A theorem that relates continuous symmetries (continuous transformations that don't affect the value of the lagrangian) to quantities conserved in time.

learn more… | top users | synonyms

6
votes
1answer
334 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 ...
11
votes
0answers
451 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+ \bar{c}^a(\partial\...
3
votes
3answers
443 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. ...
1
vote
0answers
226 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 L}{\...
2
votes
1answer
162 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 ...
7
votes
1answer
715 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 ...
10
votes
3answers
3k 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 ...
11
votes
1answer
3k views

Trick for deriving the stress tensor in any theory

In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case ...
8
votes
1answer
3k 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 ...
3
votes
1answer
514 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 $$[\hat{x}^i,\hat{p}_j]~=~i\hbar~\...
2
votes
1answer
86 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 ...
3
votes
2answers
261 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
2answers
208 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 ...
8
votes
2answers
598 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 F_{...
7
votes
1answer
2k 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 ...
2
votes
1answer
143 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 $\delta \phi = \alpha \phi$, ...
8
votes
1answer
460 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 ...
8
votes
2answers
1k 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
433 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 under ...
3
votes
2answers
841 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
78 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 ...
61
votes
8answers
5k 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
2answers
812 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 ...
2
votes
1answer
629 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) $$ \Theta^{...
2
votes
1answer
780 views

Symmetry and conservation laws related to baryon number, lepton number and strangeness

According to Noether's theorem, Every continuous symmetry of the action leads to a conservation law. For example, conservation of linear momentum corresponds to translational symmetry, conservation ...
15
votes
2answers
2k views

Symmetrizing the Canonical Energy-Momentum Tensor

The Canonical energy momentum tensor is given by $$T_{\mu\nu} = \frac{\partial {\cal L}}{\partial (\partial^\mu \phi_s)} \partial_\nu \phi_s - g_{\mu\nu} {\cal L} $$ A priori, there is no reason to ...
1
vote
1answer
385 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 ...
4
votes
1answer
4k 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 ...
7
votes
4answers
642 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 ...
5
votes
1answer
392 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)? (...
4
votes
1answer
416 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(- \frac{\hbar^2}{...
4
votes
3answers
428 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 ...
1
vote
1answer
183 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 ...
4
votes
1answer
942 views

Noether current for the Yang-Mills-Higgs Lagrangian

I am trying to calculate the Noether 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
2answers
447 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 ...
30
votes
5answers
3k 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
318 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
1k 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 ...
32
votes
3answers
3k 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 ...
10
votes
3answers
3k 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 $x$ ...
36
votes
4answers
17k views

Why can't energy be created or destroyed?

My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed. Why is that? Is there a theory or scientific evidence that proves his statement true or ...
2
votes
2answers
581 views

Which symmetry is associated with conservation of flux?

Which symmetry is associated with conservation of flux (e.g., in electromagnetism)? For example, when working with Gauss's law in electromagnetism, net flux through an arbitrary volume element ...
6
votes
2answers
764 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 ...
9
votes
2answers
236 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 ...
18
votes
1answer
1k views

Gauge redundancies and global symmetries [closed]

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 ...
27
votes
5answers
6k 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 ...
1
vote
3answers
1k views

Energy non-conservation for time-dependent potentials

Written in a book I read that the "total energy is not preserved when the potential depends explicitly on time", i.e. $U=U(x,t)$. Is there any proof or explanation for this?
5
votes
1answer
902 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 ...
20
votes
1answer
1k 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 ...
46
votes
1answer
6k views

What conservation law corresponds to Lorentz boosts?

Noether's Theorem is used to related the invariance under certain continuous transformations to conserved currents. A common example is that translations in spacetime correspond to the conservation of ...