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39

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 ...


32

Warning: this is a long and boring derivation. If you are interested only in the result skip to the very last sentence. Noether's theorem can be formulated in many ways. For the purposes of your question we can comfortably use the special relativistic Lagrangian formulation of a scalar field. So, suppose we are given an action $$S[\phi] = \int {\mathcal ...


17

Mass is only conserved in the low-energy limit of relativistic systems. In relativistic systems, mass can be converted into energy, and you can have processes like massive electron-positron pairs annhillating to form massless photons. What is conserved (in theories obeying special relativity, at least) is mass energy--this conservation is enforced by the ...


16

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 ...


16

1) If you want a Noether theorem for information, there is no such thing. Trying to obtain it from a symmetry law, by Noether's theorem can't work, simply because information is not a quantity that can be obtained for instance by the derivative of the Lagrangian with respect to some variable. Information is not scalar, vector, tensor, spinor etc. 2) ...


15

Noether's theorem says that symmetries lead to conservation laws, not the converse. Conservation of mass doesn't follow from any of the obvious symmetries of nonrelativistic motion. Those symmetries are translations in space (leading to conservation of momentum), translations in time (conservation of energy), rotations (conservation of angular momentum), and ...


15

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 ...


14

Here's what I perceive to be a mathematically and logically precise presentation of the theorem, let me know if this helps. Mathematical Preliminaries First let me introduce some precise notation so that we don't encounter any issues with "infinitesimals" etc. Given a field $\phi$, let $\hat\phi(\alpha, x)$ denote a smooth one-parameter family of fields ...


13

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. ...


12

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) ...


12

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 ...


11

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 ...


11

CPT seems to imply it. You can reverse the system evolution by applying charge, parity and time conjugation, so the information about the past must be contained in the present state. That implies conservation of information by the evolution. This may not be the answer you wanted, because it does not imply unitarity, but it is the only relationship between ...


11

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 ...


11

Indeed, nothing is wrong with Noether theorem here, $J^\mu = F^{\mu \nu} \partial_\nu \Lambda$ is a conserved current for every choice of the smooth scalar function $\Lambda$. It can be proved by direct inspection, since $$\partial_\mu J^\mu = \partial_\mu (F^{\mu \nu} \partial_\nu \Lambda)= (\partial_\mu F^{\mu \nu}) \partial_\nu \Lambda+ F^{\mu \nu} ...


11

The idea of partitioning energy into different forms like "mechanical energy" or "chemical energy" and such is actually arbitrary. More or less by definition, energy is that which is conserved unter time translations by Noether's theorem. If what you call "mechanical energy" has changed, then there is another term in the Noetherian energy that has changed ...


10

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 ...


10

It's intuitively clear that the energy most accurately describes how much the state of the system is changing with time. So if the laws of physics don't depend on time, then the amount how much the state of the system changes with time has to be conserved because it's still changing in the same way. In the same way, and perhaps even more intuitively, if the ...


9

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 ...


8

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 ...


8

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 ...


8

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 ...


8

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 ...


8

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 ...


8

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 ...


8

The trick is given in equation 4.4 of the attached article: First couple the theory to gravity, (by introducing a metric tensor in the integration measure and for each index raising) obtaining the action: $S = \int_M d^4x \sqrt{-g} \mathcal{L}$ Then vary the action with respect to the metric tensor: $T_{\alpha\beta} = \frac{1}{\sqrt{-g}} \frac{\delta ...


8

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, ...


8

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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