If we work in cartesian coordinates, we say for instance, that
D_x \phi = \left( \frac{\partial}{\partial x} + i g \sum_a T_a A^a_x \right) \phi
where g is the gauge coupling, and \{T^a\} are the generators of the gauge group, and \{A^a_\mu\} is the gauge vector field.
But what happens when...
In the usual way to do QFT, we find the Green functions to the quadratic part of the lagrangian (usually with Feynman boundary conditions), and use this in the computation of n point (usually time ordered) correlation functions. Suppose one manages to solve instead the full nonlinear equation...
Weinberg in volume 1 of his QFT text says we do not observe any non-zero eigenvalues of A = J_2 + K_1; B = -J_1 + K_2. He says the "problem" is that any nonzero eigenvalue leads to a continuum of eigenvalues, generated by performing a spatial rotation about the axis that leaves the standard...
I made a sign error with my Lie brackets -- both of them should read
\left[\frac{d}{d\theta},\frac{\partial}{\partial x}\right] = -\frac{b}{a} \frac{\partial}{\partial y}
\left[\frac{d}{d\theta},\frac{\partial}{\partial y}\right] = \frac{a}{b} \frac{\partial}{\partial x}
But that'd mean that...
I've been trying to understand exactly how the Lie derivative parallel transports a vector, by working out an explicit example: Lie dragging \partial/\partial x at (a,0) on the x-y plane anticlockwise along the ellipse
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
I choose to parametrize the ellipse using...
If one is given two known functions G[x,y] and J[y], is there an explicit transformation that could be constructed to give us either one of the following integrals from the other?
[tex] \int dz G[z,y]^n J[z] [/itex]
[tex] \left( \int dz G[z,y] J[z] \right)^n [/itex]
Here n is an integer.
Thanks!
Actually, I have a basic question of my own. Within this ADM formalism, what sort of gauge choices are possible? With and without coupling to matter? For GR, how does one decide whether a particular choice of gauge is legitimate, ADM or not?
Also, I've thought that the lapse function and shift...
D is the covariant derivative with respect to the spatial metric h_{ij}, whereas \nabla is the covariant derivative with respect to the full spacetime metric. Up to a choice of sign convention, K_{ij} = (1/2N)(\dot{h}_{ij}-D_i N_j - D_j N_i)
I understand what you are driving at, but saying the above derivation is incorrect is incorrect.
Our goal here is quite simple. We just want to find out what is the first order variation of det g. The answer does not depend on whether we wish to view our metric as a tensor field or just a...
I was using \delta/\delta g_{\mu\nu} to mean \partial/\partial g_{\mu\nu} -- I think this is just a notational issue, not a mathematical/physical one. The delta function is not necessary because the question of what the first order variation of det g does not need to involve an action.
There are 2 methods of deriving this result.
One is to use the fact that for any matrix A
\textrm{det} A = \textrm{det} \exp[\log[A]] = \exp[{\rm Tr}[\log[A]]]
Hence if we consider g \to g+\delta g = g(1+g^{-1}\delta g), we have
\sqrt{g} \to \sqrt{g} \sqrt{1+g^{-1}\delta g} = \sqrt{g}...
Taylor Expansion
Thanks for the replies so far. I have T. Frankel's Geometry of Physics, and will be looking at it in due time.
I'd like to ask a more specific question regarding this Lie vs. covariant derivatives. Say I have 2 vector fields V, W and a metric g. I also have the integral curve...
Is there any relationship between the Lie (\pounds) and covariant derivative (\nabla)?
Say I have 2 vector fields V, W and a metric g, the Lie and covariant derivative of W along V are:
\pounds_{V}W = [V,W]
V^\alpha \nabla_\alpha W^\mu = V^\alpha \partial_\alpha W^\mu + V^\alpha...
In introductory mechanics courses we derive the equations of motion in curvilinear coordinates, especially the m (d^2/dt^2)x, by expressing the coordinate basis vectors in terms of their cartesian counterparts, and then differentating them with respect to time.
For example, in 2D polar...