A theorem that relates continuous symmetries (continuous transformations that don't affect the value of the lagrangian) to quantities conserved in time.
26
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7answers
2k views
Is there something similar to Noether's theorem for discrete symmetries?
Noether's theorem states that, for every continuous symmetry of a system, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
17
votes
5answers
812 views
Is the converse of Noether's first theorem true: Every conservation law has a symmetry?
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.
Is the converse true: Any conservation law of a physical ...
17
votes
7answers
625 views
Is there a symmetry associated to the conservation of information?
Conservation of information seems to be a deep physical principle.
For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.
We may wonder if there is an underlying ...
16
votes
4answers
1k views
What symmetry causes the Runge-Lenz vector to be conserved?
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...
16
votes
1answer
211 views
Why does charge conservation due to gauge symmetry only hold on-shell?
While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
15
votes
4answers
360 views
When can a global symmetry be gauged?
Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be ...
13
votes
6answers
2k views
Can Noether's theorem be understood intuitively?
Noether's theorem is one of those surprisingly clear results of mathematical calculations, for which I am inclined to think that some kind of intuitive understanding should or must be possible. ...
12
votes
5answers
1k views
What is the conserved quantity of a scale-invariant universe?
Consider that we have a system described by a wavefunction psi(x). We then make an exact copy of the system, and anything associated with it, (including the inner cogs and gears of the elementary ...
12
votes
2answers
4k views
Why can't energy be created or destroyed?
My physics instructor told the class, when lecturing about energy, and that it can't be created or destroyed. Why is that? Is there a theory or scientific evidence that proves his statement true or ...
11
votes
3answers
550 views
Swimming in Spacetime - apparent conserved quantity violation
My question is about the article Swimming in Spacetime.
My gut reaction on first reading it was "this violates conservation of momentum, doesn't it?". I now realize, however, that this doesn't allow ...
11
votes
4answers
941 views
If all conserved quantities of a system are known, can they be explained by symmetries?
If a system has $N$ degrees of freedom (DOF) and therefore $N$ independent1 conserved quantities integrals of motion, can continuous symmetries with a total of $N$ parameters be found that deliver ...
8
votes
3answers
843 views
What is the symmetry which is responsible for preservation of electrical charges?
Another Noether's theorem question, this time about electrical charge.
According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For ...
8
votes
2answers
225 views
Translation Invariance without Momentum Conservation?
Instead of the actual gravitational force, in which the two masses enter symmetrically, consider something like $$\vec F_{ab} = G\frac{m_a m_b^2}{|\vec r_a - \vec r_b|^2}\hat r_{ab}$$ where $\vec ...
8
votes
2answers
77 views
More general invariance of the action functional
I will formulate my question in the classical case, where things are simplest.
Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the ...
8
votes
2answers
830 views
What's the interpretation of Feynman's picture proof of Noether's Theorem?
On pp 103 - 105 of The Character of Physical Law, Feynman draws this diagram to demonstrate that invariance under spatial translation leads to conservation of momentum:
To paraphrase Feynman's ...
7
votes
3answers
405 views
Noether theorem with semigroup of symmetry instead of group
Suppose You have semigroup instead of typical group construction in Noether theorem. Is this interesting? In fact there is no time-reversal symmetry in the nature, right? At least not in the same ...
7
votes
3answers
906 views
Energy momentum tensor from Noether's theorem
in the book Quantum Field Theory by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p.22):
Assume a lagrangian density depending on the spacetime coordinates $x$ ...
7
votes
2answers
275 views
Conjugate Variables, Noether's Theorem and QM
What is the underlying reason that the same pairs of conjugate variables (e.g. energy & time, momentum & position) are related in Noether's theorem (e.g. time symmetry implies energy ...
7
votes
0answers
139 views
supersymmetric Noether theorem and supercurrents — invariance requirements
Consider $\mathcal{N}=1,d=4$ SUSY, $n$ chiral superfields $\Phi^i,$ Kaehler potential $K,$ superpotential $W$ and action ($\overline{\Phi}_i$ is complex conjugate of $\Phi^i$)
$$ S= \int d^4x \left[ ...
6
votes
2answers
276 views
Does Noether's theorem also give rise to quantities conserved over space?
Noether's theorem gives rise to quantities that are conserved over time. But does it also give rise to quantities that are conserved over space?
6
votes
1answer
247 views
Why is color conserved in QCD?
According to Noether's theorem, global invariance under $SU(N)$ leads to $N^2-1$ conserved charges. But in QCD gluons are not conserved; color is. There are N colors, not $N^2-1$ colors. Am I ...
6
votes
1answer
180 views
Introduction to Gauge Symmetries: Good, Bad or Ugly?
I'm trying to come up with a good (as in intuitive and not 'too wrong') definition of a gauge symmetry.
This is what I have right now:
A dynamical symmetry is a (differentiable) group of ...
5
votes
2answers
370 views
What's the importance of Noether's theorem in Physics
The Noether's theorem that I want to mention is the following: Noether's theorem.
I know the importance of Noether's contribution to modern algebra. Can anyone write about Noether's theorem in ...
4
votes
2answers
306 views
How to apply Noether's theorem
Say I have a point transformation:
$$x' ~=~ (1 +\epsilon)x,$$
$$t' ~=~ (1 +\epsilon)^2t,$$
and Lagrangian
$$ L ~=~ \frac{1}{2}m\dot{x}^2 - \frac{\alpha}{x^2}.$$
How do I go out about showing ...
4
votes
1answer
512 views
invariance of lagrangian in Noether's theorem
Noether's theorem needs the lagrangian to be invariant.
However, given a lagrangian $L$, we know that the lagrangians $\alpha L$ (where $\alpha$ is any constant) and $L + \frac{df}{dt}$ (where $f$ is ...
4
votes
1answer
160 views
Trick for deriving the stress tensor in any theory
In D. Tong's notes on string theory (http://www.damtp.cam.ac.uk/user/tong/string/four.pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the ...
4
votes
1answer
284 views
Noether theorem and classical proof of electric charge conservation
How to prove conservation of electric charge using Noether's theorem according to classical (non-quantum) mechanics?
I know the proof based on using KleinโGordon field, but that derivation use ...
4
votes
3answers
374 views
Must all symmetries have consequences?
Must all symmetries have consequences?
We know that transnational invariance, for example, leads to momentum conservation, etc, cf. Noether's Theorem.
Is it possible for a theory or a model to have ...
4
votes
2answers
199 views
How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace?
How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace?
4
votes
1answer
433 views
Conservation of quantum Noether current
The Noether current for a set of scalar fields $\varphi_a$ can classically be written as:
$$j^\mu(x)=\frac{\delta \mathcal L(x)}{\partial(\partial_{\mu}\varphi_a(x))}\delta \varphi_a(x)$$
The ...
4
votes
1answer
199 views
Constants of motion vs. integrals of motion
Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
4
votes
4answers
349 views
Is it intuitive that the conserved quantity from time symmetry is what we know as energy?
Is there an easy (aka intuitive) way to understand that the conserved quantity from time translation symmetry is just what we call energy?
In other words, we use two definitions of energy. One is ...
4
votes
1answer
251 views
Is momentum conservation for the classical Schrödinger equation due to non-relativistic or due to some more exotic invariance?
I had no problem appliying the Neothers theorem for translations to the non-relativistic Schrรถdinger equation
$\mathrm i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) \;=\; \left(- ...
4
votes
1answer
125 views
Noether's identities
I have some questions about the Noether's second theorem (generally not covered by field theory books):
What is the most general Noether identity for (classical) field theories?
Why are Noether ...
4
votes
0answers
73 views
Noether currents for the BRST tranformation of Yang-Mills fields
The Lagrangian of the Yang-Mills fields is given by
$$
\mathcal{L}=-\frac{1}{4}(F^a_{\mu\nu})^2+\bar{\psi}(i\gamma^{\mu}
D_{\mu}-m)\psi-\frac{1}{2\xi}(\partial\cdot A^a)^2+
...
4
votes
0answers
315 views
Gauge redundancies and global symmetries
It is often said that local (gauge) transformation is only redundancy of description of spin one massless particles, to make the number degrees of freedom from three to two. It is often said that ...
3
votes
2answers
450 views
Is there a conserved quantity that enforces planar orbits in central force motion?
From what I remember, one of the first steps in finding the equations of motion for an orbiting body is to argue that the body's motion has to be restricted to a plane, because the central force has ...
3
votes
2answers
124 views
CPT Violation and Symmetry / Conservation Laws
Ok, so I remember reading that every conservation law has a corresponding symmetry (i.e. conservation of momentum is translational symmetry, conservation of angular momentum is rotational symmetry).
...
3
votes
2answers
452 views
What is the symmetry that corresponds to conservation of position?
We know that conserved quantities are associated with certain symmetries. For example conservation of momentum is associated with translational invariance, and conservation of angular momentum is ...
3
votes
1answer
125 views
What conservation law corresponds to this local $U(1)$ symmetry of the CCR?
It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by
...
3
votes
3answers
192 views
Where do the conservation laws come from?
I know the conservation of energy comes from Noether's theorem via the time-translational symmetry, and if I remember correctly, the conservation of momentum comes from space-translational symmetry.
...
3
votes
1answer
261 views
How do you know if a coordinate is cyclic if its generalized velocity is not present in the Lagrangian?
Goldstein's Classical Mechanics says that a cyclic coordinate is one that doesn't appear in the Lagrangian of the system, even though its generalized velocity may appear in it (emphasis mine). For ...
3
votes
1answer
318 views
Noether current for the Yang-mills-higgs lagrangian
I am trying to calculate the Noether's current, more specifically, the energy density of the Yang-mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, ...
3
votes
1answer
152 views
Does spontanous symmetry breaking affect Noethers theorem?
Does spontanous symmetry breaking affect the existence of a conserved charge?
And how does depend on whether we look at a classical or a quantum field theory (e.g. the weak interacting theory)?
...
3
votes
0answers
143 views
Symmetrizing the Canonical Energy-Momentum Tensor
The Canonical energy momentum tensor is given by
$$T_{\mu\nu} = \frac{\delta {\cal L}}{\delta (\partial^\mu \phi_s)} \partial_\nu \phi_s - g_{\mu\nu} {\cal L} $$
A priori, there is no reason to ...
2
votes
2answers
852 views
Noether's theorem vs. Heisenberg uncertainty principle
In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle.
Because both the ...
2
votes
3answers
220 views
Noether's current expression in Peskin and Schroeder
In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence.
But if we ...
2
votes
2answers
137 views
Does a constant factor matter in the definition of the Noether current?
This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is:
Consider a field Lagrangian with only ...
2
votes
1answer
68 views
Energy-momentum conservation without translation symmetry?
As I checked, the energy-momentum tensor defined as ${T^\mu}_\nu=\frac{\partial {\cal L}}{\partial(\partial_\mu \phi)}\partial_\nu \phi-{\cal L}{\delta^\mu}_\nu$ at the solution $\phi$ of equation of ...
2
votes
1answer
58 views
Obtaining the conserved current of the Lagrangian making the parameter depending on $x$
To calculate the conserved current due to an internal symmetry of the system (expressed by the Lagrangian density) we can proceed as follows: if it is invariant under
$\delta \phi = \alpha \phi$, ...
