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Ether is a somewhat vague term that is used in different ways. The point of the ether originally proposed to explain the propagation of light is that it constitutes an absolute rest frame. So for example you would know if you were moving relative to the ether because you would measure different velocities of light in and opposite to your direction of motion. ...
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Both $p^2$ and the sign of $p^0$ are invariant under infinitesimal Poincaré transformations (we're really looking at representations of the Poincare algebra or rather projective representations of the connected component of the Poincaré group here).
Hence any representation that involves more than one value of $p^2$ and $\mathrm{sgn}(p^0)$ is not ...
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Ref. 1 only shows [via a non-abelian generalization of the abelian eq. (13.31)] that the physical Hilbert space ${\cal H}_{\rm phys}$ [defined to be BRST-closed and have zero ghost number] can not have longitudinal gluons by using the Lorenz gauge condition and the EL equation for the Lautrup-Nakanishi auxiliary field $F^a$.
Longitudinal gluons are possible ...
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Following the answer by @ACuriousMind I believe I got the point and decided to post a version which is a little bit more expanded with what I understood.
Let $f(p)$ be some function defined in momentum space. We can define the function acting on the momentum operators $P$ by the usual procedure using the eigenstate basis of the commuting set $\{P^\mu\}$:$$f(...
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Your diagrams that you have in the first order are all equivalent. However you shouldn't lose sight of the fact that Feynman diagrams are about pairing fields in Wick's theorem. There are 3 ways to pair the four $x$ fields in $x^4$ so this Feynman diagram will carry a factor of 3. We don't actually draw 3 diagrams but this combinatorial factor is still ...
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Each component of $\Psi$ satisfies the Klein-Gordon equation, and so we can write (cf. this PSE post)
$$
\Psi_\alpha(x)=\int\frac{d^3p}{\sqrt{(2\pi)^32E_p}}\Big(a_\alpha(p)e^{-ip\cdot x}+b^\dagger_\alpha(p)e^{+ip\cdot x}\Big)
$$
for some operators $a_\alpha,b_\alpha$. If we now require $\Psi$ to satisfy the Dirac equation, we get the algebraic conditions
$$
(...
answered Dec 7 at 18:25
AccidentalFourierTransform
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zzz's answer concerns QFTs as used in particle physics. However, OP also mentions phase transitions, thus statistical field theory, where the relationship between CFT and scale-invariant RG fixed points is more subtle.
Indeed, for local QFTs where unitarity is assumed, there are theorems showing that scale invariance implies conformal invariance, see e.g. ...
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