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...
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-...