Questions tagged [diffeomorphism-invariance]
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157 questions
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Gauge invariance in General relativity [duplicate]
Suppose a theoretical physicist wants to construct a theory to explain some newly discovered phenomenon. The new theory is expected to follow certain rules or fundamental principles. There are four ...
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A confusion over how fields transform under active diffeomorphism
I have a confusion about how fields transform under active diffeomorphisms. Let me illustrate that confusion with the example of a particle sitting at point $p \in M$ on the manifold $M$. We can model ...
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Explanation of statement concerning static gauge in flat metric
I am reading Tong's notes about string theory, the second chapter, and I encountered this part that I don't know how is derived. We are considering the worldsheet $(\tau,\sigma)$ whose gauge we set to ...
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Sign issue in the Nambu-Goto constraint algebra
When trying to show that the momentum constraint in the Nambu-Goto string actually generates world-sheet spatial diffeomorphisms, I encountered the following sign issue which I was not able to resolve:...
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Global conformal gauge
I'm reading the Blumenhagen-Lust-Theisen book on string theory. On page 18, They want to discuss whether a global conformal flat metric can exist, namely
$$
h_{\alpha\beta}=e^{2\phi}\eta_{\alpha\beta}
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Can the Levi-Civita symbol be part of a geometric Lagrangian so that the corresponding action is symmetric under diffeomorphisms?
Just a quick semantic clarification before I start. By Levi-Civita symbol I mean the totally antisymmetric object $\varepsilon_{abcd}$ that's made of only $0$'s, $1$'s, and $-1$'s.
It is understood ...
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Proving the Maxwell action is conformally invariant
I want to show that the Maxwell action $$S = -\frac{1}{4}\int d^4 x F_{\mu\nu} F^{\mu\nu}$$ is invariant under conformal transformations in $d=4$.
For this I considered the proof given in Zee's book ...
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Group generating general covariance, does $ \operatorname{Diff}(M) $ reduce to $ GL(k, \mathbb{R}) $? [duplicate]
as far as i know, one of the key ideas of general relativity is that of general covariance, which states that the laws of physics are invariant under general coordinate transformations (not only ...
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Can field dependent diffeomorphisms cause a coordinate independent field transformation?
I request someone to please help me in understanding what a field dependent diffeomorphism means? If we claim that a particular change (a change which is dependent on the field) in the metric is ...
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Why do scalars scale?
The diffeomorphism invariance of scalars is often written as:
$$ \phi'(x') = \phi(x).\tag{1}$$
However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, ...
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Definitions of different types of symmetries
I'm a math student and I started studying physics last year. I'm sorry if this question has been asked before but I'm completely confused about it. In page 30 of the book "String theory and M-...
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The meaning of stress tensor conservation in general relativity [duplicate]
In general relativity one has the Hilbert stress-energy tensor defined as
$$T^{\rm matter}_{ab} = -\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm matter}}{\delta g^{ab}}~,$$
which is covariantly conserved i.e ...
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In general relativity, is gauge invariance the same as coordinate invariance?
I always understood that gauge invariance of general relativity comes from the fact that the physical observables and states are the same regardless of the coordinates we choose to express them in. I ...
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How to see the diffeomorphism invariance of a particular metric
I understand how to show in general, that under the diffeomorphism $x^\mu\to x^\mu+\epsilon^\mu (x)$, the metric tensor changes as $$g'_{\mu\nu}(x')=g_{\mu\nu}(x)-\partial_\mu\epsilon_\nu(x)-\partial_\...
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Field transformation under conformal transformation
In 1 (see references below), I'm trying to derive how a spinless field transforms under a conformal transformation, specifically eq. (2.41). CFT references/lectures are the most confusing I've seen ...
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