All Questions
Tagged with covariant-derivatives or differentiation
1,900 questions
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From material derivatives to partial derivatives in the wave equation
Consider the Cauchy momentum equation:
$$\rho \frac{d^2 \mathbf{u}}{d t^2} = \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{f}$$
where $\rho(\mathbf{x},t)$ is the density, $\mathbf{u}(\mathbf{x},t)$ ...
- 225
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2
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36
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Why must the total time derivative only be a linear function of velocity? [duplicate]
I'm hung up on page 7 of Landau & Lifshitz Course on Mechanics. They claim,
$$L(v'^2) = L(v^2)+\frac{\partial L}{\partial v^2}2\textbf v\cdot \epsilon \tag{p.7}$$
The second term on the right of ...
3
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1
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118
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Covariant derivative acting on Dirac delta function
Pardon my naive computational question. In my calculations, I encounter the following expression:
\begin{equation} \label{eq1}
\frac{\delta}{\delta g^{\gamma \epsilon}(z)} \left( g_{\mu \alpha}(x) \...
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0
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63
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Four gradient relation
I'm doing an exercise in QFT and I have to calculate the energy-momentum tensor for the Klein-Gordon Lagrangian and by doing it I got the following term:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
- 141
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0
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Mass Conservation in Kinetic Theory
In chapter 9 (The Boltzmann Equation) of Schwabl's 2006 text 'Statistical Mechanics', the author has the following statement of conservation of mass,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
9
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4
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4k
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Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant but acceleration is constant?
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant, but rate of change of velocity is constant?
Like speed is only the magnitude, so ...
26
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21
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What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves
Imagine a car that's at rest and then it starts moving. Consider these two moments:
The last moment the car is at rest.
The first moment the car moves.
The question is: what happens between these 2 ...
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3
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2
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340
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Understanding the definition of the covariant derivative
I'm currently working my way through the book "Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry" and I think I have a very fundamental ...
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1
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Need help in understanding Tangential Acceleration [closed]
I am studying Circular motion and I am confused about tangential acceleration and tangential velocity. I am studying uniform circular motion and it says the tangential acceleration is $0$ in uniform ...
3
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1
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67
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"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
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The definition of the Lie Derivative
I am aware that an answer to an almost identical question already exist, however, I found the already existing answer not helpful (at least to my current question).
Carroll defines, in his book, the ...
0
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1
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53
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Derivative for the Maxwell field [closed]
I'm struggling with the following expression, which occurs in the derivation of the Maxwell Lagrangian in field theory.
$$\frac{\partial(\partial_{\mu}A^{\sigma})}{\partial(\partial^{\nu}A_{\lambda})}...
0
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1
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70
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Why are Weyl's Equations composed of only first-order derivatives?
I'm studying the Weyl's Equations from Section 1.5 of Perkins' Introduction to High Energy Physics.
The author says this:
Dirac set out to formulate a wave equation symmetric in space and time, ...
- 1,637
3
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1
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480
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Second derivative of unit vector
We know that the second derivative of unit vector (the vector from a point toward the source) is proportional to the Electric field caused by the source in a particular point.
If we imagine that our ...
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4
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2
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243
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Leibniz rule and Nakahara's definition for functional derivatives with respect to Grassmann variables
In Nakahara's book "Geometry, Topology and Physics" in section 1.5.7 (I'm reading the second edition) he defines the functional derivative with respect to Grassmann variables. He does so in ...
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
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1
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Is Stress a Derivative?
On page 289 of the text "Fundamentals of Fluid Mechanics" by Munson et al., the authors give the following definition of the normal stress acting on the surface of a fluid element:
At any ...
5
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4
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386
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Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
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1
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1
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Grassmann Numbers, anticommutation and derivative rules
If $\psi(t)$ is a complex Grassmann number and $\psi^*(t)$ is its complex conjugated. The following is true:
$$\frac{\partial (\psi^*\psi)}{\partial \psi}=-\psi^*\frac{\partial \psi}{\partial \psi}=-\...
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1
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Relationship between covariant derivative and metric tensor
In general relativity, the covariant derivative of the coordinate vector is a tensor, equal to $$x^{\mu}_{:\rho} = x^{\mu}_{,\rho} + \Gamma^{\mu}_{\rho\nu}x^{\nu},$$ is it meaningful to equate this ...
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1
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481
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Doubt in Verlet's Algorithm
In studying the temporal evolution of a system according to the deterministic model, we begin by considering a Taylor series expansion for the displacement $r$. First, we consider a positive variation ...
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0
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59
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What's the difference between $dx$ and $\delta x$? [duplicate]
In the process of defining crystal momentum $\hbar k$, I found these formulas below.
By the definition of group velocity,
$$v_g=\frac{d\omega_{nk}}{dk}=\frac{1}{\hbar}\frac{dE_{nk}}{dk}$$
Also if an ...
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0
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How to prove that a Lie algebra-valued differential form is exact for the covariant derivative [migrated]
Given a differential $p$-form $\omega^A$ over a smooth manifold with values on some Lie algebra, I wanted to know how could one prove that it can be written as an exact form for the exterior covariant ...
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1
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What happens to $g^{\alpha\beta}_{,\sigma}=-g^{\alpha\mu}g^{\beta\nu}g_{\mu\nu,\sigma}$ when $g_{\mu\nu}\rightarrow \eta_{\mu\nu}$ (weak field limit)?
The equation
$$g^{\alpha\mu}_{\,\,\,\, ,\sigma}\,g_{\mu\nu} + g^{\alpha\mu}\,g_{\mu\nu,\sigma} = (g^{\alpha\mu}g_{\mu\nu})_{,\sigma} = \delta^\alpha_{\nu,\sigma} = 0 $$
gives the useful relation
$$g^{\...
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6
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Is integration physical, but differentiation is not? [closed]
There are electrical (e.g. analogue computers), and even mechanical (ball-pen) methods to generate the integral of a given function.
On the other hand, naively differentiating a physically given ...
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0
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62
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A trick for derivatives of thermodynamic quantities [closed]
Starting from
$$dU=TdS-PdV$$
We can write, for instance $U(T,V)$ and $S(T,V)$ to obtain:
$$\left(\frac{\partial U}{\partial T}\right)_VdT+\left(\frac{\partial U}{\partial V}\right)_T dV=T\left(\frac{\...
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2
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675
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How can I calculate derivative of eigenstates numerically?
I want to calculate $\langle n | \partial_{k_x} n \rangle$ where $| n \rangle \equiv | u_{n,\mathbf{k}} \rangle $ is the $n$-th Bloch eigenstate of a $6\times6$ Hamiltonian $H\equiv H(\mathbf{k})$. ...
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Is this mathematically correct that gradient of deformation gradient is equal to deformation gradient?
The deformation matrix is defined as follows, where $x$ is the current location and $X$ is the reference location. It shows the relationship between current $x$s with regard to original $X$s,
$$F = \...
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5
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1
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330
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Divergence of vector field term-wise
In a spacetime $(M, g)$ the following identity for the divergence of a vector field $X$ holds
$$ \nabla_{\mu} X^{\mu} = \frac{1}{\sqrt{-\det g}} \,
\partial_{\mu} \big( \sqrt{- \det g} \ X^{\mu} \big)...
- 695
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1
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Why take the derivative of variables such as area, mass, and radius?
I'm taking a module on stars and the solar system; I've attached notes from our first lecture- hydrostatic equilibrium. I'm confused about the notation $\mathrm{d}$ for $\mathrm{d}A, \, \mathrm{d}r$, ...
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Is 4-velocity a Vector in the Sense of Covariant Derivative along Worldline
The definition of 4-velocity $U^{\mu} \equiv dx^{\mu}(\tau)/d\tau$, however, we've learnt that the covariant derivative for a vector along a curve parametrized by proper time is,
$$\frac{DA^{\mu}}{D\...
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2
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Perfect gas relation in differential form [closed]
I have a problem to understand the transformation of the perfect gas relation:
$$ \rho\cdot R\cdot T = P $$
into its differential form:
$$\frac {dp}{p} = \frac {d{\rho}}{\rho} + \frac {d{T}}{T}$$
How ...
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1
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75
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Derivative wrt retarded time
I am confused by the following statement in footnote of Griffiths 4th edition (page 446):
$$\frac{\partial }{\partial t_r} = \frac{\partial }{\partial t},$$ where $$t_r=t - \frac{\mathscr{r}}{c}$$ ...
1
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1
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98
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Proving a Superfunction Identity
I am trying to figure out the proof of the identity given between equations (1.11.7) and (1.11.8) in ref. [1], i.e.
\begin{align}
\Phi'(e^{-K}\,z\,e^K)=e^{-K}\Phi'(z) \tag{1}
\end{align}
where $z=(...
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3
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
2
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1
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422
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How can the divergence of a cylinder with uniform magnetic field be non-zero?
When I'm calculating the divergence of a cylinder with uniform magnetic field ($B=K=\text{constant}$) according to the formula of divergence in cylindrical coordinates I'm getting the same constant ...
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0
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Conflicting Solutions for Calculating Apparent Speed of Jogger's Image in Convex Mirror
I’m facing a challenge with a physics problem due to conflicting solutions across different sources, and I'd appreciate some clarification.
Problem Statement:
Suppose, while sitting in a parked car, ...
0
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0
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38
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Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is
$$
\nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda}
$$
It has the following identity:
$$
\nabla_{\mu}U^{\mu} = \frac{\...
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1
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84
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Where did $1/2$ of this come from? [duplicate]
Work done by an external force $F$ upon a particle displacing from point 1 to point 2 is defined as
$$
W_{12} = \int_1^2 F \cdot dr
\, .$$
Kinetic energy and work-energy theorem: According to Newton's ...
1
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1
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How to "rectify" this wave-equation derivation for Longitudinal waves?
To derive the differential equation for longitudinal waves, my professor proceeded like this:
We are using the concept of $N$-coupled oscillators. Consider a slab of length $l$ and cross sectional ...
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1
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Can a wave function discontinuous in the time variable be a solution of the Schrödinger equation?
It is well known that wave functions that are discontinuous in the space variable cannot be solutions of the Schrödinger equation, because the Schrödinger equation is a second-order differential ...
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2
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1
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What is the meaning of this complex derivative with respect to a wave function?
In quantum optimal control papers such as (Loading a Bose-Einstein Condensate onto an Optical Lattice, https://arxiv.org/abs/cond-mat/0209195) and (Introduction to the Pontryagin Maximum
Principle for ...
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Equations with fractional derivatives
Assume we have an equation which represents the flux of some quantity as $q = -D \dfrac{\partial T}{\partial x}$ (Eqn. 1), where the diffusion coefficient $D$, variable $T$, $x$ and $q$ have some ...
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Differential form of Planck's Distribution Law interpretation
So I didn't encounter differentials that often until now, I was taught that the seperate parts of $dy/dx$ for example are not supposed to have any sort of independent existence - ok.
(Calculus, 4th ...
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1
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How to calculate jerk in uniform circular motion?
We can calculate the centripetal acceleration in circular motion by the equation v^2/r. But how do we calculate the jerk (which is acceleration over time)?
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1
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Derivative of the product of a scalar function and a vector valued function
According to Berkeley Physics Course, Volume 1 Mechanics,
The time derivative of a vector valued function can be derived from the formula:
$$
\mathbf{r}(t) = r(t)\mathbf{\hat{r}}(t)
$$
where the ...
- 25
1
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1
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71
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Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as:
$\chi^{(2)}:\textbf{E}\textbf{E}$
or
$\nabla\textbf{E}:\partial \...
- 173
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1
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53
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Why is linear charge density $dq/dl$ and not $q/l$?
If linear charge density is charge per unit length then shouldn't it be $q/l$. Why is it $dq/dl$ instead? Wouldn't that mean it is only being calculated for a small element and not the whole length?
2
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2
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95
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Covariant derivative of a Wilson line
Does the covariant derivative of a Wilson line given by $$W[A; z_0, z] = {\cal P}e^{-i\int^z_{z_0} dz ~A^af_{abc}}$$
vanish, i.e. $$D_zW[A; z_0, z] = 0~?$$
- 1,415
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0
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62
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Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
