# 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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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\frac{\delta S}{\delta \phi(x)}$ ...
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### 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 $$\mathcal{F} = F - \int dr \rho ^{(1)}(r)\phi (r)$$...
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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 $$S_0 = \frac{1}{2} \... • 778 1 vote 0 answers 53 views ### Derivation of Noether's theorem by Gateaux derivative Noether's theorem states that if:$$\ \int_{a}^{b} F(x, y, y') \,dx = \ \int_{a_{new}}^{b_{new}} F(x_{new}, y_{new}, y_{new}') \,dx_{new} $$for any a, b and y(x), and when x and x_{new} ... • 11 0 votes 0 answers 24 views ### Variations in a vector field [duplicate] When we derive Maxwell's equations from the Lagrangian that contains the Maxwell field tensor F_{ij}, I ran into a small confusion. With the Lagrangian being L = F^{ij}F_{ij}, taking variations of ... • 582 0 votes 3 answers 210 views ### How would one calculate the inverse of Dirac delta function? This is a question I've met recently while doing calculation. To make the point clear, let's just consider that a scalar field, say \phi(x), as well as a functional, say$$B(\phi(x))=f(x)\phi(x).$$... • 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 ... • 521 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{\... • 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 \frac{\delta f\big[x(t),\dot{x}(t)\big]}{\delta x^i(t')}=\... • 3,699 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 ... 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 ... 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? 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)}\}$$... • 333 3 votes 1 answer 84 views ### Deriving Schwinger-Dyson equation in the book Gauge/Gravity Duality: Foundations and Applications In the book Gauge/Gravity Duality: Foundations and Applications by M. Ammon and J. Erdmenger they derive the Schwinger-Dyson equation by considering the generating functional J[Z] and a change of ... • 33 7 votes 1 answer 336 views ### Does the "Euler-Lagrange operator" (\gamma,L) \mapsto (\partial_x L)\circ d\gamma- (\partial_v L\circ d\gamma)' have some geometric interpretation? Euler-Lagrange's equations for a Lagrangian L read$$\frac{d}{dt}\frac{\partial L}{\partial \bf v} = \frac{\partial L}{\partial \bf x} .More precisely, the statement is that a trajectory \gamma:\... • 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 ... • 8,987 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 \... • 1,782 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 ... • 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 ... • 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).$$I've ... 3 votes 3 answers 68 views ### Oscillatory motion As I was studying about simple harmonic motion(example pendulum), then I came up to a sin graph as well as a formula that is y = sin2πt/T. I then taken the example of pendulum to understand as to how ... • 123 2 votes 1 answer 115 views ### Confusion about dimensions of a functional, its functional derivative and its variation Let's take a functional F[\phi] as defined in this answer$$ F[\phi] = \int d^4x \, \phi\, \partial^2 \phi $$whose dimensions are, if the coordinates have dimensions of a length, as it's customary, ... • 2,197 4 votes 3 answers 615 views ### How to calculate functional derivative correctly? Let \phi be a real scalar field and J an arbitrary source function. Consider$$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(\...
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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)-...