# Tag Info

Accepted

Accepted

### Discrepancy in Results with Self-Adjoint Operator on a Special Hilbert Space in 2D Geometric Algebra

The scalar product corresponding to the quadratic form $\|u\|^2 = u^{\ddagger}u$ is $(u, v) = \mathrm{re}(u^{\ddagger}v)$, where $\mathrm{re}(a + xe_1 + ye_2 + be_{12}) = a$. Self-adjointness of $g$ ...

### Why does this trick make the oscillating exponential integral converge?

I guess you encountered this result in the context of a physics topic, where the argument for such a trick is usually omitted or implicit. In fact, two "formalisms" are here merged together ...
Accepted

### Inverse Proportionality

The second law does occur often in the sciences. It occurs in the context of conservation, e.g. conserved mass/charge etc. Think of Chemistry: if $x$ is the mass of one reactant and $y$ is the mass of ...
Accepted

### Show that the real K-G equation, $(\Box + m^2)\phi=0$ is the EOM for the action $S=\frac12\int d^4x(\partial^\mu{\phi}\partial_\mu{\phi}-m^2\phi^2)$

We will use the following definition of functional differentiation: \begin{equation} \frac{\delta S}{\delta\phi(y)}=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\left(S[\phi+\epsilon\delta^{(4)}(x-y)]-S[\...
### Show that the real K-G equation, $(\Box + m^2)\phi=0$ is the EOM for the action $S=\frac12\int d^4x(\partial^\mu{\phi}\partial_\mu{\phi}-m^2\phi^2)$
Let ${M}$ be the spacetime manifold. Let there be an action functional \begin{align*} {S}[\Phi]=\frac{1}{2}\int_{M}\mathrm{d}^{4}{x}{\,}\sqrt{-\mathrm{det}({g}({x}))}{\,}\bigg({g}^{\mu\nu}({x})\frac{\...