Questions tagged [functional-derivatives]

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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^\...
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43 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", ...
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109 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 $$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$ ...
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197 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)...
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865 views

How to calculate the second functional derivative of the action of a one-particle system?

Given the Lagrangian $$L(q,\dot{q})=m\dot{q}^2/2-V(q)$$ and the corresponding action $$S[q]\equiv\int_0^t dt' (m\dot{q}^2/2-V(q)),$$ I need to be able to evaluate the second functional derivative $\...
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155 views

Commutativity Variation and derivative

When in general is true that $\nabla_{\mu}\delta=\delta\nabla_{\mu}$ where $\nabla_{\mu}$ is covariant derivative? I was thinking this when one has to find the equations of motion for example in ...
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142 views

Variation in Entropy in Einstein's Brownian Motion Paper

In Einstein's Brownian motion paper, he derives a formula for the diffusion coefficient of suspended particles by assuming the system is in dynamic equilibrium and thus, for a variation $\delta x$ (...
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2answers
67 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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1answer
123 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 ...
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1answer
85 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' ...
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40 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(...
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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 (\...
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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 ...
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132 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{...
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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 ...
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2answers
222 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 $$ \delta S = \int d^4x\, \delta \mathcal{L}. \tag{2} $$ ...
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92 views

Second-order functional dervative of the Yang-Mills action by DeWitt

DeWitt calculated a second order functional dervative of a Yang-Mills action in Table I p. 1201 in his paper PR 162 (1967) 1195. The action is the usual one, $$S=-(1/4)\int d^4x F_{a\mu\nu}F^{a\mu\nu}...
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85 views

Eigenfrequencies of a truss

I want to calculate the eigenfrequencies of a 3D truss using the finite element method. The beams should be modelled as Timoshenko beams and I want to use arbitrary shape functions. I know how to use ...
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51 views

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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32 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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214 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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161 views

Functional Differetiation of a complex functional

Suppose I have a simple functional $$F=\int{dx\;\phi^{*}(x)\phi(x)}\tag{1}.$$ Assuming $\phi(x)$ and $\phi^{*}(x)$ are independent and I take a functional differential with respect to $\phi(x)$ and $\...
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187 views

Doubts taking the second functional derivative of the Klein Gordon action

I have very little background with functional derivatives and I would like to clarify some issues. I am trying to compute the second functional derivative of the Klein Gordon action expressed in real ...