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9

I apologize, this is my third correction to my answer. This question is very subtle indeed. I hope this answer is the ultimate one! First of all, if you want to take advantage of Lie's theorem you mention (some time called third Lie theorem), the Lie algebra has to be real, as it must be the Lie algebra of a real Lie group. Then, if you are interested in ...


7

There is one more option. You can check that $aa$, $\{a,a^+\}$ and $a^+a^+$ form Lie algebra $sp(2)\sim sl(2)$. Then you can add $a^+$ and $a$ treating them as supergenerators. These are words that tell you to take anticommutators of $a$ and $a^+$ as I did in the first line. Then you get a $5$-dimensional superalgebra, which is $osp(1|2)$. There is a ...


4

If you are interested in physical applications you could also include: Bratteli-Robinson: Operator algebras and quantum statistical mechanics It is a two-volume quite complete book, mathematically minded, discussing lots of applications of operator algebras theory to several physical systems, especially arising from statistical mechanics. Haag: Local ...


4

You have many comments to the effect that "topology is needed to describe continuity, calculus concepts, the notion of "looks like", homeomorphism and so forth". And these are all altogether right, but I'm getting that your question is about the global picture. Also, the following is mainly about a toplological or differentiable manifold; Joshphysics's link ...


3

It seems like they were able to rigorously prove the existence of N-body choreographies by using interval Krawczyk method to show that a minimum exist to the variational problem solved in the subspace of the full phase space satisfying some symmetry conditions. Following the links given I found this paper where they explain the method. It's not exactly a ...


2

Strictly speaking, if there can be defined charts covering a set (an atlas), you can give that set the topology induced by defining the charts to be bicontinous. That is, a set is open iff it's the domain of a chart in the maximal atlas. If your set already has a topology, the topology induced by the atlas will agree with that one under some conditions. (I ...


2

Non trivial holonomies have been proposed for quantum computation, see this article http://arxiv.org/pdf/quant-ph/0007110v2.pdf The basic idea is this: Suppose you have a sistem prepared in the ground state of an Hamiltonian $H(\lambda)$, where $\lambda$ is a set of parameters. If you slowly change this parameters the state evolves remaining in the ground ...


2

Physics is about studying the behavior of nature at an elementary level, gathering experimental data, and organizing the data into consistent mathematical models. By which I mean the mathematics in the models are consistent. The physical postulates that determine the mathematical model ( as additional axioms for the interpretation of the mathematics in ...


2

For a convex function you can do the following: For each point on the graph of the function, draw the line tangent to the function at that point. That point can now be identified by its original $x$ and $y=f(x)$ coordinates, or by specifying the slope of that tangent line and its corresponding y-intercept. Each point maps to one and only one line, and ...


2

We are asking how the rod moves through the viscous medium if we apply a force to it. Since the Reynolds number is so low, inertial forces must be small, and the externally applied force must be balanced by a viscous force. Also since we are in the low Reynold's number limit, the viscous force is linear in the velocity of the object. Thus there must be a ...


2

The trace defined as you did in the initial equation in your question is well defined, i.e. independent from the basis when the basis is orthonormal. Otherwise that formula gives rise to a number which depends on the basis (if non-orthonormal) and does not has much interest in physics. If you want to use non-orthonormal bases, you should adopt a different ...


2

Yes, e.g. all three Mandelstam variables $$ s~:=~(p_1+p_2)^2 ~\approx~ (m_1+m_2)^2 + \frac{m_1m_2}{2} ({\bf v}_1-{\bf v}_2)^2 ~>~0,$$ $$ t~:=~(p_1-p_3)^2~\approx~ (m_1-m_3)^2 - \frac{m_1m_3}{2} ({\bf v}_1-{\bf v}_3)^2 ~>~0,$$ $$ u~:=~(p_1-p_4)^2~\approx~ (m_1-m_4)^2 - \frac{m_1m_4}{2} ({\bf v}_1-{\bf v}_4)^2 ~>~0,$$ are strictly positive in ...


2

Sepration of variables is indeed a delicate topic in partial differential equations. As of today we don't (to the extend of my knowledge) have a complete theory on the conditions that make it possible. The usual posture is to have existence and uniqueness theorems for the solutions of a given PDE and using some ansatz from separation of variables, by finding ...


1

The logic goes like following. We can guess solution in forms of $X(x) Y(y) Z(z)$ for a particle in 3-dimensional box. We can find such solutions. The question is, do we miss any solution? The function $X(x)$ is eigenfunction of self-adjoint operator $$H_x = -\frac{1}{2} \frac{ \partial^2}{\partial x^2} + V(x) \tag{1} $$ $V(x)$ is the potential of infinity ...


1

The standard model of particle physics is a theoretical framework that encapsulates almost all elementary particle data to date. The full Lagrangian takes pages. In your comment: @Danu I understand the 6/7 Wightman axioms but fail to capture how does the concepts of fundamnetal particles, quarks-leptons, or bosons mediating forces etc. come from those. ...


1

1.Since $[,]$ is non-degenerate, there should be such an eigenvector $\eta$ corresponding to the eigenvalue $\lambda'$ for the eigenvector $\xi$ corresponding to the eigenvalue $\lambda$ that $[\xi, \eta] \neq 0$, which is possible only if $\lambda' = \bar{\lambda}$. Thus, the two-dimensional invariant plane $\pi_\lambda$ is nonnull. 2.Since $\xi$ is a ...


1

This is an interesting question, and although I don't know a rigorous answer, we can discuss some typical cases. Usually, the inverse exists, but the cases where this inverse does not exist are not necessarily pathological (sound models can have the problem that the inverse does not exist). For standard field theories (say, $\phi^4$, O(N) models, classical ...


1

I) First some terminology. Consider a symplectic manifold $(M;\omega)$. In a local chart $U\subseteq \mathbb{R}^{2n}$, the symplectic two-form reads $$ \tag{1} \omega~=~\frac{1}{2} \omega_{ij}~\mathrm{d}x^i \wedge \mathrm{d}x^j ,$$ and the corresponding Poisson bi-vector $$ \tag{2} \pi~=~\frac{1}{2} \pi^{ij}~\partial_i \wedge \partial_i, $$ where $$ ...


1

You can compute the trace of an endomorphism using any basis (including non-orthogonal ones). In Dirac notation, you show this by inserting the identity expressed in the new basis and re-arranging: $$\begin{align*} \sum\limits_{|s\rangle \in B} \langle s^*| \rho |s\rangle &= \sum\limits_{|s\rangle \in B} \langle s^*| \left( \sum\limits_{|t\rangle \in ...


1

I am not quite sure I understand your question. In your second equation, $E$ is treated as a variable and not as function anymore, so why do you want to find $E$ function? The density of states is basically a function counting the number of state in $x\in\mathbb{X}$ that give the same energy: $\Omega(E_0)=|\{x:E(x)=E_0\mbox{ and } x\in\mathbb{X}\}|$ ...


1

When you write $\omega(X,Y)$ this gives the impression that $\omega$ is a 2-form! It is $d\omega$ that is a 2-form, if $\omega$ is a 1-form. The action of the 1-form $\omega =\omega_\mu dx^\mu$ on the vector field $X = X^\mu \partial_\mu$ is simply $$\omega(X) = \omega_\mu X^\mu.$$ (Since you use index notation, you probably know about relativity and this ...


1

You do integrate over all paths in configuration space, but beware : differentiable paths contribute to a measure of 0 in the integral. The real contribution comes from fractal paths of dimension 2 (cf "The Dimension of a Quantum-Mechanical Path" by Abbott and Wise). This "spreading" of the path is the equivalent in path integrals of the Heisenberg ...


1

I want to add my personal understanding of the concept of reference frame. In the articles: Marmo, G., Preziosi, B. (2006). The Structure Of Space-Time: Relativity Groups. International Journal of Geometric Methods in Modern Physics, 03(03), 591-603. Marmo, G., Preziosi, B. (2005) Objective existence and relativity groups. Symmetries in Science XI, ...



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