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Questions tagged [functional-derivatives]

Generalization of the notion of derivative to functionals, i.e., to functions that take other functions as an argument. Functional derivatives are particularly useful, for example, in field theory.

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Using the functional derivative and delta function for proving The Fokker-Planck Equation

I am reading "Lectures on Phase Transitions" by Nigel Goldenfeld, specifically Chapter 8, where the Fokker-Planck equation is derived. I found the following part of the proof, but there are ...
M.Wael Youssef's user avatar
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A Limited Sense of Path Integral Respecting Classical EOM

In Weinberg QFT V1, we got the general path integral of time-ordered operators product, equation (9.1.38), $$\langle{q',t'|\text{T}\left\{\mathcal{O}_A\left(P(t_A),Q(t_A)\right)\mathcal{O}_B\left(P(...
Ting-Kai Hsu's user avatar
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Chain rule with functional derivative?

I posted the same question on math exchange but no answer yet, so I post it also here: "I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier ...
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Using functional derivatives and Euler-Lagrange to obtain wave equation in 3d elastic media

I'm trying to solve exercise (1.5) from Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur wherein we consider a 3D elastic material whose potential energy is given by $$ V = \frac{\...
Keshav Balwant Deoskar's user avatar
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Is there a practical distinction between functions of state and functionals in thermodynamics?

In thermodynamics, and more precisely when talking about continuous systems, some sources [1, 2] introduce functionals of state: $$F[s(x), \dots]:=\int_VdV(x)f(s(x),\dots,x)$$ In order to derive ...
GvPStack's user avatar
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Free scalar field deriving Ehrenfest using the path integral

In his lecture notes on String theory, David Tong derives Ehrenfest theorem using the path integral: $$S = \frac{1}{4\pi \alpha'}\int d^2\sigma\ \partial_\alpha X\ \partial^\alpha X\tag{4.19}$$ $$ 0 =...
Jens Wagemaker's user avatar
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Fermionic propagator [closed]

Given the fermionic generating functional $$Z[\eta]=\ det^{\frac{1}{2}}(K_{ij})e^{-\frac{i}{2}\eta_{i}G^{ij}\eta_{j}},\tag{1}$$ where $$G^{ij}=K^{-1}_{ij}$$ is the Green function of our theory, then ...
Michael 's user avatar
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Question about functional derivative computation in Quantum Field Theory for the Gifted Amateur

I'm confused about the evaluation of the functional derivative of Equation 1.12, $$J[f] = \int [f(y)]^p \phi(y) dy$$ on page 13 of Quantum Field Theory for the Gifted Amateur in Chapter 1. Here are ...
aadithyaa's user avatar
3 votes
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How to Relate the Functional Derivative to Infinitesimal Change in Noether's Theorem

When the Euler-Lagrange equation or the expression for Noether current are derived the term infinitesimal change is often used. For example, we write $\phi\rightarrow \phi + \delta\phi$ and say that $\...
ICOR's user avatar
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Energy momentum tensor of a scalar field by variation of metric

For the scalar field $\phi$, $$ L = \frac{1}{2}\left(\partial_\mu \phi \partial^\mu \phi + m^2 \phi^2\right) $$ The energy momentum tensor calculated using noether's theorem is given by $$ T^{\mu \nu} ...
Ratul Thakur's user avatar
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Evaluating functional derivatives

I am new to evaluating functional derivatives and I am having difficulty evaluating the following derivative: $$I = \frac{\delta}{\delta x(t)}\frac{\delta}{\delta x(t')}\int_{u_i}^{u_f}\frac{du}{2}\...
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Variation of a function

I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
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How do I get a derivative of the field inside of the path integral?

I am trying to find the 3-gluon vertex rule in QCD by finding the amplitude of a 1-2 gluon scattering process. I want to find the generating functional of the interaction by taking the functional ...
bradas128's user avatar
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Is It Functional Derivative in Transformation?

In section 2.4 of Conformal Field Theory by P.D.Francesco, and others, it has discussed the infinitesimal transformation on spacetime and field, $$x'^{\mu} = x^{\mu} + \omega_{a}\frac{\delta x^{\mu}}{\...
Ting-Kai Hsu's user avatar
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Vibration of a continuous uniform chain and the normal modes

The question is: A vertically hanged chain with the upper end attached to a fixed point. I try to find the normal modes under the small $\theta$ condition. Consider the mass $\mathrm{d}m$ with ...
Polarrr301's user avatar
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Do functional derivatives commute in a specific example?

Edit: this question is related to other already asked questions, like Symmetry of second functional derivatives , but a clear and definitive answer has never been given, and I am here giving a ...
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Is there physical significance to the Gateaux derivative of entropy?

Is there any physical "force" generated by a difference in entropy or information between two regions? There is some confusion between thermodynamic entropy and information entropy, but one ...
Jackson Walters's user avatar
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Functional derivative of a Green function

I'm trying to prove that, given the Hamiltonian $\hat{H} + \int d\mathbf{x} \hat{n}(\mathbf{x})\varphi(\mathbf{x}, t)$, where $\varphi(\mathbf{x}, t)$ is some external field and $\hat{S}$ the ...
Gyro's user avatar
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Evaluation of functional derivative of effective action

I'm trying to understand a calculation in appendix A of this paper https://arxiv.org/abs/2204.04197, however I don't understand how they end up with equation (125) and I think I am going wrong in the ...
furious.neutrino's user avatar
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1 answer
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Calculating Poisson brackets in classical non-relativistic Hamiltonian field theory

Summary of the question: How can I prove the equal-time Poisson bracket relations for the classical Hamiltonian field theory? I.e $$[q(x,t),H(t)]_\mathrm{PB}=\dot{q}(x,t)\tag{1}$$ for a field $q$ and ...
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How to evaluate the action of a fractional differential momentum operator?

I need to understand how a fractional operator works if before being applied on a test function it acts on another (well known)function: $$(-i\hbar\frac{\partial}{\partial x}\cdot\delta(x))^\epsilon\...
Cuntista's user avatar
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Something wrong in Saint Venant–Kirchhoff model?

According to the Saint Venant–Kirchhoff model, the strain-energy density function is defined as $$ W(\boldsymbol{E})=\frac{\lambda}{2}[\rm tr(\boldsymbol{E})]^2+\mu\rm tr(\boldsymbol{E}^2) $$ $\...
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For any unitary $U = e^{iA}$, question about the matrix element $A_{pq}$ of matrix $A$ and its functional derivative [closed]

Any unitary operation $U \in U(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (...
Jon Megan's user avatar
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Some questions about the derivation process of many-body Green's functions (specifically, the Hedin equations)

The response of the Green's function under the action of an external field can be represented by the density correlation function L: $$L_{s_1 s'_{1} , s_2 s_{2'} }(12) = - i \hbar \frac{ \delta G_{s_1 ...
Liang's user avatar
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Why partial derivative in Lagrange is partial?

When trying to arrive to Euler-Lagrange equation, Susskind does terrific job but I have one problem. Let's consider motion in only $x$ direction with respect to time. $$x(t) = \hat x(t) + \epsilon f(t)...
Giorgi's user avatar
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1 answer
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Functional derivative of the generating functional with respect to the source term

To be specific, let us use the notations in W. Metzner et al., Rev. Mod. Phys. 84,299 (2012). The generating functional $G[\eta, \bar\eta]$ is given by [Eq. 4] \begin{align} G[\eta, \bar\eta] = - \ln \...
a-little-bit-science's user avatar
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Functional derivative of Ricci tensor respect to metric [closed]

I am working on following functional derivative $$ \dfrac{\delta R_{\mu\nu}}{\delta g_{\alpha\beta}}=C^{\alpha\beta}_{\mu\nu} $$ Intuitively, it should be nonzero but I can not work it out.
Joseph Li's user avatar
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119 views

Propagator from free scalar path integral

Let $\langle{0|}T\phi(x)\phi(y)|0\rangle$ be the vacuum expectation value of the 2 point correlator for the free scalar field. Page six in these notes say that we can calculate this correlator by ...
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1 vote
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Dimensions of functional integration [duplicate]

I'm confused by something super simple. When taking functional variation (e.g. of the action) in the context of field theory, I often see $$ {\delta \phi(x) \over \delta \phi(y)} = \delta(x-y) \ .$$ ...
Rudyard's user avatar
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Semi-classical limit of Feynman path integral

I am reading Blau's note on The Path Integral Approach to Quantum Mechanics. I am troubled for the derivations of semi-classical limit of Feynman path integral, which is located on Page.50 of Blau's ...
Daren's user avatar
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4 votes
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How is the source term chosen when using path integrals?

Suppose I would like to compute (time ordered) vacuum expectation values for a quantum field theory by using the path integral approach. Using the Lagrangian for the theory, we define a generating ...
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Functional derivative and entropy

I just started to read about (entanglement) entropy for CFTs. I have a confusion that probably originates from a misunderstanding about functional derivatives. For instance, in (4.21) in this review ...
Nikolo J Bar's user avatar
3 votes
1 answer
78 views

Functional derivative for boson CFT on torus [closed]

Let us consider a bosonic CFT on torus: \begin{eqnarray} S=\int dzd\bar{z}\frac{1}{2}\partial X\bar{\partial} X.\tag{2.1.10} \end{eqnarray} From Page 35 of Polchinski Vol. 1, I do the same function ...
Yuan Yao's user avatar
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Why does a complete four-divergence term in a Lagrangian density not affect the equation of motion in special relativity? [duplicate]

The classical theory of fields by Landau and Lifshitz, page 68 says: As for the quantity $\epsilon^{iklm}F_{ik}F_{lm}$ (§ 25), as pointed out in the footnote on p. 63, it is a complete four- ...
procrastinator37's user avatar
1 vote
1 answer
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Minimizing the potential energy in a hyperelasticity problem

I am currently using the FEniCS/DOLFINx package to simulate deformations on a mesh volume. Following this tutorial, I am using the following equation to find $u$ such as $L(u)=0$: $$L = \vec{\nabla} \...
ofares's user avatar
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Does this particular notation for derivatives imply anything in particular? [duplicate]

In some physics textbooks (and in those of other sciences that use physics, like soil science), I've seen some derivatives written as: $$\frac{\delta f}{\delta t} $$ Which is a bit strange. Does this ...
agaminon's user avatar
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6 votes
3 answers
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Fourier transform of a functional derivative

Suppose that $x(q)$ is the Fourier transform of the function $x(r)$, where $r$ is the real-space variable and $q$ is the Fourier-space variable. Then, suppose that $E$ is an energy functional which ...
ShinyPebble's user avatar
3 votes
2 answers
621 views

Why is it possible to neglect higher order terms in the variation of the action?

In order to get the Euler-Lagrange equations, we should find the variation of the action $\delta S$ and to neglect higher-order terms: $$\delta S=\int L(q+\delta q,\,q'+\delta q',\,t)dt-\int L(q ,\,q',...
EB97's user avatar
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Mixed second functional derivative not symmetrical with respect to order of differentiation

I have a functional $E$, which is a functional of two different functions and their gradients: $$ E[\psi_1,\psi_2] = \int d^3\mathbf{x} ~ \varepsilon\{\mathbf{x}\}$$ where I'm using $\varepsilon\{\...
Jospeh's user avatar
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Functional derivative of free energy

I found a paper regarding the statistical field theory of electrolytes. In that paper, free energy functional is defined as $F = F[\phi_a, \phi_c, \psi_e(\phi_a, \phi_c), \psi_a(\phi_a, \phi_c), \...
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3 votes
1 answer
231 views

Measure of Functional Integral in Path Integral Formulation

I have a question regarding the prefactor $\sqrt{\left(\frac{m}{2\pi i \hbar \Delta t}\right)}$ in $$\left<x'|e^{-iHt}|x\right> = \int D[x] \exp(\frac{i}{\hbar}\int dt' L(x, \dot{x})),$$ where $$...
Neophyte's user avatar
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1 answer
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What's the difference between the conjugate momenta in the classical mechanics and in field theory?

In the classical mechanics the conjugate momenta was typically a derivative of the Lagrangian, i.e. \begin{equation} p_i=\frac{\partial L}{\partial \dot q_i}.\tag{1} \end{equation} However, in the QFT ...
ShoutOutAndCalculate's user avatar
3 votes
1 answer
269 views

Functional derivatives: Fréchet, Gateaux derivatives and Euler-Lagrange operators

What is the relationship between Fréchet derivatives, Gateaux derivatives and the usual Euler-Lagrange operator (ELO)? Is the ELO the Fréchet derivative, the Gateaux derivative or does it depends on ...
riemannium's user avatar
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Functional Derivative Calculation

Given the functional: $$ F[\phi] = \int_V \frac{k_B T}{a^3}\phi\ln(\phi) \ ds = \int_V I(\phi) ds $$ I want to find the functional derivative. I believe this would result in: $$ \frac{\delta F}{\delta ...
Mjoseph's user avatar
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In functional derivative the starting point confusion

how can one define the functional derivative $$\delta F= F[f+\delta f]-F[f].$$ Is it by definition or any physical reason holds for it.
Salam's user avatar
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2 answers
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Evaluating conjugate momentum from a given Lagrangian density

I have the following Lagrangian density $\mathcal{L}$ where $$ \mathcal{L}=\frac{1}{2}\left(c[\partial_{t}\phi(x,t)]^{2}-\frac{1}{l}[\partial_{x}\phi(x,t)]^{2}+\frac{1}{\omega_{J}^{2}l}[\partial_{x}\...
kowalski's user avatar
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1 answer
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Is there such a thing as infinitesimal electric field?

I am interested in calculating some response properties, namely, susceptibility and polarizability. In principle, susceptibility should be the functional derivative of the electron density to a ...
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215 views

Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} (\dot{u}(x,t))^2 - ca(u'(x,t)^2\tag{1}$$ with respect to $u$, ($\delta L / \delta u$). The ...
M. Uon's user avatar
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3 votes
1 answer
368 views

What does it exactly mean by right and left functional derivatives?

In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions. I do not quite see what it means by that. For example, let us ...
Keith's user avatar
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General relativity algebraic manipulation help

I'm having difficulty understanding a lot of the fundamentals behind the algebra of general relativity. I have a specific question I'm trying to understand but any pointers about how any of it works ...
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