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Neutrinos are weakly interacting quantum mechanical point particles, with very small mass. Refraction is a classical mechanics phenomenon, happens to waves traveling in a medium and it is a collective synergy of many photons impinging on the field of the atoms and molecules of the medium. Individual photons are not refracted but are scattered. In synergy ...


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Applying $\partial_1\overline{\partial}_1$, we have the following. First term:$$\partial_1\overline\partial_1 G_1 = -\pi\alpha' \eta^{\mu_1\mu_2} \delta^2 (z_1-z_2,\overline z_1-\overline z_2) X^{\mu_3} (z_3,\overline z_3) X^{\mu_4} (z_4,\overline z_4) $$$$+\text{ } {\rm permutations~of~indices~} (2,3,4).$$In the second term, $z_1$ can be the logarithm or ...


2

The limit $\alpha'\rightarrow 0$ is understood in terms of an expansion in $\alpha'$, where the leading order term is given by supergravity. This does not mean that one cannot have solutions with curved spacetime. As $\alpha'=\ell_s^2$, where $\ell_s$ is the string length, this limit tells us what happens if we remove "stringiness", and as it turns out, we ...


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The operator $\delta(\gamma(z))$ is meant to be the operator dual to the state $|0\rangle_{NS}$ according to the state-operator correspondence. One could call it $E(z)$ or anything like that. But Polchinski uses the notation $\delta(\gamma(z))$ with this "nested" structure because the operator described in the previous paragraph may also be interpreted as ...


2

This seems to be a simple matter of confusion regarding which variables are held constant. His notation $$\frac{\partial X}{\partial\tau}(\tau,\sigma_*)=0$$ is misleading. What he really means is $$\frac{\partial}{\partial\tau}\left(X(\sigma_*)\right)(\tau)=0$$ In other words, we fix $\sigma$ to be one of the end points and look at how it changes with ...


1

Higher dimensional dynamical objects in string theory exist, and are called branes. They arise, for example, as the objects the endpoints of open strings lie on.


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Firstly, string theory is a mathematical hypothesis that is currently speculative. It is a possible candidate for a quantum theory of gravity - a unification of QM with Einstein's theory of gravity - General Relativity. The idea of superstring theory emerged from another theory called supergravity. Supergravity was an attempt at a supersymmetric theory of ...


1

As Wikipedia explains, $E_7$ refers to several, closely related real and complex Lie groups and Lie algebras. All the various $E_7$ Lie groups (algebras) are Lie subgroups (subalgebras) of the complex Lie group $E_7$ (algebra $e_7$), respectively. The latter has complex dimension $133$ and rank $7$. Specifically, $E_{7(7)}\equiv E_{7(+7)}\equiv E_{7,7}$ ...


1

Some years ago, Gerard 't Hooft posted "How to Become a Good Theoretical Physicist", which is more inclusive than just string theory but which you'll probably still find a valuable list. Here's what he recommends for mathematics: "Primary Mathematics": Natural numbers: 1, 2, 3, … Integers: …, -3, -2, -1, 0, 1, 2, … Rational numbers (fractions): ...


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Details have to be filled in, but I think the general idea went along these lines: the variation of the action with respect to the metric $g_{\mu\nu}$ is given by $$ \delta_gS \sim \int T^{\mu\nu}\delta g_{\mu\nu}\,. $$ Now specialize to particular variations of $g_{\mu\nu}$, the diffeomorphisms. For an infinitesimal diffeomorphism of the form $x^\mu\to ...


1

The action principle holds by assumption. It is assumed that all equations of motion follow from this principle with the appropriate action. By introducing an auxiliary tensor field $h_{\alpha\beta}$, one may write down the so-called Polyakov action $$S_\mathrm{Poly}=-\frac{T}{2}\int_\Sigma ...


1

I) The action principle of a theory is the usually taken as the first principle of a theory, and therefore it can strictly speaking not be derived. Nevertheless, the Nambu-Goto action is a natural a generalization of the following line of thought: In a Riemannian space $(M,g)$ [with Euclidean signature], a geodesics are (locally) the shortest path between ...


1

A priori, it is hard to know without having any experience with dimensional reductions. One has to get a feeling for how certain quantities change under the process, e.g. how components of the higher-dimensional gauge fields may turn into adjoint scalars, how spinors behave. An interesting thing to note is that symmetries of the lower-dimensional theory ...


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It can be rather involved. A lot of technical progress as been on this subject leading up to the modern conformal bootstrap work. Something you can exploit is that these functions should behave like correlation functions and thus are eigenfunctions of the conformal Casimir. That gives you differential equations which in some cases, especially in $D=2$ and ...



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