Questions tagged [covariance]
How a quantity behaves under a change of basis vectors. This tag covers relativistic covariance, as well as contravariant and covariant tensors not necessarily in the context of relativity. DO NOT USE THIS TAG for statistical covariance.
106
questions with no upvoted or accepted answers
11
votes
0
answers
214
views
Can cut-off regularisation cause a Poincaré anomaly?
Momentum cut-off regularisation leads to non-covariant results, i.e., it breaks the Poincaré covariance of the theory. Is there any guarantee that Poincaré covariance is always restored when we remove ...
5
votes
0
answers
134
views
Can one quantize Chern-Simons theory in the covariant phase space formalism?
The covariant phase space, which coincides with the space of solutions to the equations of motion, gives a notion of phase space which does not rely on a decomposition of spacetime of the form $M=\...
- 3,875
4
votes
0
answers
106
views
How does the bulk-to-boundary propagator transform under diffeomorphisms?
In AdS/CFT, the bulk-to-bulk propagator can be obtained as the limit of the bulk-to-bulk
propagator with one point approaching the boundary. For example in the scalar case
\begin{equation}
K_{\Delta}(...
- 98
4
votes
0
answers
93
views
What is the form of the conformal covariance equation in a 2D Lorentzian CFT?
The naive expectation is that a $2D$ Lorentzian correlator should obey a covariance equation of the form
$$A_n(z_{i})=\Bigg(\prod_i (c z_i+d)^{h_i}(\bar{c}\bar{z}_i+\bar{d})^{\bar{h}_i}\Bigg)A_n\Big(\...
- 2,240
3
votes
0
answers
48
views
The Role of the Kaehler Manifold in Supergravity
Actually, I already asked a similar question Coupling of supergravity to matter which has remained unanswered. So this time I will be less general.
In the very interesting paper arXiv:2212.10044 [...
- 9,710
3
votes
0
answers
65
views
Counting independent components of Lorentz tensor
Say I have Lorentz tensors $A^{\mu\nu}$ and say this Lorentz tensor is symmetric under $\mu \Leftrightarrow \nu$ and there are only $p^\mu$ and $q^\mu$ as the physical Lorentz vectors involved. If so, ...
- 1,035
3
votes
0
answers
109
views
Is there a both manifestly covariant and unitary formalism of Quantum Field Theory?
The Lagrangian formalism is only manifestly covariant and the Hamiltonian formalism is only manifestly unitary.
In classical field theory, there exists the De Donder Weyl formalism, which is ...
- 6,332
3
votes
0
answers
159
views
Gaussian path integral in QFT in curved spacetime
We have the expression for a Gaussian integral
$$I(A,b)=\int d^nx~ \exp\left[-\frac{1}{2}\sum_{n,m}x_n A_{nm}x_m+\sum_{n}x_n b_n\right]=I(A,0)\exp\left[\frac{1}{2}\sum_{n,m}b_n\left(A^{-1}\right)_{nm}...
- 2,266
3
votes
2
answers
410
views
Autocorrelation and variance: can the fluctuation-dissipation theorem actually be written in terms of fluctuations?
I am considering the theorem in a statistical mechanics context, but I suppose the question could be extended to other fields where it applies as well.
If we have a system with property $A$ and apply ...
- 1,003
3
votes
0
answers
169
views
Symmetries in Relativistic mechanics and Field Theory and Lorentz Invariance
In Non-Relativistic Lagrangian mechanics usually we didn't impose any constraint on the Action/Lagrangian, rather than to be respectively a functional/function (or 3 form on Manifolds).
In QFT and ...
- 107
3
votes
0
answers
391
views
Understanding the four-dimensional volume form in Action of Lagrangian
Into the following part below, I don't understand what is precisely a "four-dimensional volume form" implied in the integral below:
For comparison, the ...
user87745
3
votes
0
answers
123
views
How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?
On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes:
Who adopts this point of view argues that it should also be adopted for forces in ...
- 7,940
3
votes
0
answers
213
views
Tensor density and the coefficient $\sqrt{-g}$
Usually it is claimed that we use the coefficient $$\sqrt{-g}$$ for the action in the curved spacetime, to make the integrand treats as a scalar but not as a scalar density under general coordinate ...
- 159
3
votes
0
answers
158
views
Variations of tensors are tensors?
Recently I posted a question about variation of metric.
I thought I understood it and talked with my friend about it.
After that he said he's not convinced because he can't prove variation of metric ...
- 343
3
votes
0
answers
144
views
Stuck on Weinberg's quick derivation of Thomas precession
In Weinberg's Gravitation and Cosmology he has a pretty concise derivation of the Thomas precession formula (Eq. 5.1.13). But I don't get the first step...
A particle with intrinsic spin is under the ...
- 866
3
votes
0
answers
158
views
Showing path integral formalism is Lorentz-invariant without resorting to Hamiltonian formalism
I think people typically say that path integral formalism is manifestly Lorentz-invariant, because Lagrangian density is Lorentz-invariant. However, path formalism is typically defined with time ...
- 51
3
votes
0
answers
864
views
Is classical electromagnetism conformally invariant? (and a bit of general covariance)
The contest is a flat $4d$ Minkowsky space.
A conformal transformation is a diffeomorphism $\tilde x(x)$ such that the metric transforms as
\begin{equation*}
\tilde g_{\tilde \mu \tilde \nu} = w^2(x) ...
- 616
3
votes
0
answers
1k
views
"Hypersurface orthogonal" component of covariant derivative of normal vector
I believe that answer to my question is rather trivial but I can't seem to get my head around it. In the context of the ADM formulation of gravity (or any other differential geometry context, I guess) ...
- 56
3
votes
0
answers
201
views
Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?
Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
- 71
3
votes
0
answers
392
views
composition of space expansion and movement as a gauge invariance
suppose i have a space-time where we have one point-like object* which we will call movement space probe or $\mathbf{M}_{A}$ for short, and it will be moving with constant velocity $V^A_{\mu}$ in ...
- 14.4k
2
votes
4
answers
340
views
Derivation of covariant derivative
I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows:
If we begin with the Dirac Lagrangian ...
- 90
2
votes
1
answer
829
views
Proof for covariant vector transformation law
(I'm asking this on the physics exchange not on the math one because i don't need an extremely rigorous explanation)
I understand the derivation for the contravariant vector transformation law is ...
- 21
2
votes
0
answers
172
views
Relation between rank-2 antisymmetric tensor and other bilinear covariants
Given a spinor $\psi$, if one defines the bilinear covariants $J=\bar{\psi} \psi$,
$J_{5}=i \bar{\psi} \gamma_{5} \psi$, the current $J_{\mu}=i \bar{\psi} \gamma_{\mu} \psi$, the axial current $J_{5 \...
2
votes
0
answers
134
views
In relation to General Relativity, Invariant form of Lorentz force
I am a High school student, so am new to this topic.
Using my own understanding I want to come up with the covariant(or invariant) form of Lorentz force. But I am unable to do it.
MY IDEA:
Let $\...
- 246
2
votes
0
answers
239
views
Re-Writing the Dirac Equation in True Covariant Form
This is a rather brief inquiry, but to get to the point it's always frustrated me that in non-relativistic and relativistic quantum mechanics spin matrices are written as a "vector of matrices" ...
- 167
2
votes
0
answers
101
views
Commutator relation of EM Field Covariant?
I read that for quantization of the EM-Field, you demand the canonical equal-time commutation relations:
$$[A^\mu(\vec{x},t), \pi^\nu(\vec{y},t)] = i \hbar g^{\mu \nu} \delta^3(\vec{x} - \vec{y}). $$
...
- 6,733
2
votes
0
answers
836
views
Notation for vectors and covectors
This is probably a very simple question, and I think I know the answer, but I cannot find a place to solidly confirm this.
So if I want to write a vector $\mathbf{V}$ in terms of its contravariant (...
- 981
2
votes
0
answers
82
views
Where is a proof that string field theory is generally covariant?
Given a space-time coordinate of a string $X^\mu(\sigma)$ dependent on the position $\sigma$ around the string. And a string field functional $\Phi[X]$, is there a proof that the equations of motion (...
user84158
2
votes
0
answers
180
views
Covariant quantization of an interacting relativistic particle
A method of covariant quantization for a free relativistic particle appears in the first part of some introductory string theory texts (Tong, Zwiebach,...). None of them (as far as I hae seen) give an ...
2
votes
0
answers
57
views
Is there a general procedure for covariantizing equations?
I am currently attempting to derive covariant forms of equations whose domains are D=3 space. I am considering Lorentzian $(\mathbb{R}^4, \Omega, x, \nabla)$, where $\Omega \subset\mathbb{R}^4$ has a ...
- 136
2
votes
0
answers
218
views
Background subtraction for a signal and error analysis
I use a CCD to see the split of a energy level due to Zeeman effect.
I have a 1 dimensional CCD of 7926 pixel of 7μm each one. My CCD analyze a region 2 dimensional, and then it steps forward 200 ...
- 2,177
2
votes
0
answers
148
views
quadripolar moment in curved space
So, i'm going over the Thorne's derivation of the quadrupolar radiation term, and they write the core term as:
$$ \frac{3 r_i r_j - 2 r^2 \delta_{ij}}{4 r^5} $$
But if i try to obtain this term by ...
- 14.4k
2
votes
2
answers
294
views
Index transposing in Einstein notation
When working with Einstein's summation convention, how do I have to transpose the indexes of the tensor?
For example, suppose I want to take the matrix product with its transpose. Which is the correct ...
2
votes
1
answer
385
views
How to be sure that a law is invariant under Lorentz's Transformation?
For starters let's talk about Maxwell's Equations; we know that Maxwell's Equations are invariant under Lorentz's Transformation, after all this is why all the relativity business got started. To ...
- 4,474
1
vote
0
answers
88
views
Derivation of covariant derivative by means of parallel transport
I've studied covariant derivative in many courses so far, but I got stuck on the definition given by the teacher notes of the exam of Topological QFT.
I think that he improperly used the name "...
- 144
1
vote
0
answers
43
views
Background field Method in non-linear $\sigma$-model: Covariantized Taylor series of geodesic between fields
In this paper, the authors try to develop an expansion of the non-linear $\sigma$-model action with covariant terms called the Background field method. We have a "fixed" background field $\...
- 1,290
1
vote
0
answers
22
views
Tramsformation of spatial components of the 4-force
I'm trying to learn special relativity by myself. I've been reading the Griffith's chapter about relativistic dynamics and electrodynamics (chapter 12), but one thing it's not clear to me. I've been ...
- 530
1
vote
0
answers
98
views
Under what representation of the Lorentz group do scalar $\textit{fields}$ transform?
I know that if I am sitting in a spacetime $M$ at point $p$, vectors live in the tangent space $T_pM$, and tensors in the tensor product space etc. If I want to consider general tensor fields, I ...
- 1,099
1
vote
0
answers
56
views
A curve in contravariant coordinate notation
I'm reading a general relativity text, the metric is expected to describe generally the spacetime in a gravitational field. In it there is for ex. the notation
$$\sum_j g_{ij} \frac{dx^j}{ds}$$
for ...
- 27
1
vote
0
answers
64
views
Contravariant or covariant tensor in electromagnetism?
I have a question about the following 2 tensors: the permittivity tensor and Maxwell's stress tensor. I was wondering if someone can explain which one is contravariant or covariant, and show why that ...
- 65
1
vote
0
answers
36
views
How should physical quantities transform under translation?
I'm asking this question in the setting of non-relativistic classical mechanics.
We know that (consequence of the covariance principle?) physically meaningful quantities should be described by tensors,...
- 1,480
1
vote
1
answer
72
views
When can we take the action between two fixed times in a relativistic classical field theory?
Peskin and Schroeder give a brief outline of Lagrangian field theory on page fifteen in their Quantum Field Theory book, where they write:
Lagrangian Field Theory
The fundamental quantity of ...
- 3,098
1
vote
0
answers
60
views
The Relationship between Coordinates and SpaceTime
I was reading a paper describing the contributions integral mathematicians and physicists have made in the advancement of physics by Michael Atiyah (https://www.jstor.org/stable/24111066), but have ...
- 11
1
vote
0
answers
79
views
Classical electromagnetism field strength with index up and down
In Classical electromagnetism, we know the Lagrangian density read
$$
\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}
$$
where
$$
F_{\mu \nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}
$$
However, I ...
- 1,401
1
vote
2
answers
149
views
On general covariance
If newton's theory could be formulated in the language of differential geometry (symplectic manifolds), what do we really mean when we say that the theory is covariant under the Galilean group when it'...
- 182
1
vote
1
answer
83
views
What does $C^{XY}_{\ell}$ mean when we weasure $a_{\ell m}$ in the sky?
In cosmology context, we have the general formula for the angular power spectrum $C_{\ell}$ :
$$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\...
user87745
1
vote
0
answers
71
views
Problems with index position of angular momentum
I want to calculate the poisson bracket of two angular momentum components:
$$\{L^i,L^j\}=\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial q^k} \frac{\partial\epsilon_{\:\:s}^{\:j\:r}q^sp_r}{\...
- 516
1
vote
0
answers
174
views
What physical properties are invariant under relativistic transformation?
Most of the familiar physical properties vary according to the relativistic observer's reference frame - speed, mass, energy, time, length, etc. Which properties remain invariant, so everybody will ...
- 7,372
1
vote
1
answer
605
views
General covariance of the Maxwell equations in 4-tensor form
Are the Maxwell equations written with the derivatives of the EM field strength tensor not generally covariant? I can't seem to prove that is.
The Maxwell equations in 4-tensor form:
$\partial_{\mu}F_{...
- 611
1
vote
0
answers
55
views
Question on four-velocity condition for timelike observers
I am a bit rusty on GR, I have the condition $u_{\mu}u^{\mu} = -1 $, in some notes I am given that we can obtain:
$$-1 = u^{2}_{t} [g_{tt} +2 \Omega g_{t \phi} + \Omega^{2} g_{\phi\phi}]. \tag{1} $$
...
- 11