Yes, you're right about the upper limit being ##1##. Also, the fact that ##\sigma << 1## implies that we might as well take the upper limit to ##+\infty##. But other than a factor in front, I don't see how that changes my answer.
From this paper, I am trying to compute the coefficients in the expansion of the Gaussian wavepacket
$$\phi(x) = \frac{1}{(2\pi\sigma^2)^{\frac{1}{4}}}\exp \Big(-\frac{(x-x_{0})^{2}}{4\sigma^{2}} + ik_{0}(x-x_{0})\Big) $$ where $$\sigma << 1$$and $$k_{0} >> \frac{1}{\sigma}$$
in terms of the...
I am trying to calculate the effective potential of two D0 branes scattering in Matrix theory and verify the coefficients in this paper: K. Becker and M. Becker, "A two-loop test of M(atrix) theory", Nucl. Phys. B 506 (1997) 48-60, arXiv:hep-th/9705091. The fields are expanded about a constant...
Does this mean that the expression for the above vertex is
$$ -\frac{g}{2}\epsilon^{abx}\epsilon^{cdx}\int d\tau \langle A_{a} (\tau) A_{c} (\tau)\rangle \langle Y^{i}_{b} (\tau)Y^{i}_{d}(\tau) \rangle $$
This is the paper that I refer to. I'm trying to figure out the ghost action (Equation 2.16) in the background field gauge. I am attempting to use Srednicki's (chapter 78) expression for the ghost field in the background gauge. However, I am missing out on a √g coefficient in front of the term...
I expanded the exponential with the derivative to get:
## Z = \Bigg(1 + \frac{1}{2} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} + \frac{1}{4} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl}...
I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor.
The given distance element is
$$ ds^2 = e^{2 \lambda} dt^2 -...
I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e.
$$ \partial_{\mu} T^{\mu \nu} = 0$$
where
$$ T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...
Say we have a matrix L that maps vector components from an unprimed basis to a rotated primed basis according to the rule x'_{i} = L_{ij} x_{j}. x'_i is the ith component in the primed basis and x_{j} the j th component in the original unprimed basis. Now x'_{i} = \overline{e}'_i. \overline{x} =...
Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$:
$$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$
And
$$ \textbf{e}^i = \nabla u^{i} $$
In the primed...
The ##\mu## is contravariant and the ##\beta ## covariant, right?
Shouldn't $$ \partial_{x}\partial^{x}$$ and $$ \partial_{t}\partial^{t}$$ have opposite signs, since we are working with four vectors?