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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}{\...
syphracos's user avatar
  • 141
0 votes
0 answers
75 views

Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]

Lagrangian for Klein-Gordon equation is given by $$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$ To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
Vivek's user avatar
  • 45
1 vote
2 answers
78 views

What is the meaning of the differential in Doran and Lasenby's discussion of Noether's theorem for spacetime transformations?

Doran and Lasenby (Geometric Algebra for Physicists, pg. 450) state that if a transformation involves spacetime dependence (this brings to my mind common examples: translation and rotation), then ...
foghorn's user avatar
  • 163
0 votes
1 answer
155 views

Finding the Euler-Lagrange equation for a scalar field

Consider a scalar field with the following lagrangian density: $$\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-V(\phi).$$ I want to find the corresponding Euler-Lagrange equation, ...
gangio's user avatar
  • 69
2 votes
1 answer
159 views

Notation and Terminology Questions from Schwartz' QFT Book

I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing. First off, on page 34 he defines a translation of a field to first order as $$...
Jeff's user avatar
  • 221
1 vote
1 answer
246 views

Four-vector differentiation (E-M Euler-Lagrange eq.)

$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
Fortinbras's user avatar
0 votes
1 answer
51 views

Derivative with respect a partial derivative in lagrangian density [duplicate]

I am reading about field theory and lagrangian densities, and I found the following lagrangian density in my book: $$ \mathcal{L} = \dfrac{1}{2} (\partial_\mu \phi)^2 - \dfrac{1}{2} m^2 \phi^2\ , $$ ...
SrJaimito's user avatar
  • 601
4 votes
1 answer
401 views

Calculating equation of motion in gauge theories: using ordinary derivatives or covariant derivatives?

For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the ...
Keith's user avatar
  • 1,706
0 votes
2 answers
274 views

Tensor Question (Klein–Gordon equation) [closed]

I have a question following the derivation of the Klein-Gordon equation from a lagrangian. From Eq. (13d), where does $\delta^\mu_\nu$ come from? I guess it's a conversion factor of some sort.
CarlF's user avatar
  • 11
2 votes
2 answers
164 views

Is there a equivalent theorem for closed form/ exact form for derivative with respect to fields

For a vector (one-form) $A_\mu$, when \begin{eqnarray} \partial_{[\mu}A_{\nu]}=0 \end{eqnarray} then, there exists a scalar $\phi$ such that \begin{eqnarray} A_\mu =\partial_\mu\phi \end{eqnarray} ...
KayS's user avatar
  • 91
0 votes
2 answers
244 views

Why partial w/respect to $\phi$ of d'Alembertian of $\phi$ = 0?

Why is $\frac{\partial }{\partial \phi}(\frac{\partial^2 \phi}{\partial x^2})$ = 0? I'm happy to read an article, but I don't know what keywords to search on. Background: Apologies if this is a math ...
midmath's user avatar
2 votes
1 answer
388 views

How to take derivative with respect to Lagrangian of complex field?

Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$. Question 1: Is $\...
ShoutOutAndCalculate's user avatar
2 votes
2 answers
442 views

How do I find the function derivative $(\delta/\delta \phi) (\partial_\mu \phi)$?

The question is simple: How do I find the function derivative of $$(\delta/\delta \phi(x)) (\partial_\mu \phi(x))~?$$ As far as I can tell, I cannot use any of the standard computational rules for the ...
Mikkel Rev's user avatar
  • 1,420
8 votes
3 answers
3k views

Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?

The Euler-Lagrange equation for particles is given by $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$ and for fields it is $$ \partial_{\mu} \frac{\...
Hermitian_hermit's user avatar
6 votes
2 answers
2k views

Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$

If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$. In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
SRS's user avatar
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