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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
29 views

Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} \dot{u}(x,t) - ca(u')^2$$ with respect to $u$, ($\delta L / \delta u$). Following my lecture ...
user avatar
  • 1
2 votes
1 answer
50 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 ...
user avatar
  • 1,069
1 vote
1 answer
63 views

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 ...
user avatar
8 votes
0 answers
154 views

Derivation of the Hypersurface Deformation Algebra

Let $({M},{g})$ be a smooth $4d$ spacetime manifold with lorentzian metric $g$ and local coordinates $\xi^{\alpha}$ and let further $({N},{q})$ be a smooth $3d$ manifold with metric $q$ and local ...
user avatar
1 vote
0 answers
48 views

Functional derivative acts on covariant derivative

I'm confusing about how functional derivatives act on a covariant derivative. I'm doing such a calculation: In ADM formalism, let $h_{ij}(x)$ be the spatial metric while $\pi^{ij}(x)$ is its momentum ...
user avatar
  • 37
0 votes
1 answer
65 views

Calculating some functional derivative

I am reading Mark Srednicki's quantum field theory, p.50~p.52 (Part I section 7). In the section, he derives a the formula for the ground state to ground state transition amplitude of harmonic ...
user avatar
9 votes
1 answer
297 views

Computing functional derivative of exchange-correlation functional

Sakurai and Napolitano's chapter on density functional theory has claims that it is "straightforward" to find $\delta U_{\text{xc}}/\delta n$ for $$U_{\text{xc}}[n]=\int d^3 x n(\mathbf{x})\...
user avatar
1 vote
1 answer
71 views

Calculating the functional derivative of $\partial_\mu\phi$ with respect to $\phi$

Given $F_\mu=\partial_\mu\phi$, I need to find the functional derivative $\frac{\delta F}{\delta \phi}$. I am not familiar with the treatment of functional derivatives outside the context of finding ...
user avatar
2 votes
0 answers
64 views

Second functional derivative and its units

Say I have a functional $I[\phi,g]$ with $\phi(p)$ and $g(p)$ functions from $\mathbb{R} \to \mathbb{R}$. Also say that this functional obeys the property: $$\frac{\delta I}{\delta g(p)} = -(g(p))^{-1}...
user avatar
  • 691
1 vote
2 answers
59 views

Understanding a difference between a functional derivative and discrete case

I can take the following functional derivative $$ C(p)=\frac{\delta}{\delta \phi(p')} \frac{\delta}{\delta \phi(-p')} \int_{-\infty}^{\infty} dp \phi(p)\phi(-p) = 2\delta(0). $$ where I am left with ...
user avatar
  • 691
1 vote
1 answer
44 views

Disappearing symmetry in gaussian functional determinant

I have the following integral $$I=\int D\varphi \; e^{-\int d^4p d^4p' \left[ -\frac{1}{2}\varphi(p) g(p) \delta(p+p') \varphi(p') \right]}.\tag{1}$$ This is the continuum limit of a gaussian matrix ...
user avatar
  • 691
0 votes
0 answers
42 views

Functional derivative for multivariable

I learned that for a function $f(x)$, the functional derivative of a functional $F[f] $ is defined as $$\frac{\delta F}{\delta f(x)}=\lim_{\epsilon \rightarrow 0} \frac{F[f(x')+\epsilon\delta(x'-x)]-F[...
user avatar
  • 3,803
0 votes
1 answer
38 views

Rewrite generating functional for scalar field case

I am reading Peskin & Schroeder and I cannot figure out the step between eq. (9.36) and (9.37). They are rewriting the term that appears in the generating functional: $$\int d^4x \left[\mathcal{L}...
user avatar
1 vote
1 answer
52 views

Functional derivative for the action $S$

From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15: Example 1.3 The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
user avatar
  • 3,803
4 votes
1 answer
112 views

Functional derivative for $J[f]=\int [f(y)]^p \phi(y)dy$

In QFT for gifted amateur pg. 13, the functional derivative for the functional $$J[f]=\int [f(y)]^p \phi(y)dy$$ is given by $$\frac{\delta J[f]}{\delta f(x)}= \lim_{\epsilon\rightarrow0} \frac{1}{\...
user avatar
  • 3,803
0 votes
1 answer
28 views

Minimum information required to measure your local physical environment

In Andy Weir's "Project Hail Mary" protagonist Ryland Grace wakes up in an environment and with a few physics experiments timing falling objects he relatively quickly determines that he is ...
user avatar
0 votes
0 answers
48 views

Feynman rules for non-local theory

For model with interaction: \begin{equation} H_{int} = \int f(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\boldsymbol{x}_{3})\hat{\varphi}_{S}(\boldsymbol{x}_{1})\hat{\varphi}_{S}(\boldsymbol{x}_{2})\...
user avatar
2 votes
1 answer
120 views

Functional Calculus in QFT

Does anybody know some good sources with detailed derivations of the main results we need to compute generating functionals in QFT (and functional calculus used in the subject in general). I find that ...
4 votes
2 answers
94 views

A question about commutation relation and functional derivatives

In wikipedia https://en.wikipedia.org/wiki/Canonical_commutation_relation. In quantum mechanics the Hamiltonian ${\hat {H}}$, (generalized) coordinate $ {\hat {Q}}$ and (generalized) momentum ${\hat {...
user avatar
  • 43
2 votes
1 answer
113 views

How to come up with Feynman rules: Proof of the multiplicity factor from functional derivative?

Consider $(\phi^*\phi)^2$ theory of complex scalar field. The goal is to come up with Feynman rules from functional derivatives, and the emphasis is on how does the symmetry factors or the ...
user avatar
  • 164
1 vote
1 answer
97 views

Functional derivatives on position and momentum spaces

I'll first give some context for the problem I'm having, but the essence of it seems to be related to only what is in the title. I've been working with the Wetterich equation for the Functional ...
user avatar
0 votes
1 answer
58 views

Variational derivative of $\Phi_a(-\partial^2 - m_0^2 - \Sigma)\Phi_a$

Let me refer to the below link http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-17-phi4on.pdf In Eq: 18.40, $\Gamma[\Phi_a, \Sigma]$ is given as, $\Gamma[\Phi_a,\Sigma] = NA_{coll}[\Sigma]...
user avatar
  • 567
1 vote
1 answer
102 views

Equation 13.20 of Peskin & Schroeder

I don't quite understand some skipped steps in the book An Introduction to Quantum Field Theory, by Peskin & Schroeder. Here is an example, I don't know why Taylor series would lead to this. In ...
user avatar
  • 37
3 votes
2 answers
92 views

Is $T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g_{\mu\nu}}$ a true tensor or a density?

The energy-momentum tensor is defined by $$T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g_{\mu\nu}}$$ where $S_m$ is the matter action $$S_m =\int d^4x\sqrt{-g}\mathcal{L}_m$$ and $\mathcal{...
user avatar
  • 157
1 vote
0 answers
77 views

Furutsu-Novikov Formula Generalisation

The Furutsu-Novikov formula gives the expectation value of a zero mean Gaussian process $z(t)$ and a functional of that process $R[z]$: $$\langle{z(t') R[z]}\rangle = \int^{t}_0 \mathrm{d}s K_2(t',s) \...
user avatar
  • 395
2 votes
1 answer
66 views

Confusion about the equations of motion for a non-local action

Given an action $$S = \int d^4x[\phi^2(x) \exp(\int d^4y F(x-y)\phi^2(y))]$$ it is straightforward enough to get the classical equations of motion, simply computing $\frac{\delta S}{\delta \phi(x)}$ ...
user avatar
  • 108
0 votes
0 answers
64 views

Functional derivatives in density functional theory

I am studying density functional theory and I am currently dealing with manipulating the intrinsic free energy, $\mathcal{F}$, which is defined as $$\mathcal{F} = F - \int dr \rho ^{(1)}(r)\phi (r) $$...
user avatar
  • 557
0 votes
1 answer
77 views

Functional and total variations in einbein action [duplicate]

I'm currently studying String theory by Becker& Becker, Schwarz textbook. The exercise 2.3 consists in verifying diffeomorphism invariance of einbein action wich is given by $$ S_0 = \frac{1}{2} \...
user avatar
1 vote
0 answers
53 views

Derivation of Noether's theorem by Gateaux derivative

Noether's theorem states that if: $$\ \int_{a}^{b} F(x, y, y') \,dx = \ \int_{a_{new}}^{b_{new}} F(x_{new}, y_{new}, y_{new}') \,dx_{new} $$ for any $a$, $b$ and $y(x)$, and when $x$ and $x_{new}$ ...
user avatar
  • 11
0 votes
0 answers
24 views

Variations in a vector field [duplicate]

When we derive Maxwell's equations from the Lagrangian that contains the Maxwell field tensor $F_{ij}$, I ran into a small confusion. With the Lagrangian being $L = F^{ij}F_{ij}$, taking variations of ...
user avatar
  • 582
0 votes
3 answers
210 views

How would one calculate the inverse of Dirac delta function?

This is a question I've met recently while doing calculation. To make the point clear, let's just consider that a scalar field, say $\phi(x)$, as well as a functional, say $$B(\phi(x))=f(x)\phi(x).$$ ...
user avatar
  • 37
1 vote
1 answer
56 views

Functional derivatives in elastic theories?

Short version: Does anyone know of good references for using functional derivatives to obtain equilibrium (or dynamic) equations in elastic theories? More background: I've frequently encountered ...
1 vote
2 answers
169 views

Functional derivative equalling the Dirac delta function QFT

I'm currently reading the book Quantum Field Theory and The Standard Model, and on the section on path integrals it talks about the variational partial derivative of the generating functional. It ...
user avatar
4 votes
1 answer
382 views

Definition of the Lagrangian in (classical) field theory

I am currently reading through Weinberg's Lectures on Quantum Mechanics. Chapter 11 deals with field theory: Correspondingly, the Lagrangian $L(t)$ is a functional of $\psi_n(\vec{x}, t)$ and $\dot{\...
user avatar
  • 383
0 votes
0 answers
51 views

Functional derivatives of tensors in phase space in the context of General Relativity

In classical mechanics the functional derivative of a scalar function $f(x,\dot{x})$ respect to the trajectory $x^i(t)$ is \begin{equation} \frac{\delta f\big[x(t),\dot{x}(t)\big]}{\delta x^i(t')}=\...
user avatar
1 vote
1 answer
61 views

What is $\epsilon$ in the $\delta$ smooth action functional of the Lagrangian?

At the beginning of the Lagrangian Mechanics Wikipedia page, it gives a $\delta$ function on the stationary point of the action $\cal S$: Given the time instants $t_1$ and $t_2,$ Lagrangian mechanics ...
user avatar
3 votes
1 answer
152 views

Noether second theorem calculus of variation formalization

Noether's first theorem as a statement about the action of a finite dimensional lie group $ G_{\rho} $ dependent upon $\rho $ parameters $ \varepsilon_{i}$ with $i = 1, \dots, \rho $; can be ...
user avatar
0 votes
2 answers
94 views

Functional derivative or Euler-Lagrange? [closed]

What is the difference between the functional derivative and the Euler-Lagrange equation?
user avatar
0 votes
1 answer
157 views

Taking functional derivatives of generating functional

I'm learning how to compute functional derivatives of generating funtionals. Suppose I have the following generating functional: $$Z[J] = \exp\{\int{dy_1 \; dz_1\; J(y_1) \Delta(y_1 - z_1) J(z_1)}\}$$ ...
user avatar
3 votes
1 answer
84 views

Deriving Schwinger-Dyson equation in the book Gauge/Gravity Duality: Foundations and Applications

In the book Gauge/Gravity Duality: Foundations and Applications by M. Ammon and J. Erdmenger they derive the Schwinger-Dyson equation by considering the generating functional $J[Z]$ and a change of ...
user avatar
  • 33
7 votes
1 answer
336 views

Does the "Euler-Lagrange operator" $(\gamma,L) \mapsto (\partial_x L)\circ d\gamma- (\partial_v L\circ d\gamma)'$ have some geometric interpretation?

Euler-Lagrange's equations for a Lagrangian $L$ read $$\frac{d}{dt}\frac{\partial L}{\partial \bf v} = \frac{\partial L}{\partial \bf x} .$$ More precisely, the statement is that a trajectory $\gamma:\...
user avatar
  • 12.2k
2 votes
0 answers
80 views

Symmetry of second functional derivatives

Note: I am asking this here over math.se or mathoverflow because functional derivatives are very much a physics-only thing. It seems to be a common assumption that second functional derivatives ...
user avatar
1 vote
0 answers
26 views

Functional Hessian of the Palatini action

Let's write the Palatini action like the following: \begin{align*} S_P&=\int \epsilon^{\alpha \beta \gamma \delta}\epsilon_{IJKL}e^I_\alpha e^J_\beta\Omega^{KL}_{\gamma \delta} \\ &=\int \...
user avatar
3 votes
1 answer
145 views

Extension of Faddeev-Jackiw first-order Lagrangian formalism to fields

In this paper, Toms discusses the method that Faddeev and Jackiw proposed for quantization of constrained theories. In section III.B, he applies this method to a field theory, but I have several ...
user avatar
  • 1,954
3 votes
1 answer
189 views

Definition of the stress-energy tensor in terms of functional derivatives in G.R

I have found confusing definitions in various places regarding the stress-energy tensor, in particular when used to derive Einstein GR equations from the principle of stationary action. Some of these ...
user avatar
  • 89
2 votes
2 answers
126 views

How to take 'non-local' functional derivatives?

I am currently in the process of getting into linear response theory in general, and I have often met functional derivatives of the type $$\frac{\partial J[f(x)]}{\partial f(y)} = \chi(x,y).$$ I've ...
user avatar
3 votes
3 answers
68 views

Oscillatory motion

As I was studying about simple harmonic motion(example pendulum), then I came up to a sin graph as well as a formula that is y = sin2πt/T. I then taken the example of pendulum to understand as to how ...
user avatar
  • 123
2 votes
1 answer
115 views

Confusion about dimensions of a functional, its functional derivative and its variation

Let's take a functional $F[\phi]$ as defined in this answer $$ F[\phi] = \int d^4x \, \phi\, \partial^2 \phi $$ whose dimensions are, if the coordinates have dimensions of a length, as it's customary, ...
user avatar
4 votes
3 answers
615 views

How to calculate functional derivative correctly?

Let $\phi$ be a real scalar field and $J$ an arbitrary source function. Consider $$S_{E}[\phi, J]=\int d^{4} x\left[\frac{1}{2}(\partial_{\mu} \phi)(\partial^{\mu}\phi)+\frac{1}{2} m^{2} \phi^{2}+V(\...
user avatar
  • 1,079
2 votes
1 answer
81 views

Derivation operators as arguments or the Hamiltonian

In a book I am reading about QFT (Quantum field theory by Mark Srednicki ,page 48), I see the following equation: $$ \int \mathcal D p\mathcal D q \exp\left[i\int_{\mathbb R} dt (p\dot q - H_0(p,q)-...
user avatar
  • 1,175