Are M2 and M5 membranes in M-Theory excited states of some underlying field, or are they fundamental? In what sense are membranes solitonic?  In the reference question, someone made a comment that had no reply and I am just as curious  "My question is, what is the M brane itself? Is there an underlying M brane field where the branes are excitations of the field? Or is the M brane simply a fundamental object made out of no fields?"  I am confused what an M Brane really is. M2 membranes are strings when you compatify a membrane on a circle, if I am not wrong, so this means all strings and branes are really just solitons?
122187
I'm really confused.  In M theory/String theory, the fundamental objects are M/D branes. However, branes by defintion are just solitons. Solitons are just waves that maintain there shape.  So if a brane is a soliton wave, then what is it a wave of? Would it be the excitation of a field?
39952
I just found  blog post, which gives an interpretation of virtual particles I haven't seen before.  Consider a 1D system of springs and masses, where the springs are slightly nonlinear. A "real particle" is a regular $\cos(kx-\omega t)$ wavepacket moving through the line, where $\omega$ satisfies the dispersion relation $\omega = \omega(k)$. When two real particles collide, the region where they collide temporarily looks really weird, as they interact nonlinearly, pulling and pushing on each other.  Formally, we can write this weird region as the sum of a bunch of $\cos(kx-\omega t)$ waves, but there's no guarantee they'll have the right dispersion relation. Thus, each term in the resulting expansion is a off-shell "virtual particle". If you add up all the virtual particles, you get the actual intermediate field state, which is just a weird ripple in the field where the two particles are interacting.  As another example, consider the statement "a static EM field is made of virtual particles". Under this interpretation, what that really means is, "a static field (e.g. $1/r^2$) is not equal to $\cos(kx-\omega t)$, but may be expanded in terms of such sinusoids", which is much less mysterious sounding. In fact, this is exactly what we do when we consider scattering off a potential in normal QM, e.g. in the Born approximation.  The above gives some intuition for what a 'virtual particle' means in classical field theory. They are the Fourier components of the field with $(\omega, k)$ not satisfying the dispersion relation, and they are useful in classical perturbation theory. However, they are not propagating degrees of freedom, so they only appear during interactions.   The picture above is entirely classical. Does this picture generalize to quantum field theory, giving a physical intuition for virtual particles there?
89757
How would you explain string theory to non-physicists such as myself? I'm especially interested in how plausible is it and what is needed to successfully prove it?
1895
What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?  Some subjects off the top of my head that probably need covering:   Differential geometry, Manifolds, etc. Lie groups, Lie algebras and their representation theory. Algebraic topology.
104002
So there's general relativity and Einstein's field equations that tell us "mass(or equivalently energy) warps space-time, and the warping tells mass how to move", but I'm still having trouble understanding how space (a thing I conceive to be 'empty' or nothingness) can be 'warped'.  After researching books, on-line articles, and this stack exchange itself I can't seem to find any descriptions or discussions that don't immediately resort to jumping into GR and its mathematics to provide a more physical intuitive understanding of what it means to for space to be warped.  Is it space, this empty thing, that's getting warped. Or rather is it some field (Higgs field?) that exists around matter that's getting warped?  Although this question appears similar to  , it's asking if it's empty space that's being warped by matter or rather a field in space that's being warped (the Higgs Field?). The other question and all 10 answers addressing it do not address fields vs. empty space - What's really getting warped?
100216
Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, then $\omega_\partial=\omega(v_{\text{out}},v_1,\dots,v_{k-1})$ orients $\partial M$, where the $v_1,\dots,v_{k-1}$ are basis vectors for a tangent space on a point of the boundary, and $v_\text{out}$ is an outward pointing vector (see here: , which follows the exposition of Hubbard &amp; Hubbard).  In practice I find this very difficult to interpret. How can one find this $v_\text{out}$ in practice?   For instance, take $M$ to be defined by $z=x^2+y^2$ for $x\ge0$ (technically a manifold with boundary, in fact a half-paraboloid). It's a fact that $\partial M$ is defined by the parabola $z=y^2$. If $M$ is oriented by $dx\wedge dy$, how does this induce an orientation on the boundary (in fact we should obtain a 1-form that induces an orientation on the boundary)?   Note: I find the whole discussion of boundary orientation to be rather clumsy in its exposition. In the case of surfaces inducing orientations on boundary curves, I was originally taught to imagine that the normal vector that induces the surface orientation is me walking on the surface. This normal should walk along the boundary such that the surface lies to its left. OK - but how does this translate to the $v_{\text{out}}$ interpretation, so that we can generalize to other dimensions easily?)  Edit: An ideal answer would merge the intuitive approach with the rigorous approach, if possible. I would grateful to anybody who could help order my (at present confused) understanding!
2038
I have looked at other questions on this site (e.g. "why does space expansion affect matter") but can't find the answer I am looking for.  So here is my question: One often hears talk of space expanding when we talk about the speed of galaxies relative to ours. Why, if space is expanding, does matter not also expand? If a circle is drawn on balloon (2d plane), and the balloon expands, then the circle also expands. If matter is an object with 3 spatial dimensions, then when those 3 dimensions expand, so should the object.  If that was the case, we wouldn't see the universe as expanding at all, because we would be expanding (spatially) with it.  I have a few potential answers for this, which raise their own problems:   Fundamental particles are 'point sized' objects. They cannot expand because they do not have spatial dimension to begin with. The problem with this is that while the particles would not expand, the space between them would, leading to a point where the 3 non-gravity forces would no longer hold matter together due to distance Fundamental particles are curled up in additional dimensions a la string theory. These dimensions are not expanding. Same problems as 1, with the added problem of being a bit unsatisfying. The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me.
86391
I understand that the vibration modes of an elementary string determines the identity of its particle. When I first heard this, I visualized a stiff circular steel band-like entity vibrating with 2, 3, 4, 5,... nodes or antinodes, the number determining the particle. This was further supported when I asked a well-known physicist how one gets from equations to physical strings, and he said he recognized the harmonic nature of the solutions.  Bu then I noticed that physicists seem to accept the image of a soft spaghetti-like string flopping about with no well-defined vibrational modes. I asked another physicist about this, and his answer was simply that a string has multiple modes, not necessarily in a harmonic progression.  So my questions are 1) How do you visualize the vibrations of a string, and 2) how are they related to particle identity?
121187
What is currently stopping us from having a theory of everything? i.e. what mathematical barriers, or others, are stopping us from unifying GR and QM? I have read that string theory is a means to unify both, so in this case, is it a lack of evidence stopping us, but is the theory mathematically sound?
2574
The following image serves to aid the reader in understanding the "privileged character" of $3+1$-spacetime.    The wikipedia article on , and the  "The priveleged character of $3+1$-dimensional spacetime" in particular, made me think a bit about the possibility that we might live in a non-integer amount of spatial and/or time dimensions.   The notion of attaching a non-negative real number to a metric space has at least mathematically already been described by such concepts as "" and "".   This may sound silly/ignorant/absurd to professional practicing theoretical physicists. To me (a layman), however, it doesn't sound much stranger than the idea of wrapping up six extra dimensions (which is, from what I understand, considered to be a serious possibility by those who study $10$-dimensional String Theory) into intricate shapes called "Calabi-Yau Manifolds".   Has any research on $(a +b)$-dimensional spacetime (where $a,b \in \mathbb{R}_{\geq 0} $) ever been done? If so, what where the findings? If not, why not?
8392
In very recent publications, two groups in Maryland  and Harvard  have separately worked on constructing  following the path paved by the previous works of Norman Yao (e.g.  ). For example Chris Monroe et al. (Maryland) have realised (or claim to have) a time crystal by entangling the electron spins of a set of ytterbium ions while keeping the system out of equilibrium by alternately hitting the ions by a laser (to apparently create an effective magnetic field) and by another laser to flip their spins, in a repetitive manner.   Now time crystals have been previously theoretically predicted to exist  and , representing an out-of-equilibrium form of matter, in contrast to the commonly well known equilibrium forms of matter (solid, gas, liquids, liquid crystals...). So it is really great to see such works emerging and the field gaining steam on an experimental level.   That said, admittedly, it is very difficult to grasp these works for someone not involved in the field, but it would be incredibly valuable if, briefly, just the idea of time crystals could be explained at a conceptual level.  As in, what qualifies them as a new form of matter? Surely this form is dependent on some external drive to be kept out of equilibrium, right?   Which then begs to ask "how can it be a naturally occurring form of matter". On the other hand, what does it have in common with the usual concept of crystals, whose whole structure can be represented by spatially replicating the unit cell.  Is the time crystal an addendum to the latter by extending its spatial symmetry to also include a time symmetry?    This is merely an attempt to get more input in order to understand some of the main ideas involved in the original works of Frank Wilczek and Norman Yao. These new experimental realizations (see first paragraph) are very exciting to find out about.
28110
Someone claiming to have studied physics is telling me that spacetime is all that exists and that this fact is the basis of modern physics. When I said matter/energy also exists in its various forms, he said that energy is merely spacetime "waving/vibrating/acting", else what did I think was doing the waving: magical fairy dust? and has presented the following Einstein quote to support his claim:  "When forced to summarize the general theory of relativity in one sentence: Time and space and gravitation have no separate existence from matter."  Does Einstein's quote support what he's saying, that space-time is all that exists, and that matter/energy is spacetime waving?
104208
Basically, I am aware of single mode and multi mode fibers. I have also been reading about transverse electric and magnetic fields , their corresponding modes and TEM modes. Now I am a little bit confused of the term "Mode".  I have in my understanding that "Mode" is just a path of light propagation in a fiber. If light propagates in an Optical fiber in a single path only, then its a single Moded fiber and the Mode count is 1. If there are multiple paths in which light can propagate, then there are multiple modes. Is it correct? and if its correct, how does this mode relate with TE and TM modes?
111171
At one point, I decided to make friends with the low-lying spectrum of QCD. By this I do not mean the symmetry numbers (the "quark content"), but the actual dynamics, some insight.  The pions are the sloshing of the up-down condensate, and the other pseudoscalars by extending to strangeness. Their couplings are by soft-particle theorems. The eta-prime is their frustrated friend, weighed down by the instanton fluid. The rho and omega are the gauge fields for flavor SU(2), and A1(1260) gauges the axial SU(2), and they have KaluzaKlein-like echoes at higher energies, these can decay into the appropriate "charged" hadrons with couplings that depend on the flavor symmetry multiplet. The proton and the neutron are the topological defects. That accounts for everything up to 1300 but a few scalars and the b1.  There are scalars starting at around 1300 MeV which are probably some combination of glue-condensate sloshing around and quark-condensate sloshing around, some kind of sound in the vacuum glue. Their mass is large, their lifetime is not that big, they have sharp decay properties.  On the other hand, there is nothing in AdS/QCD which should correspond to the sigma/f0(600), or (what seems to be) its strange counterpart f0(980). While looking around, I found this discussion: . The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).   gives strong evidence for an actual pole;  gives a more cursory review. The location of the pole is far away from the real axis, the width is larger than the mass by 20% or so, and the mass is about 400MeV. The authors though are confident that it is real because they tell me that the interpolation the interactions of pions is safe in this region because their goldstone properties dominate the interactions. I want to believe it, but how can you be sure?  I know this particle was controversial. I want to understand what kind of picture this is giving. The dispersion subtraction process is hard for me to visualize in terms of effective fields, and the result is saying that there is an unstable bound state.  Is there a physical picture of the sigma which is more field theoretical, perhaps even just an effective potential for pions? Did anyone who convinced himself of the reality of the sigma have a way of understanding the bound state properties? Is there an analog unstable bound state for other goldstone bosons? Any insight would be welcome.
56618
This has been asked about here: , and  said      Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ doesn't have a neighborhood with any structure like an open ball/half-ball.    This is actually an exercise from An Introduction to Manifolds by Loring W. Tu and is not mentioned in an .        I have spent almost 2 hours thinking about an exercise that looked like it would take only 15 minutes and even tried pasting lemma (it's a good thing Professor Tu has solutions unlike ): The result of all that thinking is that I don't think $[0,1) \cup \{2\}$, $(\varepsilon,1) \cup \{2\}$ or $\{2\}$ is homeomorphic to any open subset of $\mathscr H^1$ or $\mathscr L^1$. I was able to show $\{0\} \subseteq \partial M$ and $(0,1)\subseteq M^0$, but I don't quite know where $2$ belongs. I believe $M$ is not locally $\mathscr H^1$. Also, I have double checked: I believe "manifold boundary" was defined for manifolds with boundary, so this isn't some trick where "manifold boundary" is actually defined for a Hausdorff and second countable space that need not be locally $\mathscr H^n$.     To generalize,     Is a half-open interval and a point not in the interval's closure a manifold with boundary?
45057
I'm having problems with that:  Prove that M is a topological manifold.  $f:U \to \mathbb R^k$, U $\subset \mathbb R^{n}$ open, f continuous  M = $\{\left(x,y\right)\in \mathbb R^{n+k} \mid x \in U, y=f(x) \}$  Can anyone help me please?
65975
I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons.  So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again!   But if it’s all virtual photons, how do we get the difference between a magnetic and electric field?  [This question needs the tag "magnetostatics" as well].