# Questions tagged [functional-derivatives]

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### Functional derivative and variation of action $S$ vs Lagrangian $L$ vs Lagrangian density $\mathcal{L}$ vs Lagrangian 4-form $\mathbf{L}$

I have seen many potential abuse of notation that prevents me from clearly understanding variational methods in QFT and GR that I want to get this settled once and for all. This may be a bit long but ...
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### Functional derivative

I am not able to derive Eq. 21 of this paper F. Zahariev, S. S. Leang, and Mark S. Gordon, "Functional derivatives of meta-generalized gradient approximation (meta-GGA) type exchange-correlation ...
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### Variation of a field action

The field action in flat spacetime is $$S = \int d^4x\, \mathcal{L}(\phi,\partial_\mu\phi).\tag{1}$$ The variation in $S$ leads to $$\delta S = \int d^4x\, \delta \mathcal{L}. \tag{2}$$ ...
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### How can I prove/understand the following functional derivative? [duplicate]

Assume that $F[h(\xi);x,y]$ be the inverse of $G[h(\xi);x,y]$ in the sense that the following identity is satisfied: \begin{equation} \int dz F[h(\xi);x,z]G[h(\xi);z,y] \equiv \delta(x-y) \end{...
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### 2PI-effective action and functional derivatives

I'm trying to work out the 2PI-effective action for complex scalar fields. Introducing a multi field index $(a,b,c...)$ the complex conjugation and all other degrees of freedoms are suppressed, and ...
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### Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
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### Notation question in calculus of variations — QFT

these two integrals below are equal, but I am not understanding where the $x'$ variable comes from. \begin{align} I_0&=e^{ i\int d^4x \left\{ \frac{1}{2}\left[ \left( \partial\varphi(x) \right)^...
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### Help with taking derivative of Lagrangian scalar model of graviton

Quick question. Given Lagrangian density $$\mathcal{L} = -\frac12 h \Box h + \frac13 \lambda h^3 + Jh ,\tag{3.69}$$ where the scalar $h$ represents the gravitational potential, and given the Euler-...
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### Issue with calculating free fermionic propagator from partition function

$\newcommand{\D}{\mathcal{D}}$ In section 14.6 of Schwartz's "QFT and the Standard Model (7$\,^{\text{th}}$ printing)" , the author calculates the exact free fermionic partition function and, from ...
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### Inverse Green's Function identity in derivation of Hedin's equations

I'm trying to work through a derivation of Hedin's Equations in Effect of Interaction on One-Electron States by Hedin and Lundqvist (1969) and I've come across an identity that is given without much ...
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### example from physics where the action of the physical trajectory has a saddlepoint? [duplicate]

It's a well established concept in various fields of physics that the action of the field / trajectory that becomes physically real, minimizes / maximizes the action functional. For the calculations, ...
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### 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 ...
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### Confusion taking functional derivatives of the generating functional (Coleman)

I'm reading Piers Coleman's Introduction to Many Body Physics and I've run into a strange issue. In section 8.4, he considers a free theory in the presence of a source $[\bar\eta, \eta]$: \begin{...
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### Problem calculating the variation of this action

If we take a Hamiltonian density to be as following $$\mathscr{H}=\frac{1}{2}\Pi^2+\frac{1}{2}\partial_i\varphi\partial_i\varphi+\frac{1}{2}m^2\varphi^2+\frac{1}{4}\lambda\varphi^4,$$ and we have ...
1answer
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### Free propagator from generating functional in momentum space

I am reading the text by Daniel Amit, "Field Theory, the Renormalization Group, and Critical Phenomena" and trying to apply what is explained in chapters 3 and 4 to a real-life problem. However, I ...
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### Why are there two definitions for the functional derivative?

I have seen two definitions for the functional derivative. Why are there two definitions? In Goldstein's Classical mechanics 3rd edition page 574 eq. (13.63), and also in a Student's Guide to ...
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### How to calculate the second functional derivative of the action of a one-particle system?

Given the Lagrangian $$L(q,\dot{q})=m\dot{q}^2/2-V(q)$$ and the corresponding action $$S[q]\equiv\int_0^t dt' (m\dot{q}^2/2-V(q)),$$ I need to be able to evaluate the second functional derivative \$\...
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### Derivative of the magnetic field to the vector potential

So the magnetic field is defined with the vector potential A as: $$\mathbf{B}=\nabla\times\mathbf{A}.$$ How would I calculate the derivative: $$\frac{\delta}{\delta\mathbf{A}}|\mathbf{B}|^2$$ I ...