In relativistic physics, the "Cauchy stress tensor" form the space-space components of the energy-momentum tensor. The time-time component is the energy density and the time-space components are the momentum density (times ##c##).
The interesting thing with GR is that when you take the...
This is just, because the classical approximations are accurate enough for these purposes (though the item concerning the Born-Oppenheimer approximation in solid-state physics is not always applicable). Nothing in this proves the necessity of a Heisenberg cut. It's hard to accept for...
I think the logic is the other way, i.e., in the approach to construct non-relativistic QM from the symmetries of the Newtonian spacetime model, you have the generators ##\hat{H}##, ##\hat{\vec{P}}##, ##\hat{\vec{J}}##, and ##\hat{\vec{K}}## for the Galilei algebra. As it turns out this algebra...
E.g., we can just use the old SI. There we had ##\Delta \nu_{\text{CS}}## and ##c## as in the new SI, but the kg was still defined by the prototype in Paris, and the A was defined via setting ##\mu_0=4 \pi \cdot 10^{-7} \text{N} \cdot \text{A}^{-2}##. Thus also ##\epsilon_0=1/(\mu_0 c^2)## was...
I never understood this argument. E.g., using a silicon detector to detect photons doesn't mean to use a "classical device". It's based on the photoeffect and thus relies on (at least semiclassical) quantum theory to be understood.
I see. So from this point of view there's some classical argument that the true Galilei symmetry is the central extension Bargmann algebra. That's anyway the case for quantum theory.
Although my name suggests otherwise, I don't speak any Dutch. I guess, however, maybe it's possible to have an...
No! Since 2019 we can't do this, because ##c##, ##h##, and ##e## are fixed within the SI to define the units used to do measurements. What's not defined but must be measured is now ##\epsilon_0##! So using the new SI it's ##\epsilon_0## that may have changed with time. So far there's no hint at...
I guess one can derive a Newtonian theory of gravitation by making the Galilei symmetry local, as one can derive Einstein-Cartan theory by making the Poincare group local starting from general relativity.
That's of course true, but one must say that these Lagrangians always describe not a fundamental system but some "effective theory". E.g., you can approximately treat the motion of the planets around the Sun of our solar system as the motion of the planet in the fixed gravitational field of the...
Noether's theorem tells us that momentum conservation follows from spatial translation invariance, which is a symmetry of Newtonian as well as special-relativistic (Minkowski) spacetime. Thus the dynamical laws on the fundamental level must obey this symmetry and that's "why" the total momentum...
There's no limit on measurement. I don't know, where this fairy tale comes from. It's often written in popular-science textbooks, but it's wrong.
What quantum theory tells us is that it is impossible to prepare a quantum system such that all observable take determined values. A state of...
You just need, e.g., a Penning trap with its magnetic field to decide which spin component (or rather which component of the magnetic moment ##\vec{\mu}## you want to measure. You can do that with an amazing accuracy. There's nothing mystic with this but very well understood (even analytically...
Well, arguing with the Newtonian approximation it makes some sense, because the two-body Kepler problem (inclusing the motion of the Sun) is equivalent to the uniform motion of the center of mass and the motion of a "quasi particle" with the reduced mass ##\mu=m_1 m_2/M## moving in the...
It's the wrong way to think about QT. You calculate the energy eigenvectors and eigenvalues, i.e., you determine in which states you particle has a definite energy. Since ##[\hat{H},\hat{p}]\neq 0##, usually these are not eigenvectors of ##\hat{p}##, i.e., the momentum has no definite value, but...
Well, it's the correct Hamiltonian modulo an additive constant. We simply have to measure the "cosmological constant" and renormalize the value of the "zero-point energy" to be in accordance with this meaasured value. All the parameters in the QFTs have to be fitted to experiment. In this sense...
It's a matter of definition, and within the SI ##c## is defined, and both ##\epsilon_0## and ##\mu_0## must be measured somehow. It's of course enough to measure one of them and then use the relation to ##c## to calculate the other.
Historically it was the other way around: the analogue of...
Hm, it's a bit hard to say from which epoch on in the table in
https://en.wikipedia.org/wiki/Chronology_of_the_universe
one can say we are "confident of our model". I'm conservative and think we can only be starting to be "confident of our model", when we can observe at least the state of the...
No, the charge of the electron is defined to be ##-e## in the SI. As detailed in #27 the ingredient of ##\alpha## that's not defined since 2019 by defining the units s, m, kg, and A, is the "permittivity of the vacuum", ##\epsilon_0## which is now to be measured. The same holds for the...
Well, this is not more than an appetizer to read a textbook to understand it. It's about 3/4 of a standard QM lecture in a short forum posting.
My recommendation as the introductory QM textbook is
J. J. Sakurai and S. Tuan, Modern Quantum Mechanics,
Addison Wesley (1993).
The more recent...
Sure, in calculation you always use angles in "radians", i.e., dimensionless numbers. I've not seen this formula with ##(M+m)## in the numerator (where does this come from)? Instead of ##(M+m)## the standard treatment gives the mass of the Sun. Then your ##\Delta \phi## is the angle of...
That becomes very clear in Dirac's formulation of QT. It describes QT on an abstract Hilbert space:
(a) A (pure) state of a quantum system is described by a ray in Hilbert space, i.e., a vector ##|\psi \rangle \neq 0## modulo an arbitrary non-zero complex number.
(b) An observable is...
I think the following nicely describes the issue with Bohr:
https://physicsworld.com/a/the-bohr-paradox/
Feynman for sure is another caliber. Here's always very clear and a role model of a "no-nonsense physicists" and also obviously a very diligent teacher as long as you restrict yourself to...
The problem with Bohr is that he is so unclear in his writing that it invites such "philosophing" about "what might the author have wanted to say", and that's why QT till today is often displayed as something mystic. I find Bohr' and Heisenberg's writings did a bad job in "interpreting QT"...
But that's correct, but still time is not a vector but either a component of a four vector given a basis ("coordinate time") or a scalar (like "proper time" of a particle).
A component is a component (a number in the field of the vector space, i.e., here in the field of real numbers), not a...
Bohr is right that the only adequate language to discuss physics is math. Only math is precise enough to communicate what we know by observation and from building mathematical theories ordering these observations in terms of fundamental general laws of nature. That's true not only for QT but...
You can as well argue in the same way you argued about the meter with the fine structure contant since it also
I think one can make the point clear from this example, using the new SI. In fact the new SI is the (almost) most transparent definition of a coherent set of (base) units we have...
How often do I have to repeat this! There is no single physical theory, where time is a vector!!! It's a real oriented parameter parametrizing the causal order of things.
In Newtonian mechanics it's just a real number, and at each point in time there's a 3D Euclidean affine space, i.e...
The "actual quantum stuff" in fact IS not impossible but it describes how Nature behaves. It's the classical description that is flawed outside the realm of its applicability. After all, that's why QT has been discovered 98 years ago by Born and Jordan, following an idea by Heisenberg!
No, now you misunderstood what I said. Of course, if you want to have a complete set of common eigenvectors of two self-adjoint operators these operators must commute. That's easy to prove: Assume that there is a common complete set of orthonormalized eigenvectors,
$$\hat{A}|a,b \rangle=a|a,b...
The divergences of higher-order perturbative contributions to the S-matrix elements in relativistic QFT come from a too sloppy use of the field operators, which are operator-valued distributions and thus cannot be so easily multiplied at the same space-time argument. This is however just...
But then you can get misleading intuitive ideas, which you must then correct when doing the correct calculations! It's better to forget wrong intuitions in learning natural sciences. In fact, it's the job description of the natural scientist to exorce wrong intuitions by doing research (and...
Another problem is the radiation reaction of point particles in classical electromagnetics. There's no consistent description. The best approximation seems to be the Landau-Lifshitz approximation to the Lorentz-Abraham-Dirac equation. QED solves this problem.
Work about the quantum (non-Markovian) Langevin equation using the Caldeira-Leggett model can imho be treated as an example for an application of the Heisenberg picture to an open quantum system already in the QM 1 lecture. See, e.g., Sect. IV in
https://doi.org/10.1103/PhysRevA.37.4419
https://en.wikipedia.org/wiki/Squeeze_operator
Of course ##r## must be dimensionless. Otherwise to put it as an argument into the hyperbolic functions wouldn't make any sense!
That's not true. An important counter-example are the annihilation and creation operators of the harmonic oscillator ##\hat{a}## and ##\hat{a}^{\dagger}## and the "phonon-number operator" ##\hat{N}=\hat{a}^{\dagger} \hat{a}##. Since ##[\hat{a},\hat{a}^{\dagger}## it's easy to show that if ##|n...
Yes, and now you can use the first equation and its adjoint,
$$\mathrm{d}_t \langle \psi(t)|=+\mathrm{i} \langle \psi(t) \hat{H},$$
to work out the time derivative
$$\mathrm{d}_t \langle \psi(t)|\hat{V}|\psi(t) \rangle = \frac{1}{\mathrm{i}} \langle \psi(t)|[\hat{X},\hat{H}]|\psi(t) \rangle,$$...
Perhaps one should add the interesting fact that the upper limit of the so observed neutron-star masses (around 2 solar masses) is an important constraint to figure out the equation of state of "strongly interacting matter". Of course this also has to do with the question, whether there are...
No problem, in everyday-life only Newtonian spacetime is correct, and there are no such troubles as time dilation, length contraction, and all this... :oldbiggrin:
I see! Indeed, the main obstacle to teach/learn QM for the first time is that it is a pretty abstract description of Nature, and that's why we have all these (partiall heated) debates in the "interpretation forum" (and in the real world since the first formulation of QT and Born's probability...
We had this question recently in the following thread:
That's true in the Heisenberg picture of time evolution. In this picture the "covariant time dervivate" is identical with taking the derivative of the observables wrt. time,
$$\mathring{\hat{A}(t)}=\frac{\mathrm{d}}{\mathrm{d} t}...
https://en.wikipedia.org/wiki/Mass_in_general_relativity#ADM_and_Bondi_masses_in_asymptotically_flat_space-times
The energy-momentum tensor is a local object, from which you can form scalars at a given space-time point, which represent observables.
I'm not sure whether ADM mass as a quantity...
Let's argue within non-relativistic QT for simplicity (and it's entirely justified for not too high ##Z##). Due to translation invariance the total momentum of an atom is conserved, i.e., the center-of-mass motion separates from the relative motion of the electrons and the atomic nucleus. The...
This is somehow ironic. Who the heck claims this? If there is one theory, which exhausts the locality principle to the extreme, it's GR. Even the fundamental space-time Poincare symmetry is "gauged", i.e., made local. The gauge group is general diffeomorphism invariance ("general covariance")...
QED can be constructed by "gauging" the mentioned global U(1) symmetry, leading to the introdoction of a "gauge connection", i.e., the electromagnetic potential. In principle can do this for any fields of charged particles, e.g., for pions (although there a more sophisticated version, called...