Hot answers tagged noethers-theorem
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At the physics 101 level, you pretty much just have to accept this as an experimental fact.
At the upper division or early grad school level, you'll be introduced to Noether's Theorem, and we can talk about the invariance of physical law under displacements in time. Really this just replaces one experimental fact (energy is conserved) with another (the ...
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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 ...
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The symmetry you are asking about is usually called a scale transformation or dilation and it, along with Poincare transformations and conformal transformations is part of the group of conformal isometries of Minkowski space. In a large class of theories one can construct an "improved" energy-momentum tensor $\theta^{\mu \nu}$ such that the Noether current ...
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In basic Lagrangian mechanics (of the sort that is covered in a sophomore-level classical mechanics class), no it doesn't. The reason is that time plays a special role in the basic Lagrangian theory: it's the only independent parameter, which everything else is expressed as a function of. This is related to the fact that the action is the integral of the ...
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If the conservation law is general, meaning that it isn't specific to one motion, but conserved in a general configuration, then the answer is yes. This follows from the theory of canonical transformations in classical mechanics.
First, consider a perfectly triangular symmetric initial condition of three particles arranged on an equilateral triangle with ...
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Go out and discover those "other explanations" (and accumulate sufficient supporting evidence, of course) and you can laugh at the dark matter specialists.
Until then dark matter is the simplest hypothesis on offer that explains multiple observations in one go (galactic rotation curves, cluster dynamics, super cluster dynamics, the bullet cluster, the ...
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The missing mass problems are several sets of observations that could be explained if there were some matter that has mass (interacts with other matter via gravity) but does not interact with light. The same distribution of this missing mass would explain all of them. All competitors that have been explored fail to explain at least one.
I only partially ...
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I think you make some quite confusing statements, so let me be a little bit too explicit.
First, for any system whose laws don't depend explicitly on time one obtains conserved energy as an integral of motion.
Central force systems are invariant under the action of $SO(3)$. This is so because both kinetic and potential energy are scalars. Noether's theorem ...
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thank you for the nice question. It directly relates to the topics of conformal field theories. I found a very nice thread in another forum where I guess your question has been answered.
Nevertheless, I will try to summarize the main points here and maybe add some points.
Symmetries in General Relativity
In general relativities, symmetries correspond to an ...
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As lurscher says, semigroups are not too interesting in physics. They're not really "symmetries". An important physics example of a semigroup in physics is the (misleadingly called) "Renormalization Group" that allows us to derive effective laws for long distances from the short-distance ones, but this "integrating out" or "flowing" is irreversible - which ...
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Here I would like to mention the notion of quasi-symmetry. In general, if the Lagrangian (resp. Lagrangian density) is only invariant up to a total time derivative (resp. space-time divergence) when performing a certain off-shell$^1$ variation, one speaks of a quasi-symmetry, see, e.g., J.V. Jose and E.J. Saletan, "Classical Dynamics: A Contemporary ...
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I) For a mathematical precise treatment of an inverse Noether's Theorem, one should consult e.g. Olver's book (Ref. 1, Thm. 5.58), as user orbifold also writes in his answer(v2). Here we would like give a heuristic and less technical discussion, to convey the heart of the matter, and try to avoid the language of jets and prolongations as much as possible.
...
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Whether your current $j^\mu$ is conserved off-shell depends on your definition of $j^\mu$. If you define it via the Dirac and other charged fields, it will only be conserved assuming the equations of motion.
However, if you define $j^\mu$ via
$$ j^\mu = \partial^\nu F_{\mu\nu}, $$
i.e. as a function of the electromagnetic field and its derivatives, then ...
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Momentum conservation doesn't automatically follow from translation invariance. That only happens because of special features of physical laws, so if you want to prove that translation invariance implies conservation of momentum, you'll need to use some principles of physics to do it. Make up new laws that break those principles and you can indeed have ...
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Global invariance under $SU(N)$ is equivalent to the conservation of $N^2-1$ charges – these charges are nothing else than the generators of the Lie algebra ${\mathfrak su}(N)$ that mix some components of $SU(N)$ multiplets with other components of the same multiplets. These charges don't commute with each other in general. Instead, their commutators are ...
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The guiding principle is: "Anomalous symmetries cannot be gauged". The phenomenon of anomalies is not confined to quantum field theories. Anomalies exist also in classical field theories ( I tried to emphasize this point in my answer on this question) .
(As already mentioned in the question), in the classical level, a symmetry is anomalous when the Lie ...
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Yes, this is the opposite of Noether's theorem. So let's call our conserved quantity $A$ (we will consider just one conserved quantity for starters) and begin with $\left \{H, A \right \} = 0$ law for conservation. Because of the connection between Poisson bracket with flows on the phase space this tells you both that $\mathcal{L}_{V_H} A$ = 0 ($A$ is ...
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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, ...
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Noether theorem is as valid in CM(*) as in QM(**). It deals with conservation laws and symmetries. In CM the variables are certain, in QM they may be uncertain.
HUP belongs to QM and gives a limitation on canonically conjugated variable uncertainties in a given state.
If some variable in QM is uncertain, it does not mean its expectation value is not ...
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The intuitive argument for Noether's theorem, which is also the best completely precise argument for Noether's theorem, appears in Feynman's popular book "The Character of Physical Law". I will reproduce the argument, but not the diagram. The diagram is two parallel squiggles with a line connecting them at the top and at the bottom. These represent a ...
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Remember that voltage is always expressed as a "potential difference." You can't measure the absolute value of voltage because everything is invariant when you add a constant voltage everywhere. That expresses a symmetry just like time translation invariance.
When you bring in the magnetic field this invariance or symmetry can be generalised to a bigger ...
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While Kepler second law is simply a statement of the conservation of angular momentum (and as such it holds for all systems described by central forces), the first and the third laws are special and are linked with the unique form of the newtonian potential $-k/r$. In particular, Bertrand theorem assures that only the newtonian potential and the harmonic ...
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1) Problem. The Kepler problem has Hamiltonian
$$ H~:=~ \frac{p^2}{2m}- \frac{k}{q}, $$
where $m$ is the 2-body reduced mass. The Laplace–Runge–Lenz vector is (up to an irrelevant normalization)
$$ A^j ~:=~a^j + km\frac{q^j}{q}, \qquad a^j~:=~({\bf L} \times {\bf p})^j~=~{\bf q}\cdot{\bf p}~p^j- p^2~q^j,\qquad {\bf L}~:=~ {\bf q} \times {\bf p}.$$
2) ...
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Nature doesn't have this symmetry because your conservation law doesn't hold, either. According to the law of inertia, object keeps on moving with a constant velocity – which is however generically nonzero. In its own rest frame, it's zero, but in other frames, the velocity is nonzero.
If one studies the motion of the center-of-mass, it is indeed moving ...
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Conservation of energy follows from invariance under translation in time, not inversion. This symmetry states that no matter when you do your experiment, it will give the same results. All isolated systems obey this symmetry (and therefore conserve energy) and no violation of it has ever been detected. (Needless to say, it would be a huge event if it were.)
...
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I) The topic of gauging global symmetries is a quite large subject, which is difficult to fit in a Phys.SE answer. Let us for simplicity only consider a single (and thus necessarily Abelian) continuous infinitesimal transformation$^1$
$$ \tag{1} \delta \phi^{\alpha}(x)~=~\varepsilon(x) Y^{\alpha}(\phi(x),x), $$
where $\varepsilon$ is an infinitesimal real ...
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Jeff Harvey has of course provided you with the perfect, standardized answer: the scale invariance boils down to the tracelessness of the stress-energy tensor. But the tracelessness is not really a "conserved quantity" in the usual sense that you may have waited for.
However, one may transform the problem to something that is a conserved quantity in the ...
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(I only know the importance, not the mathematical treatment &c, but I doubt you want that)
Significance
Noether's theorem let's us obtain conservation laws. Conservation laws are pretty much the life of physics. If you want to calculate the outcome of any process, you have to see what is conserved in the process. Without these laws, you'd be left with ...
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