Hot answers tagged discrete
13
The answer to all questions is No. In fact, even the right reaction to the first sentence - that the Planck scale is a "discrete measure" - is No.
The Planck length is a particular value of distance which is as important as $2\pi$ times the distance or any other multiple. The fact that we can speak about the Planck scale doesn't mean that the distance ...
13
Put into one sentence, Noether's first Theorem states that a continuous, global, off-shell symmetry of an action $S$ implies a local on-shell conservation law. By the words on-shell and off-shell are meant whether Euler-Lagrange equations of motion are satisfied or not.
Now the question asks if continuous can be replace by discrete?
It should immediately ...
6
There are several forms of discreteness in quantum theory. The simplest one is the discreteness of eigenvalues and the associated countable eigenstates. Those arise similarly to the discrete standing waves on a guitar string. The boundary conditions only allow certain standing waves that nicely fit into the enforced region in space. Even though the string is ...
6
Position quantization in vacuum is forbidden by rotational, translational, and boost invariance. There is no rotationally invariant grid. On the other hand, if you have electrons in a periodic potential, the result in any one band is mathematically the theory of an electron on a discrete lattice. In this case, the position is quantized, so that the momentum ...
6
You mentioned crystal symmetries. Crystals have a discrete translation invariance: It is not invariant under an infinitesimal translation, but invariant under translation by a lattice vector. The result of this is conservation of momentum up to a reciprocal lattice vector.
There is an additional result: Suppose the Hamiltonian itself is time independent, ...
5
Let's try and make things more precise, step-by-step.
There's no such thing as "particle-wave duality": the name-of-the-game is "Quantum Field Theory". This paradoxical notion of a possible "duality" only happens when you don't use the appropriate framework to describe your Physics. Therefore, it makes no sense to speculate on what would happen if ...
5
Charge discreteness is the statement that charge comes in packets which are of size 1 electron charge. You can understand this as a consequence of the fact that everything is made of particles with definite charge, but then it leaves the question of why the antiproton and the electron have the same charge.
All particles have opposite charge to their ...
5
The answer is essentially what Kostya has pointed out:
Position is quantized but has a continuous spectrum of (generalized) eigenvalues because the canonical commutation relations on position and momentum forbid that both of them be bounded operators (and act on finite-dimensional state spaces) by Stone-von Neumann theorem. This means that, given general ...
4
It depends on what one defines as "position".
In crystals, for example, there exists a three dimensional grid on which the atoms are allowed , stacked in unit cells, so there is quantization in space to be observed, and quantum mechanical solutions are involved . More numerous are the interference solutions of qm waves which also display a quantization of ...
4
So, suppose you have an eigenstate of $\hat{x}$:
$$\hat{x}|\psi\rangle = x|\psi\rangle$$
Now let us act with $\hat{x}$ on $e^{i\hat{p}\delta}|\psi\rangle$, and use this formula (I have $\hbar=1$):
...
4
Most of the people I know who think about quantum gravity study string theory, and they aren't assuming that spacetime is "quantized". They have a slightly more subtle picture in mind, in which space and time are continuous (in so far as they make sense at all) but physical effects prevent you from seeing features which are smaller than (roughly) the Planck ...
4
If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples:
The finite number of states in a compact region of phase space. See e.g. this Phys.SE post.
The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular ...
4
You have re-discovered the Arrow Paradox from Zeno of Elea (5th century BC).
Physics doesn't deal with that. There is obviously movement, and all we do is trying to resemble and predict it with mathematical models. That is how we face the question of understanding nature.
You may find a better answer among philosophers than physicists for that question. ...
3
Considering that the answers to the question you link are available for reading you will see that position is also quantizable, as crystal structure unequivocally demonstrates.
Quantum mechanics has a mathematical formulation that intrinsically allows for quantization of any variable entering the formulation, depending on the boundary conditions set to the ...
3
I think I got the answer now. The main idea is this: When we gauge continuous symmetries we identify all the states $$A^\mu=A^\mu+\partial^\mu\chi$$ (which are continuously many) as a unique physical state.
When we gauge a discrete symmetry (let's assume it's generated by $\theta$) we identify all the states
$$|\Psi\rangle=\theta^n|\Psi\rangle$$
where ...
2
As was said before, this depends on what kind of 'discrete' symmetry you have: if you have a bona fide discrete symmetry, as e.g. $\mathbb{Z}_n$, then the answer is in the negative in the context of Nöther's theorem(s) — even though there are conclusions that you can draw, as Moshe R. explained.
However, if you're talking about a discretized symmetry, i.e. ...
2
Let me try to address your questions, even though just the first one seems quite heavy in itself.
Spacetime Discreteness: let me give you links to references that are relevant to your questions, than i'll make some general comments. An Introduction to Spin Foam Models of Quantum Gravity and BF Theory; Spacetime in String Theory; The quantum structure of ...
2
Quantum mechanically, the velocity
$$\hat{{\bf v}}~=~\frac{\hat{{\bf p}}}{m}$$
is (like any physical observable) a Hermitian operator. (In this case a vector-valued Hermitian operator.) Possible values/outcome ${\bf v}\in \mathbb{R}^3$ of the velocity are given by the corresponding eigenvalues of the operator $\hat{{\bf v}}$.
Whether the eigenvalues are ...
2
There's a fundamental problem with speaking about whether or not "velocity is quantized," say for an electron in the hydrogen atom. If we were to model of time-evolution of the state of the particle as a trajectory in three dimensional space, then we could define its velocity and perhaps start asking questions about it, but in quantum mechanics, the state ...
2
I think you all your steps are correct, I would suggest adding units though, otherwise adding $\vec{x}$ and $\vec{v}$ can be misleading.
For step 3. You can just calculate $a = F/m$, where $F$ can either be gravitation $F=m g$, so it does not depend on $x$, or for example for a spring dependent on $x$, so $F = -k x$, here you need the next position from ...
2
Your question is still somewhat a matter of philosophy and the particular interpretation of quantum mechanics you follow because we can't yet probe probe to the tiny distances needed to truly understand this.
You question is somewhat similar to Zeno's paradoxes which can be solved mathematically via calculus. Unfortunately calculus assumes everything is ...
2
As far as we can tell (up to energy scales we've measured so far), spacetime is a nice and smooth manifold. It might happen that the smoothness is approximate and spacetime is discrete at a much more microscopic scale, or it could turn out that spacetime is smooth all the way through. Short answer: We don't know.
About the notion of energy quantization: ...
1
OP wrote (v1):
If their magnetic charges are rationally related everything's alright, but if they aren't [rationally related, then] satisfying both quantization conditions is impossible.
Correct.
So in order to explain electric charge quantization you have to assume either magnetic charge quantization or that there's exactly one magnetic monopole ...
1
This is a comment turned into an answer.
Classically one can define an angular momentum of a straight track with respect to any axis as
The real meaning though comes from rotational states about a central axis. In this case a potential exists which constrains the particle to revolve about the axis.
Quantum mechanically one defines an angular momentum ...
1
Let us talk about the first question, first: Can "moving objects have infinitely decimal place velocities?"
Most certainly yes!! The simplest example is the motion of a particle on a circular orbit of radius R. Let us assume the radius of the circle is $R=10m$ exactly (as exact as we can make it,) and let the partcle make ten revolutions per second, i.e. ...
1
If you want you can go back to Planck's derivation of the black body energy spectrum, otherwise known as Planck's law, as well as Einstein's use of Planck's work in his explanation of the Photo Electric Effect (which garnered him the Nobel prize) in order to first understand some of the experimental motivation. However, to understand the roots of quantum ...
1
Both the first linked video to the first version of the question(v1), and the second link video in later versions of the question, is about deriving Euler-Lagrange equations from the principle of stationary action.
1) Susskind does not mention time discretization in the first video. Time $t$ is there a continuous real parameter throughout the video, and the ...
Only top voted, non community-wiki answers of a minimum length are eligible
