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OP last question essentially reads (v1): What's the meaning of the product of normal-ordered operators $$: \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): ~?$$ Strictly speaking, it is a radially ordered product of normal-ordered operators $${\cal R} \left[ : \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): \right].$$ However, ...

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Considering scattering amplitudes it's commonly believed that string theory is UV-finite, so it gives you the exact answer that you are seeking, at whatever energy level, without needing approximations (at least in principle, in practice we know mainly perturbation theory). Intuitively, the UV finiteness is due to the finite size of the string. To be clear, ...

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OK, I have tried to solve this one ever since I posted the question. Since three people have starred it, I will post what I have found. Probably the best way to regularize the determinant is to use the Riemann zeta-function regularization. We start by writing the product of eigenvalues (with dependence on $T$ which we want to acquire) as an exponent: $$... 1 There is nothing to add to the first part of Jake Lebovic's answer. With regard to the second part of the question -- how to compute the OPE of two stress tensors -- one uses Wick's theorem. Normal ordering means one does not contract together the individual fields making up the normal ordered operator, in this case the two \partial X's making up ... 1 You want to take the derivative with respect to both z and w. Take$${X^\mu }\left( z \right){X^\nu }\left( w \right) \sim - {1 \over 4}{\eta ^{\mu \nu }}\ln \left( {z - w} \right)$$and use the following derivative$${{{\partial ^2}} \over {\partial w\partial z}}\left[ {{X^\mu }\left( z \right){X^\nu }\left( w \right)} \right] = {\partial \over ...

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To my knowledge it is not forbidden by the standard model laws. The point is that at low energy scale this process is strongly subleading with respect to other decay's channels, for instance to an electromagnetic decay in two photons. At high energy the strength of gravitational force grow and that process could be one of the main channels of decay. To be ...

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It should be noticed the difference between aplying the Holographic Principle in to a 3D or a 4D world. First it avoids the idea of infinite dimensions or "reality is unknow (infinite) until you look at it" by allowing fast comunications. Then if you start with a some sort of +4D world it returns to a 3D world except "inside the particles themselfs" wich ...

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Since the worldsheet theory is conformal, you are allowed to "shrink the boundaries to a point". So the usual viewpoint is that the worldsheet are boundary-less with certain points on them corresponding to the former boundaries. The cylinder, for instance, becomes a twice-punctured sphere - the punctures are the places where one inserts the vertex operators ...

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The states of string theory are quantum states. They represent a "vibration" of the string in the same sense that a particle in standard QFT represents a "vibration" of the quantum field. That is, they do not represent actual "physical" vibration at all. In particular, the states do not describe actual physical position, vibrations or whatever of the ...

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The fundamental String of IIA String theory is the M2 brane of M-theory wrapped on the M-theory circle at weak coupling limit. The weak coupling limit is when the radius of the compactified circle is very small. More generally at weak coupling limit the best description is the IIA string theory while at strong coupling limit the M-theory description.

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There are no fundamental strings in M-theory. M2 branes can end on M5 branes and these appear as strings in the world volume of the M5 branes but these strings are of solitonic nature.

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I) It seems the resolution to OP's question lies in the difference between the Levi-Civita symbol, which is not a tensor and whose values are only $0$ and $\pm 1$; and the Levi-Civita tensor, whose definition differs from the Levi-Civita symbol by a factor of $\sqrt{|\det(g_{\mu\nu})|}$. II) The 2D Euler-density is $$E_2~=~ \frac{1}{8\pi} ... 1 M2 and M5 branes are fundamental objects of M-theory. You can see this from the superalgebra of D=11 SUGRA, which is the low energy limit of M-theory. Moreover M2 and M5 couple respectively electrically and magnetically to the RR 3-form A_3 of D=11 SUGRA. In the compactification limit for the eleventh dimension, from M-theory you get the IIA ... 3 This started as a comment in reply to CuriousOne's comments, but is getting too long , so I will answer the question if it does not become duplicate. Observations are the basic definition of "real" in physics, as with life in general. Observations start from classical particles, and in the last centuries a consistent mathematical model of mechanics, ... 3 Consider two vector fields a^\mu(x) and b^\mu(x) in a space-time. The local angle between the two vector fields is given by$$ \cos\theta(x) = \frac{ a(x) \cdot b(x) }{ \|a(x)\| \|b(x)\| } = \frac{ g_{\mu\nu}(x) a^\mu(x) b^\nu(x) }{ \left| g_{\alpha\beta} a^\alpha(x) a^\beta(x) \right|^{\frac{1}{2}} \left| g_{\rho\sigma} (x) b^\rho(x) b^\sigma(x) ...

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