The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [functional-derivatives]

The tag has no usage guidance.

107 questions
Filter by
Sorted by
Tagged with
50 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 $S[\phi]$ ...
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 ...
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?
54 views

61 views

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", ...
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....
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 ...
79 views

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 ...
52 views

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 $f$ ...
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 $... 2answers 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 $$\tau_2(x,y) = -\frac{\delta^2}{\delta\eta_x \delta \... 2answers 161 views Functional derivative for the same function expressed before and after Wick rotation This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf On page 17, they ... 1answer 125 views Functional derivative I am not able to derive Eq. 21 of this paper F. Zahariev, S. S. Leang, and Mark S. Gordon, "Functional derivatives of meta-generalized gradient approximation (meta-GGA) type exchange-correlation ... 1answer 86 views How to take into account the symmetry of the metric tensor when doing the Functional derivative in GR? [duplicate] I have a Straightforward question. When the functional derivative of the Ricci scalar to get the GR field equations. As the derivative is done using the metric which is symmetric do I have to ... 2answers 202 views Why does Fermat's principle (optics) not apply to all paths? Feynman's statement of Fermat's Principle regarding optics is the following, "a ray going in a certain particular path has the property that if we make a small change (say a one percent shift) in ... 1answer 91 views What does lowercase-delta mean in Noether's first theorem? Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ... 1answer 57 views Variational principle, functional gradient Given the energy functional$$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}... 1answer 185 views Notation question in calculus of variations — QFT these two integrals below are equal, but I am not understanding where the$x'variable comes from. \begin{align} I_0&=e^{ i\int d^4x \left\{ \frac{1}{2}\left[ \left( \partial\varphi(x) \right)^... 0answers 246 views Issue with calculating free fermionic propagator from partition function\newcommand{\D}{\mathcal{D}}$In section 14.6 of Schwartz's "QFT and the Standard Model (7$\,^{\text{th}}$printing)" , the author calculates the exact free fermionic partition function and, from ... 2answers 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 ... 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 ... 0answers 24 views 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, ... 0answers 41 views Functional derivative of a symmetrized field I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(... 1answer 83 views Variation of Fermionic Field Operator Suppose we have a Hamiltonian containing some interaction term$$V = \sum _{\sigma \sigma '\sigma ''\sigma '''}\iint d^3rd^3r'\hat{\psi }_{\sigma}^\dagger (\textbf{r})\hat{\psi }_{\sigma'}^\dagger (\... 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}.... $In appendix A of this paper https://arxiv.org/abs/hep-th/9908172 it is said that the basic ... 1answer 139 views Second variation of a functional I am trying to find the second variation of the Hartree energy functional$E_{H} [\rho]$: $$\dfrac {\delta^2 E_{H}}{\delta \rho (r)\delta \rho (r')}=\dfrac {\delta^2}{\delta \rho (r)\delta \rho (r')}\... 2answers 133 views How do I find the function derivative (\delta/\delta \phi) (\partial_\mu \phi)? The question is simple: How do I find the function derivative of$$(\delta/\delta \phi(x)) (\partial_\mu \phi(x))~?$$As far as I can tell, I cannot use any of the standard computational rules for the ... 0answers 135 views Confusion taking functional derivatives of the generating functional (Coleman) I'm reading Piers Coleman's Introduction to Many Body Physics and I've run into a strange issue. In section 8.4, he considers a free theory in the presence of a source [\bar\eta, \eta]: \begin{... 1answer 255 views Free propagator from generating functional in momentum space I am reading the text by Daniel Amit, "Field Theory, the Renormalization Group, and Critical Phenomena" and trying to apply what is explained in chapters 3 and 4 to a real-life problem. However, I ... 1answer 64 views Problem calculating the variation of this action If we take a Hamiltonian density to be as following$$ \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, $$and we have ... 3answers 174 views Why are there two definitions for the functional derivative? I have seen two definitions for the functional derivative. Why are there two definitions? In Goldstein's Classical mechanics 3rd edition page 574 eq. (13.63), and also in a Student's Guide to ... 1answer 51 views Existence of Functional with some Functional Derivative [closed] How could I do this? I tried to manipulate, but I don't know how I could get to a contradiction or something else. Show that there is no functional S = S[\phi] that satisfies: (where \epsilon_{... 1answer 213 views Where are the delta functions in Peskin & Schroeder eq. (11.67)? In the linear sigma model, the Lagrangian is given by \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) +\frac{1}{2}\mu^2\sum_{i=1}^{N}\left(\... 2answers 313 views Derivatives in Euler-Lagrange for fields Starting with the lagrangian density$$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 + \... 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: $$\Gamma[\varphi] := \inf_j \ F[\varphi, j] = F[\varphi, j_\varphi] \... 3answers 274 views Help with taking derivative of Lagrangian scalar model of graviton Quick question. Given Lagrangian density$$\mathcal{L} = -\frac12 h \Box h + \frac13 \lambda h^3 + Jh ,\tag{3.69}$$where the scalar h represents the gravitational potential, and given the Euler-... 0answers 111 views Under what circumstances is the functional derivative (of an action functional) an actual function? In general, for a functional F[\phi], the functional derivative is$$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$... 3answers 444 views Correct derivation of Einstein's equations from the Hilbert action I have been trying to understand general relativity from a first-principles perspective in my spare time, and I have been unable to find a convincing derivation of the Einstein equations. The most ... 2answers 261 views Variation of a field action The field action in flat spacetime is$$ 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} $$... 0answers 207 views Functional derivative of a convolution I am wondering how to compute the functional derivative of an ordinary scalar function with itself, say$$[f\star f](x)=\int dy~f(y)f(x-y).$$My attempt would give$$\delta_{f}[f\star f](x)=2f(x-y)... 0answers 91 views 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^\...
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 ...