But, in this case, doesn’t it matter than ##v## can potentially become an imaginary number, and the limits of the integral would no longer be just infinity? We would get an added imaginary term to the limit, e.g. ##\infty + 2\pi i##.
More explicitly, ##v= arcsinh(\frac{mc^{2}t}{hz}),## where...
I am not seeing how the v goes away in the third equal sign of equation (1.8). It seems to be that it must be cos(z*sinh(u+v)), not cos(z*sinh(u)).
In the defined equations (1.7), the variable "v" can become imaginary, so a simple change of variables would change the integration sign by adding...
The book that I am following, "Group Theory in a Nutshell", basically says since a rank-two tensor can be decomposed into a direct sum of 1, 3, and 5 dimensional representations, we will call them irreducible. Could you suggest a way to prove that the 3 and 5 dimensional anti-symmetric and...
When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...
I am trying to show that given the following stochastic differential equation: ##\dot{x} = W(x(\tau))+\eta(\tau),## we have
##det|\frac{d\eta(\tau)}{dx(\tau')}| = exp^{\int_{0}^{T}d\tau \,Tr \ln([\frac{d}{d\tau}-W'(x(\tau))]\delta (\tau - \tau'))} = exp^{\frac{1}{2}\int_{0}^{T}d\tau...
I was trying to say that if ##\partial_{\nu}\phi## is in some vector space and ##\partial^{\nu}\phi## is located in a different vector space, both connected through a linear map ##g^{\mu \nu}##, I would think that we would have to explicitly choose the vector space that our operator...
In Lagrangians we often take derivatives (##\frac{\partial}{\partial (\partial_{\mu}\phi)}##) of terms like ##(\partial_{\nu}\phi \partial^{\nu}\phi)##. We lower the ##\partial^{\nu}## term with the metric and do the usual product rule. My question is why do we do this? Isn't...
When deriving ##\Pi(\vec{x},t)## for the Klein-Gordon equation (i.e. plugging ##\Pi(\vec{x},t)## into the Heisenberg equation of motion for the scalar field Hamiltonian), we come across a term that is the following
##\int_{-\infty}^{\infty}d^{3}y...
Thanks!
So taking this route we end up with $$\delta^{3}(\vec{p}-\vec{q}) = \delta^{3}(\vec{p'}-\vec{q'})\frac{E'}{E},$$
where the ##\frac{dp'_{3}}{dp_{3}}## term became ##\frac{E'}{E}.## When using the Dirac delta identity above, we should technically evaluate ##\frac{dp'_{3}}{dp_{3}}## at...
From page 22 of P&S we want to show that ##\delta^{3}(\vec{p}-\vec{q})## is not Lorentz invariant. Boosting in the 3-direction gives ##p_{3}' = \gamma(p_{3}+\beta E)## and ##E' = \gamma(E+\beta p_{3})##. Using the delta function identity ##\delta(f(x)-f(x_{0})) =...
I uploaded a screen shot. The pdf is called "Neutrino masses and mixings and..." by Alessandro Strumia and Francesco Vissani. The exact page that screen shot comes from is page 31.
For a Majorana neutrino in matter we have the equation $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})\nu_{L} = m\overline{\nu_{L}}.$$ A is to be considered constant.
Squaring, in the ultra-relativistic limit one obtains the dispersion relation
$$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ i.e.
$$p...
Homework Statement
A system consists of a mass m moving in one dimension and attached to a rigid wall by a spring having stiffness constant ##K##, as shown. The mass is subjected to a constant force ##F##, and is in equilibrium with the surroundings at a temperature ##T##. The partition...