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The actual number of degrees of freedom of the electromagnetic field is 2, per point in 3-dimensional space. One can see this starting from the formulation in terms of potentials, which feature 4 comp …

You haven't really defined $(d/dy)^{-1} $, but let's pick, for instance, $(d/dy)^{-1} f(y) = \int_{0}^{y} dy' f(y')$. Then, your two expressions for $H(y)$ in general are not equal. A simple counterex …

The spin degree of freedom is generally not represented by pseudodifferential operators (though such a representation can be constructed a posteriori, as pointed out by Francois Ziegler in comments). …

A few hours ago, a question was posed asking for the eigenvalues and eigenvectors of the Dirac operator $$ H=\begin{pmatrix} x & 0 & -i\partial_{x} & \bar{z} \\ 0 & x & z & i\partial_{x} \\ -i\partia …

Denote the standard harmonic oscillator eigenfunctions (i.e., the eigenfunctions of $-\partial_{x}^{2} +x^2 $ with eigenvalues $\lambda_{n} =2n+1$) as $\psi_{n} (x)$. Introduce also the standard raisi …

Most systems you see around you are subject to a restoring force (otherwise, they'll go find an equilibrium elsewhere). Most restoring forces are linear as long as you're not too violent with the syst …

It seems to me that one can smear out your example of the delta measure at the origin. Parametrize Minkowski space in terms of the coordinates $x^{\mu } $. Pick four suitable fixed 4-vectors $a_0^{\mu …

Many of the standard classical mechanics examples have continuous symmetries. Most are invariant under translations in time. Many are invariant under spatial translations and rotations, e.g., isolated …

This is presumably simply an issue of notation - \begin{eqnarray} U\otimes I \left|\Psi\right\rangle = \frac{1}{\sqrt{2}} \left( (U_{11} \left| 0 \right\rangle_A + U_{21} \left| 1 \right\rangle_A ) \o …

No, if you want to have a 1D defect in $d$ spatial dimensions, you need $\pi_{d-2} (M)$ to be nontrivial. The fact that your Lagrangian has nontrivial $\pi_{1} (M)$ implies, when extended to 4 spati …

In physics, you certainly want to do more than just find the energy eigenvalues that you get from the 2-point functions. You also want to evaluate matrix elements of operators in the corresponding eig …

Just to add something about Gauss' law to the excellent previous answer: The Hamiltonian of a pure gauge theory, as initially derived from the Lagrangean, typically has the structure $H=E^2 +B^2 - A_0 …

Paraphrasing L.I.Schiff, "Quantum Mechanics", the $S$-matrix $S=\langle \beta | \alpha^{+} \rangle $ is the amplitude of the final asymptotic state $\beta $ contained in what became of an initial asym …

It seems to me that the fissures between (sub-)disciplines are somewhat more complex than the simple functional analysis vs. quantum theory dichotomy that Landsman emphasizes. The study of foundation …

To supplement the excellent answers previously given, some more remarks situated in the canonical quantization formalism, which is equivalent (at physicist level) to the path integral formalism referr …