General relativity - What is the physical meaning of the connection and the curvature tensor. The simplest way to explain the Christoffel symbol is to look at them in flat space. Normally, the laplacian of a scalar in three flat dimensions is: \nabla^{a}\nabla_{a}\phi = \frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}+\frac{\partial^{2}\phi}{\partial z^{2}} But, that isn't the case if I switch from the (x,y,z) coordinate system to cylindrical coordinates (r,\theta,z).
Now, the laplacian becomes: \nabla^{a}\nabla_{a}\phi=\frac{\partial^{2}\phi}{\partial r^{2}}+\frac{1}{r^{2}}\left(\frac{\partial^{2}\phi}{\partial \theta^{2}}\right)+\frac{\partial^{2}\phi}{\partial z^{2}}-\frac{1}{r}\left(\frac{\partial\phi}{\partial r}\right) The most important thing to note is the last term above--you now have not only second derivatives of \phi, but you also now have a term involving a first derivative of \phi. \nabla_{a}\nabla^{a}\phi = g^{ab}\partial_{a}\partial_{b}\phi - g^{ab}\Gamma_{ab}{}^{c}\partial_{c}\phi This is getting long, so I'll leave this at that.
Statistical mechanics - Ising model for dummies. The Ising model is a model, originally developed to describe ferromagnetism, but subsequently extended to more problems. Basically, it is an interaction model for spins. Imagine you have a system which is a collection of N spins. Each spin S_i has two possible states +1 or -1. Here you can imagine already a possible extension to more states. But I'm getting ahead of myself. The next step is to define the energy of the system. E= - \sum_i h_i S_i - \sum_{i \neq j} J_{ij} S_i S_j The first term can be interpreted as the contribution to the energy of the interaction of a spin with a local magnetic field.
The second term represents interactions between spins within the system. What I still haven't specified is how the spins are structured. E= - J \sum_{n} S_n S_{n+1} \; , if all spins can interact equally strongly. Now, to each configuration of the system corresponds a certain energy. P(\{S\}) \sim e^{-E(\{S\})/kT} where T is the temperature. Z=\sum_{\{S\}} e^{-E(\{S\})/kT} H.E. Steps for going from Polyakov action to low-energy effective action (SUGRA?) in String Theory.
To answer your question including the full supergravity, explicitly enough, one needs an amount of space not much smaller than the usual textbooks. And supergravity usually appears in their second volumes only. ;-) However, one may divide the question to the proof of supersymmetry, and the proof of gravity in string theory. The proof that superstring theory obeys spacetime supersymmetry is a separate thing.
A combination of gravity and supersymmetry immediately implies that the spacetime action has to be one of supergravity - this is pretty much by definition of supergravity, even though this verbal exercise doesn't really construct the action in components. So the remaining question is why string theory predicts gravity. It's because the world sheet action describes the proper area of a world sheet that propagates in a background spacetime. This spacetime may be infinitesimally changed by changing the background metric.
Classical mechanics - How can I measure the mass of the earth at home. Yes we/you can. I recall seeing a famous video of a homemade version of the Cavendish torsion balance experiment from the early 1960's, made I think for the PSSC high school course. Basically, the physicist hung a torsion balance from a high ceiling by a long (>10 m?) Piece of computer data tape (chosen because it would not stretch). He carefully minimized air currents. I found an Italian dubbed version of the video on Youtube. It looked really crude but qualitatively it worked. I also found some other do it your self experimenters with crude equipment, experimental tips (try fishing line) and different masses. The best summary and historical exposition I found is at . Editorial (I'll move this positive rant to meta soon) - given the obviously widely varied audience on this site, I would very much like to see more questions like this one relating to amateur or home experiments.
Classical mechanics - Physical meaning of Legendre transformation. Legendre transformations are commonly used in thermodynamics (to switch between different independent variables) and classical mechanics (to switch between the Lagrange and Hamilton formalisms). But you rightly ask: what exactly is a Legendre transformation? Where does it come from? What makes it work? In (1D) classical mechanics, for example: if we have a Lagrangian L(q,\dot{q}[,t]), why can we define a variable p = \frac{\partial L}{\partial\dot{q}} and expect to be able to construct a new function (the Hamiltonian) H(q,p[,t]) = p\dot{q}-L(q,\dot{q}[,t]) that behaves well? Let's look at the Lagrangian and Hamiltonian as a guiding example.
One thing I will do, however, is leave out the explicit time dependence. So what do we need for a Legendre transformation? Well, first of all we need two variables v, p that are single-valued functions of each other. Figure 1. \begin{align} \frac{\partial L}{\partial v} &= p \\ \frac{\partial H}{\partial p} &= v \end{align} Why does it work? Because or. Quantum mechanics - Electromagnetic Field as a Connection in a Vector Bundle. I can fully understand your confusion since it is natural that you feel overwhelmed by this new viewpoint on the theory. The answers given by Eric and Marek are just fine and I will not directly talk about principal bundles, local trivialization and the like. I want to present a very intuitive approach here. I would suggest that you go one or two steps back and try to understand the notion of a covariant derivative in classical differential geometry.
There, the covariant derivative D assures that if you derive some quantity F on a manifold, say some surface, this new quantity DF will also lie "on the manifold" (actually, something related to it like the tangent space). The following example will hopefully illustrate the issue what it means that something has to "stay on the manifold". Mass-point on a surface Ok, lets do the most simple example one could think of, the motion of a free mass point on a surface in Newtonian mechanics. L = T = \frac{m}{2}\mathbf{v}^2 second, since g is symmetric. Applications of Algebraic Topology to physics. The calculation of extended charges moving in time, requires finite difference calculations that only a computer can perform. This is one of the principle reasons why 20th century Physicists chose to use the point charge model. Quantum Mechanics (QM) was invented, as a probability theory, to incorporate the point property and attempt to explain the wave like phenomena.
In QM, the probability density of the charged entity is of course the square magnitude of the wave function, much like the energy density of the electromagnetic (EM) field is the squared magnitude of the field densities. No matter how small the region is, there is a chance of finding the charge entity, which implies the charged entity is a point charge. As any physics student knows, this implies infinite self energy, and no physical explanation for angular momentum, magnetic spin or radiation (energy exchange). If you wish to study Modern Physics, then please visit www.commonsensescience.org. Mathematics - Crash course on algebraic geometry with view to applications in physics. One good source my undergraduate adviser recommended to me are the lecture notes of Candelas on Complex Geometry. They are written with string theory in mind and cover a lot of basic ground. I am not sure, if they are available online. Griffiths and Harris is very good, but probably not suitable as your only source for self-study.
Just to get an idea what ideas were needed in string theory 25 years ago a look at chapter 12,14,15,16 in the second volume of Green, Schwarz, Witten might be helpful. Especially 14 and 15 should be interesting to you, even if you did not take a course in string theory yet. By now there are of course a lot of other applications of ideas from algebraic geometry to the study of string theory beyond those ordinarily found in textbooks. Mathematical physics - A pedestrian explanation of conformal blocks.
I did a bit of reading about this, and it turns out that conformal blocks are actually quite relevant to my research! So I figured it was worth the time to investigate in some more detail. I've never studied conformal field theory formally, but I hope I'm not writing anything outright wrong here. (I lost my first draft and had to reconstruct it, which is why it's taken so long) In conformal field theory, it's common to represent coordinates on a two-dimensional space by using complex numbers, so \vec{r} = (x,y) becomes \rho = x + iy.
In this notation, the theory is invariant under the action of a Möbius transformation (a.k.a. conformal transformation), \rho \to \frac{a\rho + b}{c\rho + d} in which a, b, c, and d are complex constants that satisfy ad - bc \neq 0. So any function of four coordinates on the plane, for example a four-point correlation function of quantum fields, x = \frac{(\rho_4 - \rho_2)(\rho_3 - \rho_1)}{(\rho_4 - \rho_1)(\rho_3 - \rho_2)} Quantum field theory - "Velvet way" to Grassmann numbers. I don't have an answer to the question "why would one want to consider such crazy stuff in physics? " since I don't know much physics, but as a mathematics student I do have an answer to the question "why would one want to consider such crazy stuff in mathematics? " What physicists call Grassmann numbers are what mathematicians call elements of the exterior algebra \Lambda(V) over a vector space V. The exterior algebra naturally arises as the solution to the following geometric problem.
Say that V has dimension n and let v_1, ... v_n be a basis of it. The thing about the naive definition of volume is that it is very close to having really nice mathematical properties: it is almost multilinear. To fix that, we need to look instead at oriented volume, which can be negative, but which has the enormous advantage of being completely multilinear and smooth. Alright, so what about the rest of the exterior powers \Lambda^p(V) that make up the exterior algebra?
Quantum field theory - Superconformal theories. The functional space of all possible field configurations of a primary field is always a representation of the conformal group. For most choices of tensors and conformal weights, this representation is irreducible, and the primary field together with its associated tower of secondary fields constructed from taking derivatives of the primary field forms a covariant representation of it.
This is called a long representation. But for some special cases, we may write down covariant constraints which drastically reduce the representation to massless modes only. Such representations are short representations. For instance, given a primary scalar field \phi with a conformal weight of 1, the constraint \square \phi = 0 is conformally covariant and gives us a short representation. Or a primary 2-form \mathbf{F} with a weight of 2, and constraints d\mathbf{F} = 0 and \partial^\nu F_{\mu\nu} = 0. The conformal group of d+1 dimensional conformal spacetime is SO(d+1,2). String theory - What is a D-brane. Branes are (usually) extended objects; p-branes are objects with p spatial dimensions.
D-branes are a special and important subset of branes defined by the condition that fundamental strings can end on the D-branes. This is literally the technical definition of D-branes and it turns out that this simple fact determines all of the properties of D-branes. Perturbatively, fundamental strings are more fundamental than branes or any other objects. In that old-fashioned description, D-branes are "solitons" - configurations of classical fields that arise from the closed strings. They are analogous to magnetic monopoles - which may also be written as classical configurations of the "more fundamental fields" in field theory. Non-perturbatively, D-branes and other branes are equally fundamental as strings. Back to the perturbative realm. So yes, D-branes also vibrate.
When we quantize a D-brane, we find open string states which are scalars corresponding to the transverse positions. Minkowski space - twistor-spacetime correspondence. The ordinary twistor space is parameterized by (\lambda^\alpha,\mu_{\dot\alpha}). Here, the \alpha is a 2-valued SL(2,C) spinor index of one chirality and the dotted index is its complex conjugate, the index of the opposite chirality. At the level of spinors, vectors are equivalent to "spintensors" with one undotted and one dotted index. V_\mu = \sigma_\mu^{\alpha \dot\alpha} V_{\alpha \dot \alpha}. This is a basic fact about the Lie algebras. If you're unfamiliar with this equivalence of vectors of "spintensors" with two indices, notice that the components of a 4-vector may be organized as v^{\alpha\dot\alpha} = \left( \begin{array}{cc} v^0+v^3 & v^1-i v^2\\ v^1+i v^2 & v^0-v^3 \end{array} \right) Note that the determinant of this matrix - a natural function of the matrix elements - is simply v^\mu v_\mu.
So far, it has only been a story about the vectors or spinors. Note that it is a set of two complex linear equations - for \alpha=0,1 - so it defines a linear object. String theory - T-duality approaches. "Any transformation that changes one theory into another" (or the same) theory is not called T-duality. It is just a "duality". A condition is that the two theories seemingly look different - otherwise the equivalence would be vacuous - but it must be true that their spectrum and the strength of interactions between their states must be totally isomorphic: physics has to be indistinguishable.
A duality is therefore just a very fancy redefinition of coordinates that can't be quite done in the classical limit so you can't really write any "explicit" redefinition. But the impact is the same - the two theories behave in the same way. For each state/object on one side, you find a counterpart on the other side - whose geometrical interpretation may be very different but whose behavior is isomorphic. T-duality in string theory Other dualities There are several other important kinds of dualities. Why the existence of dualities was surprising. $N=4$ supersymmetric yang-mills theory and S-duality.
A. The action of N=4 SYM (Super Yang-Mills theory) in d=4 is the simple dimensional reduction of the 9+1-dimensional SYM, the maximal dimensional SYM that exists. The latter is S = \int d^{10} x\mbox{ Tr } \left( -\frac{1}{4} F_{\mu\nu}F^{\mu \nu} + \overline{\psi}D_\mu \gamma^\mu \psi \right) where D is the covariant derivative and \psi is a real chiral spinor in 9+1 dimensions which has 16 real components, leading to 8 fermionic on-shell degrees of freedom. The dimensional reduction reduces d^{10}x to d^4 x but it also renames 6 "compactified" spatial components A_\mu as six scalars \Phi_I in d=4. The derivatives in the corresponding 6 directions are set to zero. If one looks what fields and interactions we get in d=4 - it's straightforward to write the action - it's one gauge field; four Weyl fermions; six real scalars.
All those fields transform as adjoint of the gauge group - the most popular ones are SU(N). B. C. Quantum field theory - Supersymmetry algebra. Mathematics - which areas in physics overlap with those of social network theory for the analysis of the graphs. Quantum field theory - What is "localisation" of instantons.