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The mass of a small fluctuation is usually defined as $$\pm m^2= \frac{d^2V}{d\phi^2}\biggr|_\text{VEV}$$ The sign depends on your conventions. This makes sense in analogy with the canonical free field potential $$V_\text{free}=\pm \frac{1}{2}m^2\phi^2$$ for which the above formula is clearly right. More generally, we can expect any (reasonably smooth, ...

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Note that the spectrum ${\rm Spec}(\hat{A}) \subseteq \mathbb{R}$ of a Hermitian/self-adjoint operator $\hat{A}$ belongs to the real axis $\mathbb{R}\subseteq \mathbb{C}$, cf. e.g. this Phys.SE post. It is therefore not surprising that a reality condition for a classical field naturally translates into a Hermiticity/self-adjointness condition for the ...

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Yes indeed there is! Firstly, the normal Hamilton-Jacobi equation goes through, so still the energy is given by the time derivative of the on-shell action. But the relevant local object that is most natural in field theory is the stress-energy-momentum tensor, which contains densities and fluxes of energy and momentum. The question of what to vary to get ...

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The first the expression $U(r) = -\frac {GM} r$ is a potential, but not potential energy. The units are velocity2. This is a widely used potential in solar system astronomy, geology, geophysics, and in aerospace engineering. For example, see ...

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Notice that $h$ and $r$ are related in the following way: \begin{align} r = R + h \end{align} where $R$ is the radius of the Earth (the distance from the center to the surface) and $h$ is the height above the surface. Then notice that \begin{align} U = -\frac{GmM}{r} = -\frac{GMm}{R+h} = -\frac{GMm}{R}\left[1 -\frac{h}{R} + O(\frac{h}{R})^2\right]. ...

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A generic state of QFT has an unfixed number of particles, that are at most countable. This is because the Fock space is constructed as a direct sum of the subspaces with any possible number of particles, from zero to infinity. Mathematicallly, let $\mathscr{H}_1$ be the one particle space, $\mathscr{H}_n$ the (symmetric or anti-symmetric) $n$-fold tensor ...

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When you introduce $\phi(x) \phi(y)$ for $x \ne y$, you postulate an action at a distance, whichever the interval between said events is: time-like, null, or what. In other words, you admit some essence that isn’t a field, but propagates through the spacetime directly, in a point-to-point fashion. I am not sure you can’t maintain causality is such theory, ...

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The uncountable number of harmonic oscillators represent "opportunities", one-particle states into which new particles may be created. But a generic field state has not only a "countable" number of particles but a finite number of particles. That's necessary at least when the particle is massive or charged, otherwise the total energy or charge would be ...

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For each continuous symmetry, infinitesimal transformations may be expressed, by a bracket involving the conserved charge operator associated to the symmetry : $$\delta_\epsilon \phi(x) = i\epsilon [Q, \phi(x)] \tag{1}$$ In our case, we must have : $$\epsilon \theta = i\epsilon [Q_\theta, \phi(x)] \tag{2}$$ A solution is then : Q_\theta = \theta ...

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