# 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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### 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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### 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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### 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 ... 101 views

### Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} (\dot{u}(x,t))^2 - ca(u'(x,t)^2\tag{1}$$ 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 ... 1 vote
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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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### 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 ...