I'm a bit confused about the condition given in the description of the symmetry transformation of the filed. Usually, given any symmetry transformation ##x^\mu \mapsto \bar{x}^\mu##, we require
$$\bar\phi (\bar x) = \phi(x),$$
i.e. we want the transformed field at the transformed coordinates to...
Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is:
$$
\delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial...
Homework Statement
Let ##x## and ##x'## be two points from the Minkowski space connected through a Poincare transformation such that ##x'^\mu =\Lambda_{\nu}^\mu x^\nu+a^\mu## and ##u:\mathcal{M}\to \mathbb{K}=\mathbb{R}## or ##\mathbb{C}##, ##\mathcal{M}## the Minkowski space. We define:
$$...
I am reading Polchinski's review on AdS/CFT https://arxiv.org/abs/1010.6134.
I have a very simple question, and please help me out. Thanks in advanced.
The question abou formula (3.19)
The scalar effective bulk action is given by
$$ S_0=\frac{\eta}{2}\epsilon^{1-D}\int d^Dx \phi_{\rm cl}...
Homework Statement
Variation during reproduction is beneficial to the species but not necessarily for the individual?
Homework Equations
Not any
The Attempt at a Solution
I only know about variation during reproduction is beneficial to the species but I don't know anything about how it is not...
Homework Statement
Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...
Homework Statement
I am not sure whether the meaning of the equation ##(3)## which used for deriving momentum is as same as equation ##(4)##.I will make a detailed description below.
The lagrangian function for a free particle is ##L=-mc^2\sqrt{1-\frac{v^2}{c^2}} \quad (1)##
The action from...
Hi folks,
I am a bit confused with the extreme condition used in the calculus of variations:
δ = 0
I don't understand this rule to find extreme solutions (maximum or minimum)
If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...
Homework Statement
So I'm deriving Lagrange's equations using Hamilton's principle which states that the motion of a dynamical system follows the path, consistent with any constraints, that minimise the time integral over the lagrangian L = T-U, where T is the kinetic energy and U is the...
Homework Statement
For a given function ##g:[a,b]→ℝ, 0 < a < b##, compute its total variation
\underset{[a,b]}{\mathrm{Var}}
(g) where ##g(x) = \sin(x), x\in[a,b].##
Homework Equations
The Attempt at a Solution
I know that between odd multiples of ##\frac{\pi}{2}##, ##\sin(x)## is monotone...
Homework Statement
Counting problems are a very tough subject to me, so if someone could give me tips, examples explaining what's really happening, that would be great.
Homework Equations
I know what permutations, variations, combinations, ... are. The problems involving only one of those...
I need a formula to calculate permutation.
For example I have a 5 numbers and I creating a 3 digit number from it.
The numbers are: 1, 1, 1, 2, 3; I could write up 13 variations, but I couldn't work out the formula.
If the numbers are: 1, 1, 2, 2, 3 the number of variations are 18 (if I wrote...
Hi there,
I have spent quite some time on PF as an unregistered user reading through various stuff and have learned a lot. Just registered now to seek help on something that's been bothering me a lot and to which I haven't managed to find a solution yet.
Please let me know if this is not the...