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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 ...


0

To quickly answer your question about the non-commutativity of matrices (and more generally operators): The uncertainty relation is entirely a consequence of non-commutativity. The uncertainly relation that you are talking about is a consequence of a much more general statement. It goes something like this... Let $A,B$ be operators (thinking of the as ...


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There is always a level of mathematical understanding. The trick is to be honest with yourself on where you can understand the articulation. The most basic introductory text that describes quantum fields with mathematics( and eventually matrix formulations) that I have encountered is Griffiths - Introduction to Quantum Mechanics. It assumes some knowledge of ...


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Good question. In my opinion, Banach-Tarski has absolutely no implications for the enterprise of describing physical reality i.e. physics. (It is still a good question though!) Here's why. Banach-Tarski just means that, okay, suppose I build a model of the actions that can be applied to a physical object. If that model has the property that every ...


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I suspect that you cannot prove it purely from experimentally/observational, and I think that approach is useless. If you think that the whole universe as a single state in state space, then there is no way you can compare with any other state. You may think that there might be a phase space with (macroscopic) parameters $(V,\{\alpha_i\})$, so we can have a ...


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In Mahan may-particle physics P15: (for bilinear Hamiltonian)It is only necessary to find the eigenvalues of the Hamiltonian matrix. Usually the matrix is of infinite dimensionality. But one may often diagonalize it exactly for many problems. Computers allow very accurate solutions for any case of interest. If all Hamiltonians had only bilinear ...


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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 ...


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Well, as the people said in the comments, the Theorems of Green, Stokes and Gauss will do the job, and are about as mathematically rigorous as you could hope for here! The two different sets of formula follow directly. I don't want to write all four of them out, you should be able to do them yourself, but for example, let's consider the Gauss Law. ...


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 ...


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 $$ ...


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 ...


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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, ...


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 ...


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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 ...


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I am not sure I understand your question either but will try something else. When you write a probability distribution $\mathbb{\pi}: \: \mathbb{R}^3 \rightarrow \mathbb{R}_+$ for the cartesian variables $x,y,z$, you can also look at the induced probability measure for the function $r^2 = x^2 + y^2 + z^2$. To figure out what is the corresponding probability ...


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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: $\Omega(E_0)=|\{x:E(x)=E_0\mbox{ and } x\in\mathbb{X}\}|$ ...


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I happen upon this old thread now. Maybe it is still worth giving an update, and more of an answer. The latest account (as of the time of this writing) of the conjectural statement in question here appears as Conjecture 1.17 in Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology in H. Sati, U. Schreiber (eds.) ...


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 ...


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Along these same lines, if our universe was found to be inconsistent, what would that mean? How would it physically manifest? This is key to seeing that "is the universe consistent?" is a meaningless question. A mathematical theory can be inconsistent but the universe can't. Also true for "if our universe was found to be contradictory, what would that ...


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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 ...


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 ...


0

The derivative of a function $f(t)$ is the function $\dot{f}(t)$ in general different than $f$, and in the general case the two are not even linearly dependent, which is simple to see if you take the Taylor expansion. It is only after you define differential equations with them that they are linked algebraically, and this is what the calculus of variations ...


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 ...


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I think a lot of the ideas discussed here are great. The mathematical motivation by @Qiaochu Yuan is spot on. Also Lubos is correct. I have personally struggled(somewhat still struggling) with this and I just thought I would say something here. Essentially, a good way to perhaps proceed would be to start with ...


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 ...


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 ...


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 ...


0

Hints: Prove that a one-parameter group $(\Phi_t)_{t\in I}$ of diffeomorphisms $\Phi_t: M \to M$ is generated by a vector field $X\in\Gamma(M)$. Prove that if the one-parameter group $(\Phi_t)_{t\in I}$ preserves the a form $\omega$ then ${\cal L}_{X}\omega =0$. Prove that ${\cal L}_{X}\omega =0$ together with the fact that $\omega$ is a symplectic ...


0

The first equation, $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ is actually a shorthand notation for its correct full form, which is $$\underset{\epsilon\rightarrow0^+}{lim}\int_{-\infty}^\infty\frac{f(x)}{x-x_0+i\epsilon}\,dx=P\int_{-\infty}^\infty\frac{f(x)}{x-x_0}\,dx-i\pi f(x_0)$$ and is valid for functions which are analytic in the ...


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 ...


1

There are a lot of rather different aspects being touched on in the question. I'll try to give some indications. But I notice that the relation of this question to actual physics is not very strong, instead the question seems to be more generally after getting a feeling for identity types in homotopy type theory (HoTT). I imagine there are other discussion ...


7

In the language of differential forms in spacetime, the field strength $2$-form $F = E\wedge\mathrm{d}\sigma + B$ gives Gauss's law for magnetism and Faraday induction: $$\mathrm{d}F = 0\text{.}$$ Meanwhile, the electromagnetic excitation $2$-form $H = -\mathcal{H}\wedge\mathrm{d}\sigma + \mathcal{D}$ provides a natural formulation of Gauss's law and ...


0

Another answer in the context of quantum mechanics. I want to show how the tensor product properties are naturally requested when dealing with the "composition of independent sistems" by looking at the observables that must be present in the composite sistem. More precisely, you start with two Hilbert spaces $A$ and $B$ describing two sistems, so that the ...


1

Kyle Kanos mentioned Geometric Algebra for Physicists. While geometric algebra is somewhat different in notation from differential forms, the basic concepts are all there, and in many ways, geometric algebra avoids some cumbersome things that differential forms does (I'm thinking of Hodge duality in particular). I think the notation is easier to relate to ...


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If you'd like to quickly obtain an understanding of the basics of differential forms, including their relation to connections, tangent bundles etc. I recommend the first 4 online lectures of the Perimeter Institute from the Gravitational Physics course (13/14, R. Gregory).


7

Very loosely speaking the reasoning is this. Imagine a two band system in which the fermi sea has one filled band with Chern number $n$ and another system with $N$ filled bands but also with Chern number $n$. Physically they have the same topological properties (for example the same Hall conductance, edge states and so on), but cannot be deformed ...



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