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Questions tagged [diffeomorphism-invariance]

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Why does diffeomorphism invariance not imply all physical theories are Poincare invariant?

It is said that diffeomorphisms are global symmetries of all theories with fixed background geometry, and local symmetry of gravitational theories i.e. theories with dynamical metric. In the case ...
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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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Resources for Faddeev-Popov method. (Specifically for diffeomorphism gauge fixing.)

I am struggling to get the same result as this paper (eq. 3.10) for my ghost field when gauge-fixing diffeomorphisms in linearized gravity. I would appreciate it if someone could point me in the ...
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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 ...
mathemania's user avatar
1 vote
3 answers
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Diffeomorphisms and pullbacks

First of all, I understand that this will be mostly a mathematics questions. However, I'm asking this in the context of General Relativity, which comes with its own language, conventions and ...
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1 answer
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Semidirect product of Diffeomorphism group and Weyl transformations

This is more a mathematical question but in my string theory lecture we always divide in the Polyakov path integral by $$\mathrm{Diff}\ltimes \mathrm{Weyl}$$ and I was wondering why there is the ...
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Metric tensor under diffeomorphisms

It's probably a stupid question but I can't understand where I am wrong. I have a manifold with metric $g(X,Y)$ and I know that under infinitesimal diffeomorphism ($x'^\mu = x^\mu + \varepsilon^\mu$) ...
Physics Koan's user avatar
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A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables [duplicate]

I would like to understand the following problem: You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows: $$S=\int_{\mathbb{R}}dtL(\...
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A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables

I would like to understand the following problem: You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows: $$S=\int_{\mathbb{R}}dtL(\...
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8 votes
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"No local gauge invariant observables in gravity"... Is it a classical or quantum statement?

I have seen different explanations to understand why there are no local gauge invariant observables in gravity. Some of them explain that diffeomorphisms are a gauge symmetry of the theory and thus ...
P. C. Spaniel's user avatar
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2 answers
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Diffeomorphism invariance for derivative of scalar fields [closed]

In GR, it is well-known that the gravitational stress-energy tensor is a pseudotensor, i.e. it is not gauge-invariant. To make it gauge-invariant one needs to take it under average integral $\langle \...
gravitone123's user avatar
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Diffeomorphism Invariance of Terms in Lagrangian which use Gauge Fields

A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms ...
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Diffeomorphism invariance in 2 dimensional metric [duplicate]

I am reading "String Theory and M-Theory" by Becker, Becker and Schwarz. In Chapter 2, the authors try to gauge fix the auxiliary field. They start from the general expression $$h_{\alpha\...
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Spin Connection, Killing Equation and Spinors under Diffeomorphism

Under a diffeomorphism $dx'^{\mu}=\frac{\partial x'^{\mu}}{\partial x^{\nu}}dx^{\nu}$ we have that the components $A_{\mu}$ of every 1-form $A=A_{\mu}dx^{\mu}$ transform as: $A'_{\mu}=\frac{\partial x^...
Aleph12345's user avatar
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Gauge invariance or diffeomorphism invariance in GR of observables?

I am confused by the definitions of a gauge transformation, a coordinate transformation and a diffeomorphism. In particular, should observables in GR be fundamentally invariant under gauge ...
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How do general coordinate transformations fix components of the metric?

In studying string theory, one sees the argument that we can gauge-fix the worldsheet metric to become the Minkowski metric. The argument usually goes that general coordinate transformations (GCTs) ...
Daniel Waters's user avatar
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1 answer
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Current of conformal transformation

Suppose we have a theory with conformal invariance that has been extended to a diffeomorphism invariant theory in a way that the resulting energy-momentum tensor is traceless on-shell (which can ...
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Composition of diffeomorphisms in the ADM formalism?

In the ADM formalism there are 4 constraints: $C_\mu(x)$, which are known as the Hamiltonian and spatial diffeomorphism constraints. In the quantum theory, $C_\mu(x)$ are the generators of ...
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Can any useful physical theories other than TQFTs be formulated on a smooth manifold without a metric structure?

The vast majority of physical theories are formulated on a spacetime that is mathematically represented by a pseudo-Riemannian manifold, i.e. a smooth manifold with a metric tensor structure. The ...
tparker's user avatar
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Is the massive Fierz-Pauli action invariant under changes of coordinates?

The action for linearlized gravity on a curved background has the form $S_{LinGrav}=\frac{1}{16 \pi G} \int d^{4} x \;\sqrt{-g}\;\left[-\frac{1}{4} \nabla_{\rho} h_{\mu \nu} \nabla^{\rho} h^{\mu \nu} +...
Nik's user avatar
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How to define volume of Weyl transformation and diffeomorphism groups?

I have a trouble in defining Weyl transformation and diffeomorphism group volumes in the formal expression of string partition function on some manifold $M$: \begin{eqnarray} Z_M=\frac{\int Dg [DX]_g}{...
Yuan Yao's user avatar
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Corrected Poisson's equation and the preferred time direction of the cosmological constant? [closed]

So if I correct (equation 2.6) the Poisson's equation with the cosmological constant. Then, I get $$ \Delta \Phi = (4 \pi G_N) \rho - \Lambda \tag{1}$$ But when I think of this from the Einstein-...
More Anonymous's user avatar
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Are different "topological" AdS spacetimes isometric?

In this paper https://arxiv.org/abs/hep-th/9808032 the author says that black holes in AdS spacetimes can have horizons with different topologies. In particular, when the black hole mass $M$ vanishes, ...
Gianluigi Tartaglione's user avatar
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1 answer
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How is the Virasoro symmetry realised on $AdS_3$?

In the context of the Holographic correspondence, $AdS_{n}/CFT_{n-1}$, it is often cited as a "confirmation" that the two symmetry groups of the theories correspond. Indeed, in dimensions $n&...
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How does an axial vector field transform under a diffeomorphism?

Does it behave like a normal vector field?
ututu's user avatar
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1 answer
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Mathematical characterisation of diffeomorphisms in General Relativity

Considering the diffeomorphism covariance/invariance of General Relativity, is it possible to characterise mathematically the various kinds of possible transformations $x'^{\mu} = f^{\mu}(x)$? All of ...
Nicola's user avatar
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1 answer
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State of the art on Modified gravity : going beyond the 2nd order differential equations, diffeomorphism invariance breaking, extra degrees of freedom

I am going to do a state of the art on Modified gravity models. I have found a talk that presents the problematic. In particular, it is said the following things : Modifying General Relativity How to ...
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1 answer
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String theory: Conformal invariance and Conformal Killing Vectors

I am confused by the relation between the invariance of the Polyakov action under conformal transformations and the Conformal Killing Vectors (CKVs) appearing during the process of quantization. Let ...
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Does compactification of a Nambu-Goto string in one direction break Diff invariance?

Assume we have a Nambu-Goto action, in phase space, for a closed string. If I compactify one coordinate of the target space, do I reduce the diff invariance of the system. We have $$S=\int d^2\sigma \...
physshyp's user avatar
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3 votes
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143 views

Why string theory action needs to be manifestly diffeomorphism invariant?

A typical QFT action is obviously not diff invariant, in other words, the integration parameters have physical significance as time and position. For example action for massless free scalar field on ...
physshyp's user avatar
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2 votes
1 answer
192 views

A Question about Diffeomorphism Invariant Action

I remember that the canonical Hamiltonian of a diffeomorphism-invariant theory, in general, is zero. For example, the geodesic equation is derived from the action of arc length $$S[g(\tau)]=\int_{a}^{...
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1 vote
1 answer
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Weyl + Diffeomorphism invariance = Conformal symmety only in flat space?

Let $(M,g)$ be a Riemannian manifold and consider a classical field theory $\phi: M \to \mathbb{R}$ given by some action functional $S$ satisfying Weyl & diffeomorphism invariance $$ S(\phi, e^\...
desos's user avatar
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2 votes
1 answer
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What is the meaning of diffeomorphism invariance?

I have read a bunch of papers and I see "diffeomorphism invariance" and I cannot understand how it works. For instance, in asymptotic safe quantum gravity, we make 2 assumptions: ...
bostorgay's user avatar
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0 answers
204 views

Gauge freedom when solving Einstein field equation

In Weinberg's Gravitation and Cosmology, Ch7, the Einstein field equations give us 6 independent equations (totally 10 equations but $∇^\mu G_ {\mu \nu} =0$ gives 4 constraints) while $g_{\mu \nu}$ ...
DarkGlimmer's user avatar
3 votes
4 answers
1k views

Diffeomorphism invariance in GR

I apologize for asking the duplicate question, but I still get confused after reading the related post In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are ...
Hao's user avatar
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4 votes
1 answer
412 views

Hamilton-Jacobi-Einstein equation

I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation: $$\frac{1}{\sqrt{g}} (\frac{1}{2}g_{pq}g_{rs} - g_{pr}g_{qs}) \...
Ayush Raj's user avatar
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1 answer
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Diffeormophism invariance of a non-local series possible?

If I want to construct Lorentz invariant forms involving a scalar field $\phi(x)$ I could have non-local terms such as: $$\int \phi(x)\frac{1}{|x-y|^2}\phi(y) dx^4 dy^4$$ or 'local' forms such as: $$\...
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2 votes
0 answers
206 views

Trying to understand the conformal gauge "derivation" in Polyakov action symmetries [duplicate]

In section 2.3 on p. 16 of the book "Basic Concepts of String Theory" by Blumenhagen, Lüst, Theisen, 3 symmetries of Polyakov action are discussed: Poincarè invariance, diffeomorphism ...
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5 votes
0 answers
297 views

Linearised diffeomorphisms on an arbitrary gravitational background Part 2

This question is a follow on from my recent post here, in the sense that I will use the notation introduced there. In that post, I considered infinitesimal diffeomorphisms of a metric $g_{\mu\nu}$ ...
NormalsNotFar's user avatar
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Linearised diffeomorphisms on arbitrary gravitational background Part 1

Consider some spacetime $\big(\mathcal{M},g_{\mu\nu}\big)$ parameterised by local coordinates $x^{\mu}$ ($\mathcal{M}$ is a smooth differentiable manifold equipped with a Lorentzian metric $g_{\mu\nu}$...
NormalsNotFar's user avatar
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1 answer
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Are fundamental symmetries thought to be exact or just an approximation?

As title states, I wish to know whether fundamental symmetries (in the most general sense of the word, e.g. gauge symmetries, Lorentz invariance, diffeomorphism invariance, not necessarily just global ...
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6 votes
1 answer
916 views

Why do we have residual symmetry when we already used all symmetry in gauge fixing of the worldsheet metric?

In Becker, Becker and Schwarz' book about string theory, the following symmetries are listed for the $\sigma$-model of the string: Poincaré transformations Reparametrizations $\sigma^{\alpha}\...
aitfel's user avatar
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9 votes
1 answer
264 views

Why is the group of gauge transformations on the frame bundle isomorphic to $\text{Diff}(M)$?

Consider the frame bundle $LM \to M$ for given Lorentzian manifold $M$. The group $\mathcal{G}$ of gauge transformations of the second kind are automorphisms $\phi:LM \to LM$ covering the identity $\...
Ben's user avatar
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1 answer
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Why do we integrate up to the invariances

Following Witten's essay What every physicist should know about string theory I understood that in the Hilbert-Einstein action is invariant under diffeomorphism in 1D and under conformal mapping in 2D....
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Why is the proper time $\tau$ the only invariant under diffomorphizem?

I'm following Witten's essay: What Every Physicist Should Know About String Theory . When applying path integral in 1D Witten mention: Part of the process of evaluating the path integral in our ...
ziv's user avatar
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3 votes
1 answer
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Prerelativity physics, Special Relativity, and General Relativity formalisms summary [closed]

In order this have a better understanding of "the big picture", a tried to do the following summary, but I can't really complete it for GR. This based on the introduction chapter of Wald's ...
xpsf's user avatar
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1 vote
1 answer
216 views

General relativity from the general linear group

I am looking at this answer: https://physics.stackexchange.com/a/225417/747. It states: Let $f\colon U\to V$ be any coordinates transformation on charts of a manifold $U,V\subset\mathcal{M}$ (i. e. a ...
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6 votes
0 answers
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No particle states in quantum gravity?

It is established that the only states in the Hilbert space of a gauge invariant theory are states which are gauge invariant. This seems like a redundant sentence, but it has nontrivial consequences ...
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