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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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:
$$(-i\hbar\frac{\partial}{\partial x}\cdot\delta(x))^\epsilon\...
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Something wrong in Saint Venant–Kirchhoff model?
According to the Saint Venant–Kirchhoff model, the strain-energy density function is defined as
$$
W(\boldsymbol{E})=\frac{\lambda}{2}[\rm tr(\boldsymbol{E})]^2+\mu\rm tr(\boldsymbol{E}^2)
$$
$\...
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For any unitary $U = e^{iA}$, question about the matrix element $A_{pq}$ of matrix $A$ and its functional derivative [closed]
Any unitary operation $U \in U(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (...
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Some questions about the derivation process of many-body Green's functions (specifically, the Hedin equations)
The response of the Green's function under the action of an external field can be represented by the density correlation function L:
$$L_{s_1 s'_{1} , s_2 s_{2'} }(12) = - i \hbar \frac{ \delta G_{s_1 ...
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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.
$$x(t) = \hat x(t) + \epsilon f(t)...
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Functional derivative of the generating functional with respect to the source term
To be specific, let us use the notations in W. Metzner et al., Rev. Mod. Phys. 84,299 (2012).
The generating functional $G[\eta, \bar\eta]$ is given by [Eq. 4]
\begin{align}
G[\eta, \bar\eta] = - \ln \...
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Functional derivative of Ricci tensor respect to metric [closed]
I am working on following functional derivative
$$
\dfrac{\delta R_{\mu\nu}}{\delta g_{\alpha\beta}}=C^{\alpha\beta}_{\mu\nu}
$$
Intuitively, it should be nonzero but I can not work it out.
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Bell State Density Operators in Continuous Systems
My question is about how density operators work in continuous systems, specifically maximally entangled states. I've outlined the math below but the TLDR is:
Does it make sense to talk about the ...
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Propagator from free scalar path integral
Let $\langle{0|}T\phi(x)\phi(y)|0\rangle$ be the vacuum expectation value of the 2 point correlator for the free scalar field. Page six in these notes say that we can calculate this correlator by ...
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Dimensions of functional integration [duplicate]
I'm confused by something super simple. When taking functional variation (e.g. of the action) in the context of field theory, I often see
$$ {\delta \phi(x) \over \delta \phi(y)} = \delta(x-y) \ .$$
...
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Semi-classical limit of Feynman path integral
I am reading Blau's note on The Path Integral Approach to Quantum Mechanics. I am troubled for the derivations of semi-classical limit of Feynman path integral, which is located on Page.50 of Blau's ...
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How is the source term chosen when using path integrals?
Suppose I would like to compute (time ordered) vacuum expectation values for a quantum field theory by using the path integral approach. Using the Lagrangian for the theory, we define a generating ...
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Functional derivative and entropy
I just started to read about (entanglement) entropy for CFTs. I have a confusion that probably originates from a misunderstanding about functional derivatives.
For instance, in (4.21) in this review ...
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Functional derivative for boson CFT on torus [closed]
Let us consider a bosonic CFT on torus:
\begin{eqnarray}
S=\int dzd\bar{z}\frac{1}{2}\partial X\bar{\partial} X.\tag{2.1.10}
\end{eqnarray}
From Page 35 of Polchinski Vol. 1, I do the same function ...
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Why does a complete four-divergence term in a Lagrangian density not affect the equation of motion in special relativity? [duplicate]
The classical theory of fields by Landau and Lifshitz, page 68 says:
As for the quantity $\epsilon^{iklm}F_{ik}F_{lm}$ (§ 25), as pointed out in the footnote on p. 63, it is a complete four- ...
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Minimizing the potential energy in a hyperelasticity problem
I am currently using the FEniCS/DOLFINx package to simulate deformations on a mesh volume.
Following this tutorial, I am using the following equation to find $u$ such as $L(u)=0$:
$$L = \vec{\nabla} \...
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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 ...
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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 ...
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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:
$$\delta S=\int L(q+\delta q,\,q'+\delta q',\,t)dt-\int L(q ,\,q',...
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Mixed second functional derivative not symmetrical with respect to order of differentiation
I have a functional $E$, which is a functional of two different functions and their gradients:
$$ E[\psi_1,\psi_2] = \int d^3\mathbf{x} ~ \varepsilon\{\mathbf{x}\}$$
where I'm using $\varepsilon\{\...
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Functional derivative of free energy
I found a paper regarding the statistical field theory of electrolytes. In that paper, free energy functional is defined as
$F = F[\phi_a, \phi_c, \psi_e(\phi_a, \phi_c), \psi_a(\phi_a, \phi_c), \...
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Measure of Functional Integral in Path Integral Formulation
I have a question regarding the prefactor $\sqrt{\left(\frac{m}{2\pi i \hbar \Delta t}\right)}$ in $$\left<x'|e^{-iHt}|x\right> = \int D[x] \exp(\frac{i}{\hbar}\int dt' L(x, \dot{x})),$$ where $$...
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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 ...
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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 ...
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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:
$$ \frac{\delta F}{\delta ...
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In functional derivative the starting point confusion
how can one define the functional derivative $$\delta F= F[f+\delta f]-F[f].$$
Is it by definition or any physical reason holds for it.
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Evaluating conjugate momentum from a given Lagrangian density
I have the following Lagrangian density $\mathcal{L}$ where
$$
\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}\...
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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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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 ...
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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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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 $$U_{\text{xc}}[n]=\int d^3 x n(\mathbf{x})\...
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Calculating the functional derivative of $\partial_\mu\phi$ with respect to $\phi$
Given $F_\mu=\partial_\mu\phi$, I need to find the functional derivative $\frac{\delta F}{\delta \phi}$. I am not familiar with the treatment of functional derivatives outside the context of finding ...
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Second functional derivative and its units
Say I have a functional $I[\phi,g]$ with $\phi(p)$ and $g(p)$ functions from $\mathbb{R} \to \mathbb{R}$. Also say that this functional obeys the property:
$$\frac{\delta I}{\delta g(p)} = -(g(p))^{-1}...
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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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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:
$$\int d^4x \left[\mathcal{L}...
user255856
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Functional derivative for the action $S$
From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15:
Example 1.3
The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
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Functional derivative for $J[f]=\int [f(y)]^p \phi(y)dy$
In QFT for gifted amateur pg. 13, the functional derivative for the functional
$$J[f]=\int [f(y)]^p \phi(y)dy$$
is given by
$$\frac{\delta J[f]}{\delta f(x)}= \lim_{\epsilon\rightarrow0} \frac{1}{\...
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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 ...
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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 ...
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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 {...
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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 ...
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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 ...
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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]...
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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 ...
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