# Tag Info

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A simple reference problem Suppose we want to analyse the problem of a forced harmonic oscillator. Denote as $\phi(t)$ the time dependent position of the oscillator. The oscillator experiences two forces, the spring force $-k\phi(t)$ and an external force $F_{\text{ext}}(t)$. Newton's law says \begin{align} F(t) &= m a(t) \\ -k \phi(t) + ... 1 To calculate explicitly the curvature and geodesic equations for the conical spacetime you need an explicit metric. The metric ds^{2}=dr^{2}+r^{2}d\phi^{2} describe a conical spacetime in the range of definition of the coordinates (r,\phi)\in (0,\infty)\times [0,2\pi-\alpha). You can notice that this metric describe a flat spacetime in the domain of ... 0 Essentially what is happening is that we deform the integration contour near the pole by adding a semicircle of radius \varepsilon centered on the pole either above or below the real axis. We can then calculate the integral via the residue theorem and take the limit \varepsilon \rightarrow 0 corresponding to the original contour. 1 It is a distinction corresponding to different types of spectral measures. The absolutely continuous spectrum corresponds to absolutely continuous measures, singular spectrum to continuous singular measures (both with respect to Lebesgue measure). Refer e.g. to Reed-Simon Chapter VII for a more detailed description. 6 OP considers the 'same-time' functional derivative (FD)\tag{1} \frac{\delta f(t)}{\delta x(t)}~:=~\frac{\partial f(t)}{\partial x(t)} - \frac{d}{dt} \frac{\partial f(t)}{\partial \dot{x}(t)} +\ldots. $$Here f(t) is shorthand for the function f(x(t), \dot{x}(t), \ldots;t). Although the 'same-time' FD (1) can be notationally useful, it has various ... 5 In general functional derivatives obey chain and product rules. If the concept troubles you you can always think of a function as a vector with an infinity of coordinates. Then a functional derivative is just a partial derivative. If F[h] is a functional of the function h(x). You can think of this as$$ h \to \vec{h} = \left(h(x_1), h(x_2), ..., ...

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Yes. Here, we are dealing with functional derivatives and these satisfy the chain rule and the product rule, which is really an important reason why it can be called a derivative to begin with. Important note: The definition that you give for the functional derivative is not the standard one, and does not satisfy its usual properties (as shown by ...

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In circuit analysis we make some assumptions and we use shorthand notations frequently. For example we assume the potential doesn't vary anywhere in the wire, even it was 1 km long. Hence we don't write the spatial coordinates like we do in electromagnetics, in that they complicate the analysis and don't give much accuracy (the dimensions of the wire and ...

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The electric potential is always a function of both spatial position and time, in both circuit theory and electrodynamics. $V$ is constant along a wire, so in circuit analysis you can take a short cut by specifying the position by just specifying which wire you're talking about, like $V_{1}(t)$, instead of specifying three spatial coordinates. $V$ doesn't ...

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When you consider a voltage $V(t)$ in a circuit, you are talking about a voltage at a specific point in the circuit - in other words, implicitly it's $V(t,x,y,z)$ - at a certain time & place. When you have a static situation (things don't change over time) it's possible to talk about a potential as a function of location: $V(x,y,z)$ describing the ...

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What you need is Kirchhoff's Matrix-Tree Theorem which expresses ${\rm det}\ A$ as a sum of trees. You can find an easy "Fermionic" proof of this theorem and a list of original references in my article "The Grassmann-Berezin calculus and theorems of the matrix-tree type" (arXiv version here if you do not have access to the journal).

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I think you should have a look at Reed&Simons classic book on mathematical physics ("Methods of Modern Mathematical Physics", 4 volumes). Excellent and clear writing style, many further references and it covers most of the important analytic methods which are used in physics. For geometry stuff, I recommend Bishops "Tensor analysis on manifolds". Very ...

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For an introductory book on the topic, consider "Numerical and Analytical Methods for Scientists and Engineers, Using Mathematica" by Daniel Dubin (ISBN-13: 978-0471266105). Whats sets this book apart from the rest is that it combines theoretical physics, teaches the math, and solves practical physics problems both by hand and by using Mathematica. This ...

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Hints: Assume that $H$ is a complex Hilbert space. Assume that $A:H\to H$ is a normal operator$^1$. Then a version of the Spectral Theorem says that $A$ is orthonormally diagonalizable. Let $(\lambda_i)_{i\in I}$ denote the set of different eigenvalues of $A$ with corresponding multiplicities $(m_i)_{i\in I}$. Let $P_i$ be the orthogonal projection ...

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The fine structure constant is given as: $$\alpha = \frac {k_{e} e^2} {\hbar c^2}$$ Immediately we have a problem in determining the rationality or otherwise of $\alpha$. The Coloumb constant, Planck constant (maybe not?) and speed of light are all either exact numbers or pre-defined. Since the elementary charge $e$ is an empirically derived constant we can ...

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This is a situation where knowing the history of the terminology can be helpful. The QFT/string theory terminology comes from algebraic geometry, where the term moduli space is used for any space whose points correspond to some kind of geometric object. The projective space $\mathbb{P}(V)$, for example, is the moduli space of lines in the vector space $V$. ...

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I) It is worthwhile mentioning that there exists a basic approach well-suited to physics applications (where we usually assume locality) that avoids multiplying two distributions together. The idea is that the two inputs $F$ and $G$ in the Poisson bracket (PB) $$\tag{1}\{F,G\} ~=~ \int_M \!dx \left( \frac{\delta F}{\delta \phi(x)}\frac{\delta G}{\delta ... 1 What you are proposing is known as the Heisenberg picture of quantum mechanics which had a primitive formulation even before the Schrödinger formulation in the form of Matrix mechanics. The linked wikipedia article is very well written, so I think me giving a detailed description would be redundant. In this picture, all the evolution is transformed into ... 1 We are very well allowed to ask for exact values of x and p. We can just measure them. What we can't expect is that if we reprepare the same state, we'll get the same answer, if we measure again. We can also measure the position and the momentum and will obtain a specific answer - once again, if we reprepare the same state, we'll not get the same ... 1 If I've understood you correctly, you want rigourous mathematical formalism to treat PDE solutions which are not differentiable or not square-integrable etc. That is, you can have those point charges with fields blown up on them, fields being not differentiable on boundaries and so on. There exists a rigourous formalism to treat such things. It is called ... 1 The stationary action principle and the Euler-Lagrange (EL) equations are very broad and general constructions. The field variables in the variational principle could in principle map into some generic manifold M. On the other hand, Euler-Poincare (EP) equations appear in the special situation where the manifold is a Lie group M=G, and the action is ... 1 Something like \textbf{E}(\textbf{r}) \propto (\textbf{r}-\textbf{r}^{\prime})/|\textbf{r}-\textbf{r}^{\prime}|^{3} due to a static point charge does not belong to the appropriate class of functions (differentiable, square-integrable, etc.) for \textbf{E} fields. In a mathematically precise sense, it is not a solution to Maxwell's equations. Still, we ... 2 One of the major issues that seems to be going on here is the notion of point and surface structures in our 3D world. When we define electrostatic fields by a distribution of point charges, we are being somewhat non-physical. If we keep zooming in on an electron, it's going to start not looking like a point charge anymore. Consider the Darwin Term in the ... 1 The relation$$\langle a|b\rangle\propto\delta(a-b)$$is nothing unusual, it is simply an orthogonality condition. If the proportionality was an equality, and in addition we had completeness, the set of states would form an orthonormal basis. The reason why the delta function shows up is that you assume your operator to have a continuous spectrum of ... 1 The steps that you wrote down till eq. 1 is in fact a simple proof of the following theorem (which can be looked up in elementary text books of quantum mechanics): Eigen functions (of a Hermitian operator or more generally a symmetric operator on a separable Hilbert space) belonging to distinct eigenvalues are orthogonal. This is always true for separable ... 0 But these fields diverge/become infinite in case of point charges, how is this justified and mathematically consistent ? If the charge q of the particle is finite, both densities have to be singular at the point where the particle is, because a regular (everywhere finite) density would need to be non-zero in a region of non-zero volume to give finite ... 2 While i was typing, two good answers were posted. Since I don't want to delete everything, I'll leave this here nontheless. Without appealing to Lie theory, one might argue by physical reasoning. The unitary operators your book has in mind depend on a continous parameter \alpha. They describe continous transformations of the quantum mechanical state ... 3 Well, quantum mechanics is famous for not being intuitive for earthlings like us, but the following couple of facts might help: Observables in quantum mechanics are Hermitian/selfadjoint operators. The spectrum {\rm Spec}(\hat{A}) \subseteq \mathbb{R} of a Hermitian/self-adjoint operator \hat{A} belongs to the real axis \mathbb{R}\subseteq ... 11 There's no escaping the Lie theory if you want to understand what is going on mathematically. I'll try to provide some intuitive pictures for what is going on in footnotes, I'm not sure if it will be what you are looking for, though: On any (finite-dimensional, for simplicity) vector space, the group of unitary operators is the Lie group \mathrm{U}(N), ... 1 Hint: Establish first that$$\delta(xy)~=~\frac{\delta(y)}{|x|}+\frac{\delta(x)}{|y|}. 1 Let there be given an n-dimensional manifold (M,\nabla) endowed with a connection \nabla. [In particular, we do not assume that the manifold M is equipped with a metric tensor.] Let there be given a curve \gamma:\mathbb{R}\to M. Here the reader should think of \mathbb{R} and M as time and space, respectively. If f: M\times \mathbb{R}\to ... 11 We have also the same notions of derivation, curl, etc... for functions that are less regular. When you write Maxwell's equations, you are writing a system of partial differential equations. To investigate them, you have to specify the type of solution you look for (in the language of PDEs: classic, mild, weak...) and the functional space you set your ... 3 \begin{align*} \sum_{n=1}^\infty n\exp[-\epsilon n\sqrt x] &=-\frac1{\sqrt x}\frac{\partial}{\partial \epsilon}\frac{1}{1-\exp[-\epsilon\sqrt x]}\\ &=\frac{\exp[-\epsilon\sqrt x]}{(1-\exp[-\epsilon\sqrt x])^2}\\ &=\frac{1}{\exp[\epsilon\sqrt x]-2+\exp[-\epsilon\sqrt x]}\\ &\simeq\frac{1}{\frac2{2!}(\epsilon\sqrt ... 1 The geometric series formula tells us \sum_{n=1}^{\infty} e^{-\epsilon n\sqrt{x}} = \frac{e^{-\epsilon \sqrt{x}}}{1-e^{-\epsilon\sqrt{x}}}. $$The derivative with respect to \epsilon of the left hand side gives -\sqrt{x} times your sum. Therefore your sum is equal to$$ -\frac{1}{\sqrt{x}}\frac{\partial}{\partial\epsilon}\frac{e^{-\epsilon ...

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Disclaimer: This is answer is given from a mathematical physics point of view, and it is a little bit technical. Any comment or additional answer from other points of view is welcome. The classical limit of quantum theories and quantum field theories is not straightforward. It is now a very active research topic in mathematical physics and analysis. The ...

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Short answer: I think the notation is the main problem here. In your second equation, the LHS $\rho\mathbf{u}$ is a function of $\mathbf{x}_0$ and $t$, while your RHS $\rho\mathbf{u}$ is a function of $\mathbf{x}$ and $t$. The subtle difference is that $\mathbf{x}_0$ should be treated as a particle label, not an actual position. As you suspected, the ...

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My references are very good reviews: Quantum inverse scattering and Algebraic Bethe Ansatz: Faddeev: How Algebraic Bethe Ansatz works for integrable model Kulish and Sklyanin: Quantum Spectral Transform Method. Recent Developments Takhtajan: Introduction to algebraic Bethe ansatz and the Books: Jimbo and Miwa: Algebraic Analysis of Solvable Lattice ...

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Your intuition that If, instead, my measurement is only partly accurate and says that the momentum of the particle is in a set $\Delta =(a_x,b_x)\times(a_y,b_y)\times(a_z,b_z)$, will the measurement collapse the wave function into $P\Psi$ (where $P$ is the spectral projector of the momentum operator on the set $\Delta$)? is exactly correct. ...

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There are, in general, infinitely many operators with equally spaced eigenvalues. Suppose a self-adjoint operator $A$ has a purely discrete spectrum (i.e. it is either compact or with compact resolvent) and denote by $\{\lambda_i\}_{i\in\mathbb{I}}$ its real eigenvalues ($I\subseteq \mathbb{N}$): then by the spectral theorem it can be written (on its domain ...

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Let the definition of the functional derivative be $$\int \frac{\delta F}{\delta \rho(x)} \phi(x) dx = \lim_{\epsilon \rightarrow 0} \frac{F[\rho + \epsilon \phi] - F[\rho]}{\epsilon}$$ Choose $$\phi(x) = \delta(x-y)$$ And complete the integration on the left-hand side $$\int \frac{\delta F}{\delta \rho(x)}\delta(x-y) dx = \frac{\delta F}{\delta ... 0 Well, I'd like to give a different perspective to the train of thought here. Geometric topology and its related fields are important in the study of elastic membranes and sheets and their d-dimensional variants. In particular, understanding topological transformations of membrane vesicles or sheets is a non-trivial and open problem. On a related note, the ... 2 The statement is simply false as it stands when adopting the standard Hilbert space formulation of QM. The true statement is that a self-adjoint operator with pure point spectrum admits a Hilbert basis made of eigenvectors. (It happens in particular, but not only, when either the operator is compact or its resolvent is.) The proof is not so simple and is a ... 2 The standard approach in numerical simulations is to do a discrete time stepping. If the motion is limited to the vertical direction, you just need to keep track of vertical position z and vertical velocity v (with positive v denoting upward speeds). Starting with given values for z and v (the initial position and initial velocity) you update the ... 2 Assuming you are using C code: #include <stdio.h> int main(void) { float position = 0, velocity = 8, time, timeStep = 0.01; float g = 9.8; for(time = 0; velocity > -10; time+=timeStep) { velocity = velocity - timeStep * g; position = position + velocity * timeStep; printf("time %f; velocity %f; position %f\n", time, velocity, ... 0 EDIT 2:There is no general procedure. It is not easy to find out how many commuting observables there are. In classical mechanics some systems are integrable. Then they have as many constants of motion as the number of degrees of freedom. For a system of N particles in d spatial dimensions times the number of degrees of freedom is Nxd. One particle and a ... 3 Valter's answer is completely correct, but I'll just briefly expand on it to address the specific values you ask about. The place to go, really, is the Wikipedia page Particular values of Riemann zeta function, which lists mosts of the values of \zeta(s) (which, as Valter explained, equals$$\zeta(s) := \sum_{n=1}^{+\infty} \frac{1}{n^s}$$when ... 6 The true fact is the following. Consider$$\zeta(s) := \sum_{n=1}^{+\infty} \frac{1}{n^s} \quad \mbox{with $s\in \mathbb C$ and } Re \:s >1\:. \tag{1}$$That function, with the said complex domain, is well defined (the series absolutely and uniformly converges) and is a complex analytic function. As a consequence of a well-known theorem on analytic ... 3 Matt Visser's How to Wick rotate generic curved spacetime is a great reference on this subject, which basically summarizes a lot of folklore on the subject. Addendum (Summary of Paper). This turns out to be an important problem in quantum gravity and QFT in curved spacetime for the obvious reason ("How do we know the usual tricks still work in curved ... 1 From wikipedia: Mathieu equation$$\frac{d^2y}{dt^2} + \left[a-2q\cos(2t)\right]y=0$$is a special form of a Hill equation with only 1 harmonic mode. Meaning the pendulum has only one harmonic or intrinsic frequency of operation (except the basic frequency). If one replaces the \left[a-2q\cos(2t)\right] term with a more general (periodic) term f(t), ... 5 The gauge connection is not unique, and this has nothing to do with the presence of matter fields. Let \Sigma be our space-time, P a principal G-bundle, and \mathcal{A} the space of connections on P. Then, gauge transformations t : P \to G, forming the group of gauge transformations \mathcal{G} have an action on \mathcal{A} given by$$ A ...

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