Z-Man

Wrong. Very different. In your boat experiment, you do not observe an interference pattern. * In the particle experiment, you usually do. Unless, as Starhawk points out, you modify the experiment to find out which slit the particle went through, then the interference pattern vanishes. *: It is, of course, in principle possible to set up a double slit experiments with boats where you do observe an interference pattern. But not if your setup includes a warm human standing by the slits who chooses to observe the boat go through or not by having his eyes open or closed.

PB666: Declare your AddNumbers method static. Then you can call it without creating an object for it, like so: Class_AddNumbers.AddNumbers(...); The closest analoge to your methods/functions in VB that you just write and call in a single module probably would just be private static functions in C#. Callable without an object, but only from other functions from the same class. Same in C++, only there, you can also just have static functions in the .cpp file without declaring them in the class header. Note: 'static' means about seven different things in C++ depending on context. And even though the two 'static's here work towards the same goal, they're totally different.

Original article was here: http://www.sciencemag.org/content/332/6034/1170 Nope, sorry. I was asking about an experiment that measures which slit the particle goes through (in the best case). That article describes an experiment where they measure the average momentum the particles have as they leave the slits. Totally different thing, perfectly compatible with the standard interpretation and quite easy to calculate using it. Still, a splendid experiment and worth a read; it boils down to them being able to measure the entire waveform, both the amplitude and phase (differential). Using a lot of repeat measurements, of course. Quote from the actual article, by the way: "For the experimentally reconstructed trajectories for our double slit (Fig. 3), it is worth stressing that photons are not constrained to follow these precise trajectories; the exact trajectory of an individual quantum particle is not a well-defined concept." Emphasis mine. Those trajectories are the ones you want to be "real".

If you're so convinced that in the double slit experiment, the particle really only goes through one of the slits, please give an example where this has observable consequences that the standard implementation of QM cannot explain. For example, can you determine which slit it goes through and still not lose the interference pattern?

Fun fact: The duration of a low orbit (negligibly far away from the surface) around a spherical object depends only on natural constants and the density of the sphere. So yes, this would work. But not on the ISS itself, as KerbMav says; air currents are going to mess everything up. Near the ISS is also not a very good place; tidal forces from Earth are going to be of equal magnitude to the gravitational forces between your bodies even if they are as close as possible. See Roche Limit. There is some leeway if you make your bowling ball out of gold, but your orbits are not going to be very stable. You'd have to do it further away from Earth.

Sorry, but you're talking mostly gibberish. In the realm of Real or Complex or Integer or Natural numbers, the answer is simply that it is not a legal operation, and that's the end of it. Yes, you can add infinity to your realm of numbers to "solve" this problem and make division a globally defined continuous operation, but you inevitably run into problems with the other axioms you want numbers to obey.And yes, twice infinity equals once infinity. Look up the Hilbert Hotel. Yep. Your rule "0 to the power of anything is 0" is not an actual rule. For Natural numbers (0 included) a and b, the only rules are: 1. a0=1 for all a 2. ab=a*ab-1 for b > 0 and all a. For integer/whole numbers, you extend rule 2 to all b and you're done. So no argument there, according to the real rules, the only valid value for 00 is 1 in those domains. Now, for Real numbers, if you want, you can define ab as a completely new function and depending on how you do it, it can be consistent to have 00 equal either zero or one. However, if you operate on real numbers and a result of your calculation is 00, that means you have at some point not done a proper limit process; and depending on what that proper limit process would be, the true answer you seek could be anything (ab is only continuous and well defined for all b if a > 0), so the value you assign to 00 is somewhat arbitrary and of little significance. Usually, one would pick 1 for consistency, then. See question 1. Not legal. Possible to define away, but you get other problems. If your calculation result is 0/0, then you again forgot a proper limit process, in which case I refer you to kerbiloid's answer.

Got the same issue in my 0.9 career game and managed to repair my persistence file. The lines in the log that tipped me off were [EXC 11:20:23.798] NullReferenceException: Object reference not set to an instance of an object Contracts.Templates.RescueKerbal.OnSave (.ConfigNode node) Contracts.Contract.Save (.ConfigNode node) Contracts.ContractSystem.OnSave (.ConfigNode gameNode) ... Over and over, I assume for every frantic click. The fix was to remove this section from persistent.sfs: CONTRACT_FINISHED { guid = 4f7b2cd2-65b1-4853-8539-c0969ab9f5a0 type = RescueKerbal prestige = 1 seed = 527793544 state = Completed agent = Kerbodyne deadlineType = Floating expiryType = Floating values = 21600,46008000,7500,30000,7500,0,15,50,23953.3750823963,9081.68962682668,46017081.6896268,35595.1529716382 body = 1 kerbalName = Elming Kerman PARAM { name = BoardAnyVessel id = Board enabled = False state = Complete values = 0,0,0,0,0 title = Get Elming Kerman aboard a vessel failOn = Any winOn = All } PARAM { name = RecoverKerbal id = Recover enabled = False state = Complete values = 0,0,0,0,0 title = Return Home and Recover Elming Kerman failOn = All winOn = All } } Which was the contract for rescuing a kerbal that I subsequently fired. I assumed that with the limited slots available, that would be a requirement for getting more rescue contracts.

They are not troubling me. Two problems in your line of argumentation: The momentum you speak of does not exist here, or is insignificant. Look at the particles that move against the warp ship with low global velocity. They also pass the center of the bubble at low velocity relative to the ship. The warp bubble simply carries them along. Sure, movement does exist and is relevant, but not momentum. More importantly, since you bring up particle physics, this very much is a scattering problem. Do you know a nontrivial particle-particle scattering problem where a pure x-t treatment gives even approximately correct results? I don't. Take classic Rutherford scattering: your prediction would be that the alpha particle is always reflected, the scattering angle is 180 degrees 100% of the time. The full 3+1d treatment is not just a refinement of that, the 1+1d treatment only covers a very small subset of the full problem and is dead wrong on the whole. Only when the scattering target is much wider than the particle beam and the wavelength of the incoming particle is larger than the microscopic length scale of the target (tunnelling problems, mirror reflection) and it is impossible for the particle to miss the target, it can be adequate. That's not the case here. See, in the reduced x-t problem, there are only three possible outcomes: the particle passes through, it gets reflected or it gets caught. Here, for the case where the ship moves > c, the reflection outcome is forbidden (Sort of. The particle could leave the bubble in the back with reversed global velocity, but that doesn't seem to happen either), leaving only two possible outcomes. There is not even room for deflection in the model, the outcome that is the most likely in all other scattering experiments (well, second most if you count 'no interaction happens'). The 1+1d reduced model simply can't claim to be universally even approximately right. Now, what we CAN read from the paper is that given the right initial state, there is the possibility that particles interacting with the bubble can spend an arbitrarily long time caught in it and gather an arbitrarily high momentum before they are released.

No. The relevant variables of the particle before the encounter are just its regular 3-momentum(*) in the reference frame the warp drive started at and the spot it hits the bubble. Both will deviate significantly from the case studied for a generic particle. *: that's even in their analysis. Particles with a negative initial x-velocity all just pass through the bubble, only those with a positive initial x-velocity get caught: "We found that when a bubble catches up to a particle with a nonzero global velocity in the same direction as the ship, ut diverges, which implies <stuff I can't paste here>."

Quote from the article: "The following analysis is restricted to the t-x plane". And by "following", they mean "whole". They only consider particles moving on the symmetry axis. Drawing conclusions from there to the general case is like claiming a sphere cannot possibly move through an incompressible medium because all the liquid on the symmetry axis will block it.

Almost just like here. If the artificial gravity comes from the ring's rotation (I'm not into the Halo canon enough to know), it does not stop if you lift of as long as the speed relative to the surface is small compared to the ring's rotation speed. Essentially, if you drop a brick from the flying airplane, it would go on in a straight line all right, but the ground would accelerate towards it. The net effect is the same as if there was gravity: ground and brick accelerate towards each other.

He is talking about the rest mass (I think. I don't like reading papers that SHOUT at ME. But the m in the final equation is that). That is indeed independent of the observer, it is an invariant quantity of an isolated system. All observers can agree on it. The quantity that is increasing with velocity is the inertial mass, the one that appears in the equation F = m a; different observers assign different values to it. To distinguish the two, the rest mass is usually labeled m0.

And even for black holes, there are (believed to be) limits on how much charge they can have: http://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric#Charged_black_holes The limit on the charge/mass ratio is of the same order of magnitude as for regular gravitationally bound objects.

What do you mean by 'timelike dimension"? In the context of QTF on a fixed background manifold, we have that as the interior of the future lightcone at every event (spacetime point). What we no longer have since 1905 is a meaningful global time coordinate. Usually, one can still at least define a global coordinate that is timelike in the sense that it is smooth and always increases along every futurebound timelike or null world line. But even that goes out of the window if there are closed timelike curves. Ultimately, you'll have to settle for the definition that "time is what the clock measures". That's why we consider the Planck time to be the smallest possible time: Any clock that could measure shorter times would need to be so massive/energetic and small that it would collapse into a black hole. And it also leads to a good definition of time travel: A situation where a non-clonable system containing a clock can be influenced by itself with a more advanced reading on the clock. And the two prescriptions for resolving time loops fit. For a suitable definition of 'itself' in the case of Deutsch's method. That a model appears static and not moving is not a new problem, either. A pendulum is described as having the motion x(t) = A sin(w t) + B cos(w t). That's one static function (with parameters). No actual movement visible, nothing is happening. Is there anything specifically new here? And I'm afraid you can't apply the Copenhagen measurement prescription inside a time loop (or, for that matter, a non-reversible MW split). It is not at all clear how it would be supposed to work; it applies to an external observer doing the measuring, and who would be external to the time loop, yet close enough to measure something caught in it? Instead, if you want a measurement from inside the loop, you have to do the complicated thing and model the measuring apparatus as part of the system. It can be a mock apparatus that could not exist in reality, it just needs to be able to measure and store the same thing. Then, you model how the measuring apparatus goes out of the time loop. Only after all is over, you measure the state of the measurement apparatus with your chosen interpretation of QM. And it can matter a whole lot more than normal what you measure and how you treat your equipment. In Deutsch"s method, it matters whether you push your apparatus through the time loop or not after it has done its measurement. In Lloyd's method, what you measure has a lot more influence on the future of the system than usual, and quantum erasers no longer work.

Well, yeah, we haven't observed anything close to it, so all we can do right now is explore the mathematical possibilities. No fixes are required, actually: What you think are "rules" for MWI are actually consequences of the way time evolution usually works. You take an in-state and via a linear and unitary operation, you transform it to an out-state. A lot of the rules of QM we take for granted derive from that: the no-cloning theorem, the no-communication-via-entagled-states theorem and conservation of total probability. And locally, time evolution is still unitary. However, globally, it seems like it goes down the drain, just like causality itself: for the larger region of spacetime under consideration, it is lost in both models. And without unitarity, the consequences also get invalidated.And yes, we are ignoring those potential problems. They may be avoidable. Energy conservation is not a problem in the given context: it's another thing that is valid locally, but does not have to be true globally. That's invalid interpretation of calculation steps as reality. Yeah, Deutsch's model looks a bit like that: "First, you take the state of the universe. You use that continuously to feed the time loop until it has settled down. Then, you take the state of the loop and set it as the initial state of the loop in the real universe, then you run that." But that's not relevant. You could just as well interpret the regular time evolution as "Next Tusedayism", because it allows you to progress time one week at a time, each week copying the end state of last week to the initial state of next week. What matters are the results that come out, and the proper way to analyze them is to apply the formalisms (which are interpretation free and don't care whether you use MWI or Copenhagen) to test systems. And then you interpret the result like you would do with any other out-state. You have to treat the region with the time loop as a black box. And to determine how events inside the loop influence the outcome, you play with different in-states or time evolutions and see how things change.The time travel aspect simply comes in because we partially feed the out-state back into the in-state; it's similar to a regular feedback loop. It is actually that, the multiple worlds/alternate universes term is just... marketing. Only very rarely is there a need to consider the whole universe. Usually, the states of the photons on your experiment table are all you need to model in your system. Time travel considerations get messy there because you need to include everything that can potentially loop.

No problem, glad I could help without actually doing the integration work myself

A couple of identities between sinh and cosh? Jolly good. Choose b so that the expression under the square root becomes one side of one of them. And I suppose you also know the derivatives and integrals of sinh and cosh themselves.

That's just a number. If you want a better hint from computer algebra, put in a variable as upper limit, or let it compute the indefinite integral. edit... wait, why is there no "a" in the result you quoted? There should be. And, well, what do you know about sinh and cosh?

Try substituting x = b sinh(t) with a fixed parameter b. For a good choice of b, everything is going to turn into cosh. Hidden hint, only use if needed: Use 1 + sinh(t)2 = cosh(t)2 to simplify the nasty square root.

The short answer: Time travel changes the rules The long answer depends on what you mean by "detection" and which time loop resolution mechanism is considered. There are at least two of them. I'm ignoring any problems they have here, for brevity. The short long answer for both is that they introduce nonlinearities into the time evolution. The long long answers: In Lloyd's model, which is the one you get when you simply apply the usual rules of Quantum Field Theory on a fixed spacetime with loops in it, you detect them only as you usually detect alternate worlds, by their interference patterns that influence the final outcome. The difference that the possible time travel makes is that even those alternatives that usually would not cause interference because they're too far apart can now interfere. The trace operation can mash them together, and it ensures global consistency. An external observer will always see the same thing coming out of the time machine than later gets in. Still, in this model, unless you actively set up paradoxes in some worlds, interference between them is as weak as ever, at least after averaging over macroscopically indistinguishable initial states. In Deutsch's model, it's simpler. You actually travel back into a different world than you started from. What an external observer sees leaving the time machine does not need to match what he later sees entering it in the future. There are different ways to interpret the equations, though, and the alternate worlds are not necessarily the ones from MWI. You can also see them as actual clones. Of course, the interpretation does not matter for the end result. Needless to say, if we get hold of a couple of closed timelike curves, we can experimentally distinguish between the two alternatives, they are not equivalent. In the context they are formulated in, there is nothing that makes one preferable over the other, though I imagine Lloyd's trace method would mend easier with whatever Quantum Gravity theory we come up with. Nobody is arguing over them here, you just keep asserting they are both incomplete based on falsehoods Again. We now quite clearly know what a measurement is, at least we can construct an operation for every quantity we want measured that definitely is a measurement. If you entangle your system with a measuring apparatus and leave the apparatus alone for the rest of the experiment (no interaction with the system or other measurements, no quantum erasing, it helps if it is a macroscopic apparatus, but not a requirement), you measured it. Likewise, a split is just writing the state as the sum of two or more states. Every measurement in the above sense comes with a well defined, useful split you can do.Yes, for a given setup, it's not always easy to tell whether it is to be considered a measurement and if yes, of what. That's also true in classical physics and general statistics, so I would not hold it against QM.

Yeah... that. I'm not even sure what Ben is trying to say. The first triangle also does not exist. It is a mathematical construct. No. Copenhagen prescribes a collapse after every measurement, not every point. There has been a lot of calm and rational (cough) debate about what a measurement is, but if done right, it loses nothing, it just transforms a superposition that can still interfere into a classical probability distribution which can't. And the transition happens when interference would become practically impossible.And MWI does allow for interference (is that what you mean by interaction? It better be) between the branches. It's very important early on in the life of new branches. Only later, interference gets increasingly unlikely. I really don't get where you get the idea from that any of these are inadequate tools for building models. You should probably look up at least a bit of the actual math involved and how it is applied to simple systems such as the spin of a single electron.

None are scientific. The question of what is real is a philosophic question. IIRC, you never gave your criteria, so I assumed "parts of the model that influence the outcome in a significant way". Which I personally consider incomplete and too strict, here is why. The way the universe is expanding according to our current best understanding implies that there is a cosmological horizon: A "sphere" around us where everything inside the sphere can potentially still influence us and everything outside can not (assuming FTL is impossible) and also cannot influence anything inside the sphere. And locally, that horizon is sweeping inwards towards us with the speed of light. As I write this, maybe another star has moved from the inside to the outside. Ten minutes ago (in its proper time), it could still send light that could reach us, now it can't. There is no reason to assume anything bad has happened to it, yet if you only grant reality to things that can possibly be observed, it stopped existing. That does not sound right to me. I'd allow extrapolation of reality across space and time within reasonable bounds. If you don't, fine by me. Everything in theoretical physics is just that. We have models with no ambition to match reality exactly, just to reproduce experimental results and make predictions. Any science talk that implies otherwise is just shorthand because constantly saying "Our current best model, the one reproducing experimental results most accurately, for this system is..." is so much more tedious that just saying "This system is...". And quantum mechanics isn't even a model, it's a framework for models. Well, when do the branches stop existing, then, and why? Same situation as with the star vanishing behind the horizon, really. Yeah, taking a purist scientific view, they don't aim for that. But here's a rule of thumb IF you want to attribute reality to the objects they operate on:If you think reality is continuous, that nothing can go from existing to not existing without a good reason, then MWI is the "correct" interpretation. It keeps superpositions "alive" for as long as you care about them. If you think you are real and by extension everything that can interact with you or interact with things that can interact with you and so on, Copenhagen is the one for you. The collapse culls off branches that do not matter anymore. If you have to do actual calculations, you use whatever fits the problem better.

The triangle analogy is good for the kind of multiple worlds laymen usually think about: The world where you won the lottery last week, or the one where we all ride to work on dinosaurs. You will never interact or observe these worlds from where we are now. Therefore, you are free to consider them not real. The multiple worlds physicists are concerned with usually do not differ so drastically. It's the world where the electron takes the right slot vs. the one where it takes the left slot. Those worlds all do influence the outcome of the experiment, the experience of the one doing the experiment. For pragmatic definitions of "real" (those where you ignore the possibility that you're a simulation of a brain in a jar dreamed up by some guy), they are as real is it gets. Yes. Well, depends on what you mean by 'maths'. MW is a2 + 2 ab + b2, Copenhagen is (a+2. They look different, but the result is the same.

No. Even for small commercial projects, you can use the free edition. You won't have access to the full feature set, but you probably would not miss it.

Ben, you misunderstand what the difference between MW and the other interpretations is. Let me give you an equivalent example: We label our electric charges so that electrons carry negative charge and protons carry positive charge. That's the analogy for the Copenhagen interpretation. But we could just as well label the electrons positive and protons negative (and switch the charges of other particles around). That would be MW. Both yield the same predictions for all experiments if you take into account that some axes will be labelled differently. Either both views are right or both are wrong. You will not find an experiment that distinguishes between the two. It's just that for some problems, causality loops being one of them, MW turns out to be the better tool for understanding what may happen. CNR, doubly relevant: