Questions tagged [functional-derivatives]

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Variation of the kinetic term in modified gravity

Assume you have an action $S=\int \sqrt{-g}(R+p(X,\phi)) d^4x$. I want to calculate the perturbations for $\phi(t,x)=\phi_0(t)+\delta \phi(t,x)$ and the standard perturbed metric $ds^2=(1+2\Phi)dt^2-(...
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Taking the functional derivative twice to the find equations of motions? for actions of this form: $S=\int \sqrt{(R\sqrt{g})^2+(L\sqrt{g})^2}d^4x$

With normal total derivatives, I observed the following interesting pattern. Start with the Euclidean norm, then take the total derivative twice: $$ \begin{align} s^2&=x^2+y^2+z^2\\ d[s^2]&=d[...
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Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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How do I show that a given Wilson loop satisfies the loop equation?

In the book Methods of Contemporary Gauge Theory by Yuri Makeenko, the loop equation in the large-$N$ limit is given by $$\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu \nu}} W(C) = \lambda \oint_C ...
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47 views

Partial time derivative of the on-shell action

I have a few questions about differentiating the on-shell action. Here is what I currently understand (or think I do!): Given that a system with Lagrangian $\mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, ...
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1answer
84 views

Notation of derivatives in field theory

Some textbooks write $$ \frac{\delta F_{\mu\nu}}{\delta(\partial_\sigma A_\kappa)} $$ which sort of implies the derivative of a functional. Some other textbooks write $$ \frac{\partial F_{\mu\nu}}{\...
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I am stuck in the derivation of Schwinger-Dyson equation for 1-point Function in Schwartz's QFT book

This is from chapter 14.7.1 in Schwartz's QFT book. I am trying to derive contact terms starting from field redefinition $\phi\rightarrow\phi(x)+\epsilon(x)$. For the 1-point function we have from ...
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60 views

Functional Derivative

In Chapter 1 of Lancaster's Quantum Field Theory for the Gifted Amateur, he gives an example of taking the derivative of a functional but doesn't show much work. I was confused how he got the answer. ...
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65 views

Calculating higher-derivatives of quantum effective potential (Weinberg QFT chapter 16 problem 2)

Couple a set of classical currents, denoted by $J$, to a set of fields $\phi$. Let $iW[J]$ be the sum of all connecte vacuum-vacuum amplitudes. The Weinberg Quantum Theory of Fields chapter 16 says ...
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Integral over a total functional derivative is identically vanishing

In following an extension course on quantum field theory, a problem popped up that my TAs couldn't quite explain to my satisfaction. I suspect the answer is really simple, so I hope somebody with a ...
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How can a current be related to a functional derivative (in Ginsburg-Landau theory)

I am faced with this task in independent study where I am told that a vector in 3-dimensional space (current) is related to the functional derivative: $$ \mathbf{J} \propto \dfrac{\delta \mathcal{F}}...
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Partial differentiation of action function [closed]

I'm trying to understand the principle of least action, and the author of the book I'm reading is presenting of derivation of the Euler-Lagrange equation. The author picks an arbitrary position $x_8$ ...
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50 views

Lagrange multiplier associated with the requirement of constant particle number

I am following Jones and Gunnarsson (1989). In their paper, readers find the following equation that is often used in many-body quantum physics, in particular density functional theory: $$ \frac{\...
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Eigenfunctionals and their application in physics

Is there any sensible meaning of the term eigenfunctionals? The object I want to describe is a solution to the following equation $$ {\mathscr D}_x F[g] = f(x) F[g] $$ where $ {\mathscr D}_x$ is an ...
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1answer
60 views

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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238 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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123 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
330 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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45 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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62 views

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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302 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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186 views

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
227 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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177 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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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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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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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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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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142 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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Functional derivative of meta-generalized gradient approximation (meta-GGA)

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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176 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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231 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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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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65 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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221 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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321 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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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
170 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
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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
103 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
180 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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159 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 ...
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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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1answer
324 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 ...