Questions tagged [functional-derivatives]

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30
votes
2answers
3k views

Symbols of derivatives

What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...
18
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3answers
5k views

Why is the functional integral of a functional derivative zero?

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e. $$\...
16
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2answers
3k views

Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{...
13
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4answers
2k views

What is the relation between (physicists) functional derivatives and Fréchet derivatives

I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books: $$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\...
11
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2answers
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Introductory texts for functionals and calculus of variation

I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good ...
9
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4answers
1k views

Is the Lagrangian of a quantum field really a 'functional'?

Weinberg says, page 299, The quantum theory of fields, Vol 1, that The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of a set of generic fields $\Psi[x,t]$ and their time ...
8
votes
2answers
2k views

Functional Derivative in the Linear Sigma Model

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(\...
8
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1answer
2k views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
8
votes
1answer
2k views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu \frac{\...
8
votes
1answer
235 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 ...
7
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2answers
2k views

Derive Schwinger-Dyson equations in Srednicki

In eq. (22.20) on p. 135 in Srednicki he defines the functional integral $$Z(J) = \int\mathcal{D}\phi\,\exp\Big[\mathrm{i}\big(S+\int\mathrm{d^4}y \,J_a\phi_a\big)\Big], \tag{A}$$ where $S$ and $...
7
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1answer
264 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
6
votes
2answers
403 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 ...
6
votes
1answer
434 views

Srednicki's Path Integrals: Ground-State to Ground-State Transition Amplitude in the Presence of a Perturbation

Srednicki's Quantum Field Theory mentions the following at the end of the unit on path integrals in non-relativistic quantum mechanics: Assume that the total Hamiltonian is of the form, $$ H = ...
6
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3answers
2k views

Is it safe to ignore derivatives of velocity w.r.t. position and vice versa?

In a certain textbook a function is given as: $$f=f(x(t))$$ And then this is differentiated w.r.t. $t$ to get: $$f_t=\dot{x}f_x$$ (Where the notation $f_u=df/du$, $f_{uu}=d^2f/du^2$, etc.) This ...
6
votes
2answers
743 views

Functional Derivation of Holonomy

I would like to know how to take the functional derivative of the holonomy, or Wilson line. I have tried it and I will show what I have done below, but before I wanted to say that I also have seen ...
6
votes
1answer
572 views

What's the Propagator in the Free Particle Case? (Path Integrals with Source Term)

If I take the Lagrangian to be, $$L(t)=\frac{1}{2}m \dot q(t)^2$$ The Euclidean Path Integral is supposed to be, $$K=\int D[q(t)] \ e^{-\int L(\dot q) d \tau}$$ If I add a source term $J(\tau)$ we ...
6
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3answers
847 views

Is there a natural (suitable) definition for functional derivative in Curved space time

If $$\delta S = \int \sqrt g F[\phi] \delta \phi\tag{1}$$ Then is it natural to define the functional derivative as follows, $$\frac{\delta S}{\delta \phi} = F[\phi].\tag{2}$$ In particular does ...
5
votes
2answers
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Accounting for metric tensor derivatives in Einstein-Hilbert action

I'm puzzling over the canonical derivation of GR from the Einstein-Hilbert action; getting the derivation to gel with an explicit treatment of the functional derivative isn't working out. So the ...
5
votes
1answer
195 views

Why can the bra and ket be varied independently?

Given a functional which depends on a function (ket), and its complex conjugate (bra), e.g. $$F[\varphi] = \langle \varphi|\hat{F}|\varphi\rangle = \int \varphi^{*}(\mathbf{r}) \hat{F} \varphi(\...
5
votes
3answers
4k views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\...
5
votes
3answers
365 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-...
5
votes
1answer
587 views

Vary action with respect to velocity

Variation of the action $S$ corresponding to a Lagrangian e.g. $L(x(t),\dot{x}(t))$ gives the Euler-Lagrange equations: $$ \frac{\delta S}{\delta x(t)} = 0 \\ \int du \left ( \frac{\delta L}{\delta x(...
4
votes
1answer
1k views

Functional derivative and units

The both sides of below equation don't give the same units, e.g. $$ \frac{\delta}{\delta \phi (\tau)}\int_a^b \phi (\tau') d\tau'=1\;. $$ where $a<\tau<b$. To show this assume that the field $\...
4
votes
2answers
776 views

Are the partial derivatives of Lagrangian in the varied action functional derivatives?

In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon ...
4
votes
2answers
1k views

Functional Derivative of action

Consider the action of free Klein-Gordon theory $S[\phi]=\frac{1}{2}\displaystyle\int d^4y(\partial_\mu\phi(y)\partial^\mu\phi(y)-m^2\phi^2(y))$ Integrating by parts in the first term gives me $S[\...
4
votes
2answers
1k views

Field theory:functional derivative involving Fourier Transform

I have to solve the following functional derivative $$ \frac{\delta}{\delta \Lambda(\mathbf{x})}\log[A-\mathbf{k}^2\Lambda(\mathbf{k})] $$ where $\Lambda(\mathbf{k})$ is the Fourier transform of $\...
4
votes
2answers
231 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 ...
4
votes
1answer
181 views

How to deal with $\nabla(\delta\Psi)$ in functional derivatives?

I am trying to compute the functional derivative of the following functional $$F[\Psi]=\int{}d^nx\Psi{}e^{(\nabla\Psi)^2}.$$ What I have tried up till now is the following $$F[\Psi+\delta\Psi]=\int{...
4
votes
1answer
1k views

Wick Contraction

I am reading Quantum Field Theory in a Nutshell by A. Zee. Zee introduces the rationale/machinery behind Feynman diagrams in three steps: Baby -> Child -> "Real". The baby problem generates ...
4
votes
2answers
214 views

Functional derivative of meta-generalized gradient approximation (meta-GGA)

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 ...
4
votes
2answers
417 views

Equivalence of functional and partial derivatives

I am trying to derive Newton's second law from the principle of least action, that is, setting the functional derivative $\frac{\delta S}{\delta x(t)}$ equal to 0. $$S = \int dt' \left[ \frac{m}{2} ...
4
votes
0answers
132 views

How do I show that a given Wilson loop satisfies the loop equation?

In the book Methods of Contemporary Gauge Theory by Yuri Makeenko, the loop equation in the large-$N$ limit is given by $$\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu \nu}} W(C) = \lambda \oint_C ...
4
votes
0answers
70 views

Eigenfunctionals and their application in physics

Is there any sensible meaning of the term eigenfunctionals? The object I want to describe is a solution to the following equation $$ {\mathscr D}_x F[g] = f(x) F[g] $$ where $ {\mathscr D}_x$ is an ...
3
votes
3answers
2k views

Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?

I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows: \begin{align} \dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x} = \left( \...
3
votes
2answers
227 views

Why is this “the” functional of Laplace's equation?

Halfway through a discussion of the finite element method for solutions to Laplace's equation, Sadiku (2000) drops in a formulation of the work functional for an electric field: Algebraically and ...
3
votes
1answer
604 views

Einstein action and the second derivatives

I have naive question about Einstein action for field-free case: $$ S = -\frac{1}{16 \pi G}\int \sqrt{-g} d^{4}x g^{\mu \nu}R_{\mu \nu}. $$ It contains the second derivatives of metric. When we want ...
3
votes
1answer
63 views

I am stuck in the derivation of Schwinger-Dyson equation for 1-point Function in Schwartz's QFT book

This is from chapter 14.7.1 in Schwartz's QFT book. I am trying to derive contact terms starting from field redefinition $\phi\rightarrow\phi(x)+\epsilon(x)$. For the 1-point function we have from ...
3
votes
1answer
180 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')}\...
3
votes
1answer
295 views

Deriving Hamilton's equations from KdV Hamiltonian

Let $f=f(q,p)$, $g=g(q,p)$ and Possion bracket $$\{f,g\}=\frac{\partial f}{\partial q}\frac{\partial g}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial g}{\partial q}. \tag{1}$$ Then Hamilton'...
3
votes
1answer
227 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....
3
votes
1answer
114 views

Free energy variations

In a paper, I found this: $\mathbf{h}=\mathbf{h}(\mathbf{r})$ is called molecular field and is defined as the variation field of the Frank free energy functional $F_{d}$ with respect to the ...
3
votes
1answer
329 views

Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
3
votes
1answer
624 views

Variational proof of the Hellmann-Feynman theorem

I use the following notation and definition for the (first) variation of some functional $E[\psi]$: \begin{equation} \delta E[\psi][\delta\psi] := \lim_{\varepsilon \rightarrow 0} \frac{E[\psi + \...
3
votes
1answer
188 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 ...
3
votes
1answer
324 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 ...
3
votes
1answer
243 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(\...
3
votes
0answers
125 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^\...
3
votes
0answers
551 views

Vacuum to vacuum transition amplitude [duplicate]

I have two questions about Vacuum to vacuum transition amplitude. Can any particle stay in $|0\rangle$? I was studying this topic from Srednicki's QFT book. He writes in eq.$(6.22)$ $$\langle0|0 \...
2
votes
2answers
597 views

Is $\frac{\partial}{\partial \Phi(y)} \Phi (x) = \delta(x-y)$ correct?

As stated in the heading: Is $\frac{\partial}{\partial \Phi(y)} \Phi (x) = \delta(x-y)$ correct? Here denotes $\Phi(x)$ denotes a scalar field. And if yes, why? Any reference where I can read about ...