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The Stress-Energy Tensor in Field Theory

The stress-energy tensor is a rank (0, 2) tensor, denoted $T_{ab},$ that contains information about how a field theory transforms under local diffeomorphisms. This post explains the properties of the stress tensor that I think are of most fundamental importance. In section 1, I take up the issue of what it means for a diffeomorphism to act on a field. I discuss push-forwards of tensor fields under diffeomorphisms, and the linear transformations that are sometimes used to supplement push-forwards in theories where doing so gives a symmetry of the Lagrangian. In section 2, I define the stress tensor in classical field theory, and derive the main equation it satisfies: that when all fields in a theory are pushed forward infinitesimally by a local diffeomorphism $\xi^a,$ the change in the action is $$\delta_{\xi} S = \int_{\mathcal{M}} \xi^b \nabla^a T_{ab} \boldsymbol{\epsilon}.$$ In section 3, I discuss the stress tensor in quantum field theory, and how it can differ from its classical

Ward Identities

Ward identities are one of the most fundamental tools for studying quantum field theory, and they're encountered in almost any quantum field theory course. You've almost certainly encountered them before, so why should I bother writing about them? Simply put: despite learning how to derive Ward identities for the first time more than 5 years ago (in my first quantum field theory class, as an undergraduate at UChicago), I didn't really understand why they were important until quite recently. This is a product of my own unique research path — I haven't ever done any research in pure QFT, working instead mostly in quantum information and classical geometry, which means I haven't ever had to really understand what's going on under the hood in field theory. I don't think this oversight is so uncommon, so I'm putting together some basic thoughts on Ward identities in this post. So, what is a Ward identity? On its face, it's an equation that tells you how

Projective representations, central extensions, and covering groups

In my post on Wigner's theorem , I explained the famous result that any symmetry transformation on quantum states can be realized as a unitary or antiunitary operator on Hilbert space. But when we study symmetries of quantum systems, we usually have in mind not a single symmetry but a full group of symmetries; Wigner's theorem tells us nothing about how the operators corresponding to different symmetries in the same group should compose with one another. Suppose, for example, that a quantum system transforms under the symmetry group $G$, and that the unitary or antiunitary operator corresponding to the element $g \in G$ is denoted by $\hat{U}_g$. Because any two unitary operators related by a phase are physically equivalent, it may not be the case that $\hat{U}_{g_1}$ and $\hat{U}_{g_2}$ compose to $\hat{U}_{g_1 g_2}$; instead, we will have a relationship like $$\hat{U}_{g_1} \hat{U}_{g_2} = e^{i \phi(g_1, g_2)} \hat{U}_{g_1 g_2}.$$ At first glance, it seems like we just made