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There are at least two different $\sigma$-algebras that Lebesgue measure can be defined on: The (concrete) $\sigma$-algebra ${\mathcal L}$ of Lebesgue-measurable subsets of ${\bf R}^d$. The (abstract …

The nature of the Maxwell equations (including their Lagrangian aspects) can be clarified by viewing these equations as part of a larger family of classical field theories in physics. For simplicity …

There is a trivial sense in which the answer is "yes": the solutions to the Maxwell equations are (formally, at least) the global minimisers to the functional $$ \int\int |\mathrm{div} E|^2 + |\mathrm …

This operation (generalised slightly by replacing $e^{i (xk-yp)}$ by $e^{i\lambda (xk-yp)}$ for a parameter $\lambda$) is known as "twisted convolution" in the harmonic analysis literature, see e.g. C …

It is perhaps helpful to distinguish between four types of mechanics here: Pure-state classical mechanics. Here, the mechanics are classical, and the system is described by a single point $(q,p)$ i …

One way to understand the symbol of a differential operator (or more generally, a pseudodifferential operator) is to see what the operator does to "wave packets" - functions that are strongly localise …

Mathematics is virtually the only profession that has the luxury of insisting on near-100% certainty. In physics, one is always making approximations and idealised assumptions, and even the known law …

I discuss these sort of issues in this blog post of mine (particularly in Section 4).

The Yang-Mills functional $\int_{{\bf R}^{1+d}} F^{\mu \nu} F_{\mu \nu}\ dx dt$ is dimensionless (scale-invariant) if and only if the spacetime dimension is four. (The integrand is a quadratic functi …

Enumerative combinatorics also provides an important source of inequalities. The most basic is that if you can show that $X$ is the cardinality (or dimension) of some set $A$, then you automatically …