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Questions tagged [functional-derivatives]

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Expanding about background field

I refer to this set of lecture notes by Hugh Osborn, equation 4.184 on p.70. We expand an action $S[\phi]$ around a background field $\varphi(x) = \phi(x) -f(x)$ If we expand the action $S[\phi]$ ...
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120 views

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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1answer
89 views

Why does the integral symbol disappear when applying a functional derivative?

it is known that variation is defined by following: but could anyone tell me why the integral symbol disappears after following functional derivative?
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Proving Poisson bracket relations $\{\phi, P^r\}=\Pi^r$ in Ticciati's “QFT for Mathematicians”

Let $\phi$ be a scalar field, and $\Pi$ be the conjugate momentum of $\phi$. Let $\cal L=\cal L(\phi, \partial_\mu \phi)$ be the Lagrangian density. Define the stress-energy tensor as $$ T^{\mu\nu}=\...
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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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1answer
159 views

Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
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1answer
34 views

Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary

Suppose we want to make an integral $S$ of the form $$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$ stationary with the constraint $y_1\left(x_1\right) = \...
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Questions about Euler-Lagrange derivation in Classical Field Theory

I'm new to classical field theory, so I have a few basic questions: From the derivation of the Euler-Lagrange equations, we have the following: \begin{align} \delta S[\phi]&=\int d^4x\delta L(...
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2answers
146 views

Different definitions of Functional Derivative

In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent. Firstly, in Wald's book General Relativity, as ...
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Derivative of $\nabla\times(\nabla\times A)$ by A

I'm trying to find out how to quantize EM field. It seems like $\vec{A}$ and $\vec{E}$ are it's canonical coordinates. For example: $$\mathfrak{H} = \frac12E^2 + \frac12(\nabla\times A)^2$$ $$H = \int ...
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Understanding the variational / functional derivative (in relation to the Euler-Poincaré equation)

I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", ...
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1answer
173 views

Path integral measure in Chern-Simons/WZW correspondence

The relationship between 3d Chern-Simons theory on the product of the disk and the real line ($D\times \mathbb{R}$) and the chiral WZW model on $S^1\times \mathbb{R}$ was shown in Elitzur et al Nucl....
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127 views

Confusion about functional derivative in path integral

If we act a functional derivative $$\frac{\delta}{\delta J(z)}$$On the expression$$\int\int d^4x d^4y \space J(x)\Delta(x-y)J(y)$$ where $\Delta(x-y)$ is Feynman propagator. What one should get is ...
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1answer
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A Question about Path Integral Measure

I want to do the following path integral. $$\mathcal{Z}=\int\mathcal{D}x e^{iS[\dot{x}]}$$ The action only denpends on $\dot{x}$. For some reason, I want to replace the integral measure $\mathcal{D}...
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42 views

Variation of a integration involving derivatives

I'm having problem with calculating the functional derivative of $F$ with respect to $\phi(x)$ while $$F = \int d^{4}x \phi^2 \partial_{\mu}\phi\partial^{\mu}\phi.$$ I want to obtain $\frac{\delta F}...
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1answer
103 views

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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1answer
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Doubt in Functional Derivative of Lagrangian

Lecture XXXIII: Lagrangian formulation of GR by Christopher M. Hirata NON-INTERACTING DUST Consider a system with a suite of particles {A} each of mass $\mu_{A}$ following some set of trajectories $...
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123 views

Two-point Green for Free Dirac Fields

I am trying to compute the $2$-point Green function $\tau_2(x,y)$ for free Dirac fields. The corresponding formula for $\tau_2(x,y)$ is given by $$\tau_2(x,y) = -\frac{\delta^2}{\delta\eta_x \delta \...
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Functional derivative for the same function expressed before and after Wick rotation

This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf On page 17, they ...
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1answer
125 views

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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1answer
86 views

How to take into account the symmetry of the metric tensor when doing the Functional derivative in GR? [duplicate]

I have a Straightforward question. When the functional derivative of the Ricci scalar to get the GR field equations. As the derivative is done using the metric which is symmetric do I have to ...
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2answers
202 views

Why does Fermat's principle (optics) not apply to all paths?

Feynman's statement of Fermat's Principle regarding optics is the following, "a ray going in a certain particular path has the property that if we make a small change (say a one percent shift) in ...
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1answer
91 views

What does lowercase-delta mean in Noether's first theorem?

Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ...
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1answer
57 views

Variational principle, functional gradient

Given the energy functional $$E[\Psi] = \frac{\langle \Psi \vert H \vert \Psi \rangle}{\langle \Psi \vert \Psi \rangle},$$ its functional gradient is $$\frac{\delta E[\Psi]}{\delta \langle \Psi \vert}...
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1answer
185 views

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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246 views

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)" [1], the author calculates the exact free fermionic partition function and, from ...
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294 views

Making use of functionals in Martin Siggia Rose formalism

I am currently studying "Critical Dynamics - A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior", and came across an issue I can't solve. If you know about functional ...
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1answer
141 views

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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Functional derivative of a symmetrized field

I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
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1answer
83 views

Variation of Fermionic Field Operator

Suppose we have a Hamiltonian containing some interaction term $$V = \sum _{\sigma \sigma '\sigma ''\sigma '''}\iint d^3rd^3r'\hat{\psi }_{\sigma}^\dagger (\textbf{r})\hat{\psi }_{\sigma'}^\dagger (\...
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2answers
101 views

Functional derivatives of inverse tensor field

The short-hand notation here is $1 = x_1 , 2 = x_2 ,... $and $\int_{1}=\int{dx_1},\int_{2}=\int{dx_2}.... $ In appendix A of this paper https://arxiv.org/abs/hep-th/9908172 it is said that the basic ...
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1answer
139 views

Second variation of a functional

I am trying to find the second variation of the Hartree energy functional $E_{H} [\rho]$: $$ \dfrac {\delta^2 E_{H}}{\delta \rho (r)\delta \rho (r')}=\dfrac {\delta^2}{\delta \rho (r)\delta \rho (r')}\...
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2answers
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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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135 views

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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1answer
255 views

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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1answer
64 views

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 ...
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3answers
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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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1answer
51 views

Existence of Functional with some Functional Derivative [closed]

How could I do this? I tried to manipulate, but I don't know how I could get to a contradiction or something else. Show that there is no functional $S = S[\phi]$ that satisfies: (where $\epsilon_{...
3
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1answer
213 views

Where are the delta functions in Peskin & Schroeder eq. (11.67)?

In the linear sigma model, the Lagrangian is given by $ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) +\frac{1}{2}\mu^2\sum_{i=1}^{N}\left(\...
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313 views

Derivatives in Euler-Lagrange for fields

Starting with the lagrangian density $$L=\frac{1}{2}((\partial_\lambda\phi)(\partial^\lambda \phi) + \mu^2\phi^2),$$ Chen and Li yield the Klein Gordon equation $$(\partial_\lambda \partial^\lambda + \...
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1answer
76 views

Concerning a functional of a functional of the former - classical fields in Quantum Action

Let $\varphi(x)$ and $j(x)$ be two field configurations. Let $\Gamma[\varphi]$ be a functional of the field $\varphi$ defined by: $$ \Gamma[\varphi] := \inf_j \ F[\varphi, j] = F[\varphi, j_\varphi] \...
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3answers
274 views

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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Under what circumstances is the functional derivative (of an action functional) an actual function?

In general, for a functional $F[\phi]$, the functional derivative is $$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$ ...
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3answers
444 views

Correct derivation of Einstein's equations from the Hilbert action

I have been trying to understand general relativity from a first-principles perspective in my spare time, and I have been unable to find a convincing derivation of the Einstein equations. The most ...
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2answers
261 views

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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207 views

Functional derivative of a convolution

I am wondering how to compute the functional derivative of an ordinary scalar function with itself, say $$[f\star f](x)=\int dy~f(y)f(x-y).$$ My attempt would give $$\delta_{f}[f\star f](x)=2f(x-y)...
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Can I do this kind of change of variables? What would be the functional determinant?

Let's say I need to solve this functional integral: $$ Z=\int \mathcal{D}x(\lambda)F[x(\lambda)] $$ Then, I want to change the integration to $\mathcal{D}v(\lambda) $. Where $v$ is a shorthand for $v^\...
2
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1answer
533 views

Why Peskin & Schroeder are taking functional derivatives of the Lagrangian density when it is not a functional?

I have some doubts regarding equation 11.58 (see below) in the QFT book by Peskin and Schroeder. If I understand correctly, they are expanding the Lagrangian density about $\phi_{\text{cl}}$ by ...