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23
votes
2answers
1k 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 ...
12
votes
3answers
2k 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. $$\...
12
votes
4answers
1k 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)+\...
9
votes
2answers
1k 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{...
9
votes
4answers
613 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 ...
9
votes
2answers
1k views

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 ...
8
votes
1answer
639 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 ...
6
votes
1answer
193 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
votes
1answer
120 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{\...
6
votes
1answer
274 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
204 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 ...
5
votes
1answer
95 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
2answers
643 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(\...
5
votes
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 ...
5
votes
3answers
438 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 ...
4
votes
1answer
664 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
1answer
64 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(...
3
votes
3answers
599 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
1answer
468 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 $\...
3
votes
2answers
578 views

Derive Schwinger-Dyson equations in Srednicki

On page 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 $J_a$ are the ...
3
votes
2answers
113 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 ...
3
votes
2answers
510 views

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 ...
3
votes
1answer
53 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
2answers
786 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 $\...
3
votes
1answer
59 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 = ...
3
votes
1answer
128 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{...
3
votes
1answer
85 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
2answers
134 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} ...
3
votes
1answer
112 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 + \...
2
votes
2answers
316 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 ...
2
votes
2answers
72 views

General form for functional derivatives

Working on the hamiltonian formalism applied to canonical field theory, how do I deduce the general form for the functional derivatives $\frac{\delta}{\delta \pi}$ and $\frac{\delta}{\delta \phi}$ (...
2
votes
3answers
1k 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^...
2
votes
1answer
76 views

Full time derivative of the Frank-Oseen energy. Mathematical problem

I am studying liquid crystal theory with the book Kleman, Lavrentovich, Soft Matter Physics. In the Ericksen-Leslie theory, Frank-Oseen energy density is: $$ f=0.5*(K_1*div^2 (n)+K_2 *(n*curl(n))^2+...
2
votes
1answer
186 views

How to relate second variation to quadratic terms?

I am working on a physics problem, but my issue is math-related. My professor skips some steps based on 'intuition' that I lack: In a conservative system, to find out the nature of equilibrium points,...
2
votes
1answer
75 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[\...
2
votes
0answers
370 views

Vacuum to vacuum transition amplitude

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 \...
1
vote
2answers
113 views

Variation of a term in the Lagrangian

I don't understand why $$\frac{\delta}{\delta\phi}\left(\frac12\partial^\mu\phi\partial_\mu\phi\right)~=~\partial^\mu\partial_\mu\phi.\tag{1}$$ If we use integration by parts, there should be a minus ...
1
vote
1answer
274 views

Derivative of the magnetic field to the vector potential

So the magnetic field is defined with the vector potential A as: $$\mathbf{B}=\nabla\times\mathbf{A}.$$ How would I calculate the derivative: $$\frac{\delta}{\delta\mathbf{A}}|\mathbf{B}|^2$$ I ...
1
vote
1answer
254 views

Why this integral is equal to zero?

Recently I have read that for gauge-invariant functional (under transformations of some $SU(n)$ group) $R(A) = R(F_{\mu \nu}^{a})$ contains only gauge field $A_{\mu}^{a}$ satisfies the identity $$ \...
1
vote
1answer
171 views

Proof of Dyson-Schwinger Equation

Assuming that the functional integral of a functional derivative is zero, so $$ \int \mathcal{D}[\phi] \frac{i}{\hbar}\left\{ \frac{\delta S[\phi]}{\delta \phi}+J(x) \exp \left[ {i \over \hbar} \...
1
vote
1answer
334 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 ...
1
vote
1answer
216 views

Derive non-linear $\sigma$ model from a theory of SU(2) matirx

It's said in Chapter VI.4 of A. Zee's book Quantum Field Theory in a Nutshell, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of ...
1
vote
1answer
137 views

matrix field theory

I am studying a field theory where the field is a matrix. The problem is that I have to calculate some functional derivative. How could we define functional derivative when the field is a matrix ?
1
vote
1answer
51 views

Principle of Stationary Action and Euler-Lagrange Equation

Principle of Stationary Action: Given a mechanical system, there exists an action $S$ such that it is extremitized, or $\delta S=0$, for the actual motion of the system. $$S = \int_{t_1}^{...
1
vote
1answer
225 views

How to calculate the functional derivative of the functional integral?

I study by myself with the QFT, in the page 197 of book of Lewis H. Ryder (2nd edition), The author wrote that he take the functional derivative of equation 6.69: $$\frac {\delta\widehat{Z}[\phi]}{\...
1
vote
0answers
53 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 $\...
1
vote
0answers
67 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 ...
1
vote
0answers
33 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 ...
1
vote
0answers
71 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 $\delta x$ (...
1
vote
1answer
105 views

Finding Hamilton's equations given a Hamiltonian

I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\int_\mathbf{R} \phi(u,u_x)dx$$ Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\...