Recent content by knobelc

Do you know a proof for this? Christian

I have now found the formal proof (hopefully). I will give the thread of the solution soon.

Let me clarify a few things. First of all, \Phi is a real scalar field (we don't consider complex scalar field hier), thus the step \Phi \rightarrow \hat{\Phi} yields a hermitian operator \hat{\Phi}. This means \Phi is an observable whose eigenvalues are measurable, i.e. \hat{\Phi} \left|...

I did this of course. If we have the simple quantum mechanical harmonic oscillator, the annihilation operator reads as \hat{a} = \sqrt{\frac{\omega}{2}} \left(\hat{x} + \frac{i}{\omega} \hat{p} \right) = \sqrt{\frac{\omega}{2}} \left(x + \frac{1}{\omega} \frac{d}{dx} \right), where in...

I have a two questions concering the vacuum state of the Klein-Gordon field. But let me first briefly introduce the topic. Suppose we are in special relativity, adopt the signature for the metric \eta_{\mu \nu} = (+,-,-,-), and work in natural units, i.e. c=1 and \hbar=1. Then the Lagrangian...

Why are my formulas not displayed correctly?

I think, I got the reason for the wrong sign. Since I used the signature g_{\mu \nu} = (+,-,-,-) my definitions of T^{\mu \nu} and \mathcal{L_{\rm tot}} are not correct. With my signature the correct expressions read as \delta S = \delta\int \mathcal{L} \sqrt{-g}\; dx^4 = +\frac{1}{2}\int...

Hi I have a small subtle problem with the sign of the energy-momentum tensor for a scalar field as derived by varying the metric (s.b.). I would appreciate very much if somebody could help me on my specific issue. Let me describe the problem in more detail: I conform to the sign convention...

Why is that? Is this just a rule in order to get the "right" result? Without any further information I don't see a reason not to write \delta \mathcal{L} = \frac{\partial \mathcal{L}}{\partial g_{ab} } \delta g_{ab} = - \frac{ \partial \mathcal{L} }{ \partial g^{ab} } \delta g^{ab} in...

I agree that \delta(g^{\mu \nu}\partial_{\mu}\phi\partial_{\nu}\phi) = - \delta(g_{\mu \nu}\partial^{\mu}\phi\partial^{\nu}\phi), but then problem arises that the way we write the Lagrangian determines the sign of the first term in the energy-momentum-tensor. How can we decide what way to...

Thanks a lot for your quick answer. Still there remains for me a paradox: First of all, the correct formula is \delta g^{\mu\nu} = -g^{\mu \alpha}g^{\beta \nu}\delta g_{\alpha\beta}. The whole paradox for me is then \delta g_{\mu \nu}\;\partial^{\mu}\phi\partial^{\nu}\phi =...

Sorry, I had some problems with the latex here. So I could not finish my first posted text. Here is the full version: Suppose you are given the Lagrangian of a scalar field \Phi(t) \mathcal{L} = \frac{1}{2} \dot{\Phi}- \nabla \Phi - V(\Phi ). By introducing covariant notation with...

Suppose you are given the Lagrangian of a scalar field \Phi(t) \mathcal{L} = \frac{1}{2} \dot{\Phi}- \nabla \Phi - V(\Phi ). By introducing covariant notation with \eta_{\mu \nu} = (1,-1,-1,-1) this reads as \mathcal{L} = \frac{1}{2} \eta^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi-...