag{1}$$ where $\phi$ is some tensor field and $L$ is a self-adjoint differential operator whose spectrum is bounded below. For example, if we make $\phi$ a real scalar field and $L$ the differential operator $$L = m^2 - \nabla^2,$$ then $S[\phi]$ is the action of a free scalar with mass $m.$ It is useful to have general techniques for studying the path integrals of free field theories, not only because free theories serve as an important testing ground for interesting physical ideas, but also because in the semiclassical limit, any quantum field theory can be approximated as free. In particular, if we consider some Euclidean quantum field theory defined by the path integral $$\int \mathcal{D} \phi e^{- \frac{1}{\hbar} S[\phi]},$$ then in the limit $\hbar \rightarrow 0$ the path integral

Fano's inequality

(Thank you to Alex May for catching an error in the first version of this post!) I had to learn Fano's inequality for a quantum information project, but found the proof as it currently exists on Wikipedia extremely confusing, largely because it uses a lot of jargon about "messages" and "estimates" that, in my opinion, obscures the underlying mathematics. So here's a simple proof. I'll probably try to edit the Wikipedia proof after I post this. Heuristically speaking, Fano's inequality is a bound on how successfully I can guess the output of a random process $X$ if I can't access $X$ directly but can access a different random process $Y$ that is correlated with $X.$ To state the bound precisely, we'll need a bit of notation. A finite random variable , henceforth just a "random variable," is a finite set $\mathcal{X}$ together with a map $p_X : \mathcal{X} \rightarrow [0, 1]$ satisfying $$\sum_{x \in \mathcal{X}} p_X(x) = 1.$$ The fu

The stress-energy tensor in field theory

I came to physics research through general relativity, where the stress energy tensor plays a very important role, and where it has a single unambiguous meaning as the functional derivative of the theory with respect to metric perturbations. In flat-space quantum field theory, some texts present the stress tensor this way, while some present the stress tensor as a set of Noether currents associated with spatial translations. These definitions are usually presented as being equivalent, or rather, equivalent up to the addition of some total derivative that doesn't affect the physics. However, this is not actually the case. The two stress tensors differ by terms that can be made to vanish classically, but that have an important effect in the quantum theory. In particular, the Ward identities of the two different stress tensors are different. This has caused me a lot of grief over the years, as I've tried to compare equations between texts that use two different definitions of the