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10

Newtonian physics is generally a good approximation in a problem as long as any significant differences in the action involved in the problem are much larger than Planck's constant (if not, quantum mechanics will be needed), the speeds involved in the problem are much less than the speed of light (if not, special relativity will be needed), and as long as ...


3

There is really no complication in arriving at equation (5) given equation (5). We have: $$ \frac{d}{d\rho}\left[\frac{\rho^3}{\sqrt{1+\left(\frac{dy_6}{d\rho}\right)^2}}\frac{dy_6}{d\rho}\right]=0. $$ We solve this differential equation. $$ \frac{\rho^3}{\sqrt{1+\left(\frac{dy_6}{d\rho}\right)^2}}\frac{dy_6}{d\rho}=\tilde{c} $$ $\tilde{c}$ being a ...


3

Well, there is the Kawai-Lewellen-Tye (KLT) relations, which says that a closed string amplitude is roughly speaking a product of two open string amplitudes. See e.g. Ref. 1. References: Z. Bern, Perturbative Quantum Gravity and its Relation to Gauge Theory, Living Rev. Relativity 5 (2002) 5; Section 3.1.


3

This is a situation where knowing the history of the terminology can be helpful. The QFT/string theory terminology comes from algebraic geometry, where the term moduli space is used for any space whose points correspond to some kind of geometric object. The projective space $\mathbb{P}(V)$, for example, is the moduli space of lines in the vector space $V$. ...


2

First of all, note that the radial operator ordering ${\cal R}$ is implicitly implied in many textbooks of CFT (e.g. Ref. 1). For instance, eq. (2.2.7) on p. 39 in Ref. 1 is discussing Wick's theorem between two operator ordering prescriptions. In this case between normal ordering $:~:$ and radial ordering ${\cal R}$. See also e.g. this Phys.SE post. The ...


2

In principle yes, but there are several conceptual and technical issues that make it unclear how this could be achieved. Even though the AdS/CFT correspondence is conjectured to be exact(with much evidence hinting at this), it is hard to prove this essentially because in order to do calculations, one still has to use approximations and perturbation theory on ...


2

To be clear what we're talking about (as I'm not totally sure this is what the question intended), I'll talk about the paradigmatic example of AdS/CFT, the equivalence between $\mathcal N=4$ Yang-Mills on the one hand, and IIB string on (asymptotically) $AdS_5\times S^5$ on the other (at general parameters: no t'Hooft limits etc). We are very much closer ...


2

It is not enough to be popular or enough to quantize gravity. A TOE has to be able to include special relativity ( Lorenz invariance) and to embed the whole standard model of particle physics. At the moment only string theories are able, have the group structures , to do this, but the specific string model is still a matter of research. this also mean ...


2

The answer depends on the thermodynamic temperature of the environment of these objects, the interaction strength with which they couple to this environment and their lifetime. The spatially largest and consequently longest lived observed quantum effects, that I am aware of, are interference fringes of light that came from galaxies that are millions of ...


2

No other thresholds comparable to quantum-classical are known, nor is there any reason known at present to suspect them. The precise threshold between quantum and classical physics is actually rather simply: it is ignorance (quantum) vs knowledge (classical). More precisely, regardless of the sizes or masses or scales involved, quantum rules always apply ...


1

Well, technically, both newtonian physics, relativity and QM work in tandem, all the time. However, some of the abstractions break down - for example, when you're dealing with an electron in isolation, it behaves well in accord with newtonian physics. The same way, even if that electron moves at half the speed of light, from the POV of the electron, it sill ...


1

I suspect that Becker & Becker are referring to asymptotic safety, a theoretical programme that attempts to describe gravity with a quantum field theory. The philosophy is that a QFT is defined and sensible at all energy scales so long as all of its couplings are always finite. The simplest way for that to be the case is if all couplings flow to UV ...


1

I think aspects of this question are a bit too broad and philosophical--asking "How to explain all the mathematical structure that arises in string theory?" reminds me of Wigner's essay puzzling about the question of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"--but as to the question of whether a completed version of superstring ...


1

When you try writing a quantu, theory of strings, you get supergravity in the classical limit. Branes are just solitonic solutions to those supergravity theories. You can classify string/brane theories based on the type of SUGRA theory you get in the classical limit. That gives you four kinds in (9+1)dim and M-theory in (10+1)dim, all related by various ...


1

1) First, looking at $(2.3.4)$, you see that $j^a$ is the coefficient of $\partial_a\rho$. An application of this $(2.3.12), (2.3.13)$. To make connection with this formalism, it is preferable to choose the variations : $X^\mu\rightarrow X^\mu-\epsilon \rho(\sigma) v^c\partial_c X^\mu$ From this, we get : $\partial_a X^\mu\rightarrow ...


1

The action is $$ S = \frac{1}{2\pi \alpha'} \int d^2 \sigma \sqrt{\gamma} \gamma^{ab} \partial_a X^\mu \partial_b X_\mu $$ The definition of the stress tensor from GR is $$ T_{ab} = \lambda\frac{4\pi}{\sqrt{\gamma}} \frac{ \delta S}{ \delta \gamma^{ab}} $$ Usually $\lambda = 1$, but different books use different conventions. I do not remember what convention ...


1

Classical fields emerge when there is a large (but not definite) number of particles in a coherent state. For a simple example, for a scalar field $\phi(x)$ we can write a state that describes a classical configuration as something like $$ \exp\left(\int d^D p\; \tilde\phi(p) a^\dagger_p\right)|0\rangle. $$ Note that this isn't an eigenstate of particle ...


1

This type of calculation can be done in Mathematica using the xTensor package (its free). There is a bit of a learning curve, but the documentation is great and they have a very active google-group. Typically you will need to write the Lagrangian explicitly as a polynomial in the Riemann Tensor. Once you do this, the VarD command (xTensor) can handle the ...



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