A theorem that relates continuous symmetries (continuous transformations that don't affect the value of the lagrangian) to quantities conserved in time.
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 ...
0
votes
1answer
39 views
Relativistic Lagrangian transformations
I need to study the relativistic lagrangian of a free particle.
It's
$\ L= - m c^2 \sqrt[2]{1- \frac{|u|^2}{c^2}} $ I need to study the translation, boost and rotation symmetry. I say it doesn't ...
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 ...
1
vote
1answer
52 views
Topological vs. non-topological noetherian charges
What (if any) is the relationship between the conserved (non-topological) noetherian charges and topological charges? Namely, is there any "generalization" of the Noether's first theorem that includes ...
-3
votes
2answers
237 views
why is dark matter the best theory available to explain missing mass problems?
Why is dark matter the best theory to explain the missing mass problem?
Why is dark matter mathematically necessary to explain the missing mass problem?
On a side not I believe dark matter is ...
26
votes
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 ...
1
vote
0answers
99 views
Question about Noether theorem
For the Noether theorem for pseudoeuclidean 4-spacetime a-current $J_{a}^{\mu}$ is equal to
$$
J_{a}^{\mu} = \frac{\partial L}{\partial (\partial_{\mu}\Psi_{k})}Y_{k, a} - \left( \frac{\partial ...
7
votes
0answers
139 views
supersymmetric Noether theorem and supercurrents — invariance requirements
Consider $\mathcal{N}=1,d=4$ SUSY, $n$ chiral superfields $\Phi^i,$ Kaehler potential $K,$ superpotential $W$ and action ($\overline{\Phi}_i$ is complex conjugate of $\Phi^i$)
$$ S= \int d^4x \left[ ...
15
votes
4answers
359 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 ...
17
votes
7answers
623 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 ...
4
votes
0answers
70 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+
...
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 ...
3
votes
3answers
191 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.
...
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 ...
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 ...
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 ...
4
votes
1answer
198 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
1answer
159 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
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 ...
3
votes
1answer
123 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
...
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 ...
1
vote
1answer
41 views
Please provide the simplest example you can think of, of generators of time evolution and generalized coordinates
I was reading the Wikipedia article about Noether's theorem and this thing popped out:
Then the resultant perturbation can be written as a linear sum of the
individual types of perturbations
...
1
vote
2answers
111 views
In Noether's theorem, what is a “classical solution of the equations of motion”?
I'm reading a book which states that:
for each generator of a global symmetry transformation, there is a
current $j^{\mu}_{a}$ which, when evaluated on a classical solution
of the equations of ...
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 Klein–Gordon field, but that derivation use ...
8
votes
2answers
224 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 ...
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 ...
8
votes
2answers
76 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 ...
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 Schrödinger equation
$\mathrm i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) \;=\; \left(- ...
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 ...
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 ...
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$, ...
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 ...
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 ...
7
votes
2answers
274 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 ...
2
votes
1answer
187 views
Application of Noether's theorem
Consider one parameter transformation: $y = y ( \tilde{y}, \alpha)$ such that lagrangian satisfies: $\tilde{L}(\tilde{y}, \alpha) = L(y ( \tilde{y}, \alpha))$. We say that equation is invariant ...
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}.$$
How do I go out about showing ...
13
votes
6answers
2k views
Can Noether's theorem be understood intuitively?
Noether's theorem is one of those surprisingly clear results of mathematical calculations, for which I am inclined to think that some kind of intuitive understanding should or must be possible. ...
3
votes
2answers
124 views
CPT Violation and Symmetry / Conservation Laws
Ok, so I remember reading that every conservation law has a corresponding symmetry (i.e. conservation of momentum is translational symmetry, conservation of angular momentum is rotational symmetry).
...
1
vote
0answers
65 views
Noether's charge due to lorentz transformation [duplicate]
Possible Duplicate:
What conservation law corresponds to Lorentz boosts?
For a relativistic free particle, What is the Noether charge generated due to Lorentz transformations? What is ...
2
votes
1answer
153 views
Why has the trace of the energy-momentum tensor to vanish for conserved scaling currents to exist?
In this paper, the authors say that the trace of the energy-momentum tensor has to vanish to allow for the existence of conserved dilatation or scaling currents, as defined on p 10, Eq(22)
$$ ...
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 ...
3
votes
0answers
142 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 ...
1
vote
1answer
144 views
Improved energy-momentum tensor
While still dealing with this issue, I've stumbled upon this answer to a question asking about the conserved quantity corresponding to a scaling transformation. It mentions that in accordance with ...
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
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)?
...
1
vote
1answer
89 views
Cyclic co-ordinates implying the constant velocity motion of center of mass of a system of particles
I'm reading the section on Central Force in my textbook (Goldstein's Classical Mechanics has a similar argument in the chapter titled "The Central Force Problem", first section), where we have the ...
3
votes
1answer
317 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, ...
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 ...
1
vote
1answer
196 views
Conserved quantum observables from symmetries *with density matrix*
I’ve read Ballentine where he derives the conserved observable operators (momentum, energy, ...) from symmetries of space-time.
Can I read up such a derivation in more detail somewhere else or even ...
5
votes
2answers
369 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 ...
