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Consider the Dyson series for the S-matrix: $$ S = \lim_{t\to\infty} U(-t, +t) = \mathcal{T}\exp\left(-i\int_{-\infty}^{\infty}\mathrm{d}^4 x \ \mathcal{H}(x)\right) $$ Expanding out the first few terms, we see that this is $$ S = \mathcal{T}\left[\color{red}1-i\int \mathrm{d}^4x \ \mathcal{H}(x) + \frac{-i^2}{2!}\iint \mathrm{d}^4x \ \mathrm{d}^4y \ \...


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The fact that the particle is described by a superposition of states, does not mean that one has to measure both states at once. That's how superposition works, one one makes a measurements and then the wavefunction collapses on one of the possible superimposing states. This means that the neutral pion, when measured, can be found in the $|d\bar{d}\rangle$ ...


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The answer is 98.5% is the contribution of the sea of virtual gluons, quarks and antiquarks (that is, the interaction between the virtual gluons, quarks and antiquarks in the sea) to the total rest mass of the proton. The masses of the quarks entering the proton when summed have mass less than 15 MeV, and the proton has a mass close to 1000. The mass of the ...


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I agree with tparker's comment: it's for absolutely no good reason at all. As discussed in this R├ęsonaances blog post, there is a Higgs-mediated force between particles, which is a bit like scalar gravity, but with a weak-force-like range. Its Feynman diagrams look just like those for other forces. I can't see any good reason to deny fundamental-force status ...


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That paragraph is still a good high level description of the state of (perturbative) string theory. However there are additional elements which complicate the story, which I think Brian Greene probably didn't want to get into. In particular: The way the extra dimensions are compactified affects the spectrum. When you go to energies above the "string ...


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Fields, I think that the last one was a really useful advice. I can add that you should recognize the derivative as the momentum operator. $W_{\mu}=\frac{1}{2}\epsilon_{\mu\alpha\beta\sigma}\sum^{\alpha\beta}P^{\sigma}$. Then you can go to the rest frame (since it's a massive particle) and evaluate the scalar. You have $P^\mu=(m,0)$ so the index $\sigma$ ...


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Your looking for: $$ \frac{d\theta^*}{dp_t} = \frac 1 {\sqrt{(\frac 1 2 M_W)^2-p_t^2}}$$ In the C.O.M frame, with $E \gg m_{\mu}$: $$E_{\mu}+E_{\nu} = 2E=2p = M_W$$ so $$\sqrt{(\frac 1 2 M_W)^2-p_t^2} =\sqrt{p^2-p_t^2}=p_L$$ Also: $$p_t = p\sin{\theta^*}$$ and $$ p_L = p\cos{\theta^*}$$ (the definitions of transverse and longitudinal), you get: $$ \frac{dp_t}...


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Consider a particle moving in the $\hat z$ direction, for simplicity. Define $\kappa= p/(E+m)$ and note it collapses to 1 for m =0. In this frame, $$ u_ \uparrow =\sqrt{E+m} \begin{pmatrix} 1\\ 0\\ \kappa\\ 0\end{pmatrix}, \qquad u_ \downarrow =\sqrt{E+m} \begin{pmatrix} 0\\ 1\\ 0\\ -\kappa \end{pmatrix}.\tag{4.65} $$ Up to normalizations, the ...


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However, my teacher says that in classical mechanics light cannot be both and for quantum mechanics we can. Does this mean we need more trust in the validity of quantum mechanics because there is a sense of ambiguity? We only trust that we know more about nature than a hundred and fifty years ago, not because of ambiguity. We have gone to much smaller ...


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By definition, creation operators applied to the vacuum state define the 1-particle states of the field theory, and by iteration multiparticle states. All particle terminology in QFT is based on this definition. In order that the particle interpretation works and produces a multiparticle Fock space, the creation and annihilation operators must satisfy ...


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In quantum field theory, whether for particle physics or other frames , (for example nuclear physics,) the fields are the plane wave solutions, i.e. without a potential, of the corresponding quantum equation, filling all points (x,y,z,t), a kind of coordinate system, whether particles exist or not. Creation and annihilation differential operators create or ...


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Big bang should have made up an equal amounts of both matter and anti matter, This is called the baryogenesis problem for models of the universe based on general realtivity and the particle standard model. In physical cosmology, baryogenesis is the physical process that is hypothesized to have taken place during the early universe to produce baryonic ...


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At lower temperatures, quarks are always bound with other quarks because of color confinement. Only color-neutral particles can exist by themselves. A free particle with a color charge has too much energy due to its interactions with other color charges, even if they are far away. In fact, the energy it takes to separate quarks is so large that new quarks ...


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