For the traveller it’s not the same distance; due to length contraction it’s a smaller distance, and so takes less time. For the observer who sees the distance as a full light year it appears to take more than a year, but the traveller’s clocks appear slowed down and so will advance by less than that. (The time experienced by the traveller will still be more than a year unless the travel speed exceeds , ie just over 70% of the speed of light.)
Category: All
Timelike vs Spacelike
If two distinct events are such that there is any inertial frame in which they have zero spatial distance between them, then there is no frame in which they are simultaneous and so they are said to be “timelike separated”. This is because the frame in which they have zero spatial separation corresponds to an observer who sees them both happening at the same place one after the other; and for any other inertial observer, the time between them is also nonzero (since for any v<c the Lorentz contraction factor is never zero).
On the other hand, any two events which are seen as simultaneous by some inertial observer (which is different from being seen simultaneously by that observer!) are said to be “spacelike separated”. But the appearance of simultaneity is relative to the observer and only happens in one particular frame. Other inertial observers won’t see the events as simultaneous but all will agree that it would take faster than light travel to see them both at the same place – eg to actually be present at both of them.
GR and Twin Paradox
General relativity theory does not “solve the twin paradox of special relativity”.
Despite being “paradoxical” in the sense of contradicting our intuition that the time ordering of separated events should be absolute, there is no “paradox” in the sense of internal contradiction in special relativity. Nor is it impossible to analyse the experience of an accelerated observer in special relativity; and in the case where one twin is turned back (eg by a rocket) this leads to the conclusion that both agree on the difference between their ages when reunited.
The only case in which general relativity is needed is when the acceleration is due to gravity (eg by slingshotting about a massive star) – and so does not lead to the feeling of applied force by the freely falling traveller. But as soon as gravity comes into the picture we are no longer talking about special relativity.
Lagrangian Described Simply?
For particles moving independently and freely subject only to specified forces, the motion of each particle obeys Newton’s law F=ma which can be written in terms of the Cartesian coordinates in the form x”=(1/m)dV/dx where V is the potential energy (and the form is the same for all components).
For particles constrained to be parts of a rigid body we might prefer to use more natural variables like the angular orientation of the body, but then the equations become more complicated (eg with centripetal and coriolis “forces” coming in, so the equation for the r component is not just
r”=(1/m)dV/dr ). And it is often not easy to work out the translation of Newton’s eq into the new coordinates.
The Lagrangian approach rewrites the equations of motion in an equivalent form which has the same structure in all coordinate systems – which turns out to be given for every component q by d/dt(dL/dq’)-dL/dq=0 with L=T-V where V is again the potential energy and T the kinetic energy (which in Cartesian coords would be (m(x’)^2)/2 ). This provides a systematic way of getting the appropriate equations rather than having to work them out in terms of the desired coordinates by a complicated conversion process from the Newtonian form.
It turns out that the Lagrangian equations are also equivalent to a variational principle (as mentioned by other responders) – namely that the actual trajectory be a stationary point (eg min or max) of the path integral of the Lagrangian function L.
Source: (1001) Alan Cooper’s answer to In layman’s terms, what is a Lagrangian? – Quora
Feynman Diagrams – Quora
Feynman diagrams are labels for a way of breaking a complex integral into parts, each of which can be evaluated by applying fairly simple rules. As such they make it much easier to evaluate expressions which originally looked very difficult, and so make it possible for theoretical predictions to be made (and checked) much more quickly (and by people with lower technical skill levels). This led to a rush of progress in the theory and application of Quantum Electrodynamics, and paved the way for much more rapid progress in applying and testing other Quantum Field Theories (such as those used to model the strong and weak interactions of elementary particles).
Source: (1001) Alan Cooper’s answer to Why did Feynman diagrams revolutionize particle physics? – Quora
Irritating Quora Answers
It is still an open question as to whether or not the theory that is commonly used for calculations in high energy physics can be interpreted as an actual field theory in the sense of being able to produce well defined predictions for experiments at arbitrarily high energy-momentum scales – or equivalently arbitrarily short distance-time scales (which is why such theories are sometimes called strictly local field theories).
In fact no interacting strictly local quantum field theory has yet been constructed in a four dimensional spacetime, and some theories which do make sense in two and three dimensions have been shown not to do so in four. But the Yang-Mills type theory that underlies the Standard Model has not been either excluded or shown to be incompatible with the conditions of such a theory.
Current approaches to actual calculations combine perturbation theory for versions of the theory with cutoffs with a non-perturbative “renormalization group” analysis of how the predictions vary with changes of cutoff and coupling parameters. These may be sufficient for most practical purposes but answers which portray that as somehow winning a competition against the question of what mathematical framework is appropriate for working at all scales are in my opinion both misleading and pathetically misguided.
Spin
Alan Feldman has a lot of good Quora answers.
Source: (1001) Search
Social and Physical Determinism | alQpr
Hermitian but non-SA in QM
need to add a physics example of a non esa Sturm-Liouville problem where different boundary conditions determine different physics for a physically intuitive reason
Decay times of excited atomic states
The time spent in the excited state is a random variable which can have any positive value but whose expectation value depends on the energy drops to lower energy unoccupied states (with the one with closest energy giving the dominant contribution). Since the drop in an isolated atom can only happen via a transfer of energy to the electromagnetic field, the actual formula results from a quantum field theory calculation involving the strength of the EM coupling constant (and would be infinite if that coupling constant were zero). But I suspect that the result turns out to be consistent with the Heisenberg uncertainty relation $#\Delta E \Delta t \gtrsim \frac{h}{4\pi}#$ and that the bound is similar for all cases of a top level excited electron in a neutral atom so that we can say the expected lifetime is inversely proportional to the energy drop to the nearest available level (and since the levels tend to get more closely spaced the higher we go this is also consistent with more highly excited states of the same atom decaying more quickly – albeit usually not directly to the initial ground state).
For more info just do a Google search for something like ‘atomic excited state lifetimes’.