Noether's Theorem
Pmb
Noether's theorem states -
(1) For every continuous symmetry of the laws of physics, there must exist a
conservation law.
(2) For every conservation law, there must exist a continuous symmetry.
There are many incarnations of this theorem. Two of which are particle
dyanmics and field dynamics. In the case of the former one special
case/example pertains to cyclic (aka "ignorable) variables in the
Lagrangian. For each cyclic variable there is a conserved conjugate
momentum. There is a more general conservation law with particle dynamics
though (see Goldstein). Then there is the field incarnation where the
conserved quantities are currents.
Here is my question: Does anyone know of an online English version and the
original publiucation date and how Noether herself stated her theorem?
John Baez has an online explaination of it here
http://math.ucr.edu/home/baez/noether.html
He gives a date of 1915. At the bottom of this page there is this link
http://www.emmynoether.com/noeth.htm
It gives a publication date of 1905
Pmb
Danny Ross Lunsford
> Noether's theorem states -
> (1) For every continuous symmetry of the laws of physics, there must exist
> a conservation law.
> (2) For every conservation law, there must exist a continuous symmetry.
Noether's theorem states nothing of the kind. First, it is math, not
physics. Second, it states that a variational integral that is invariant
under a one parameter family of transformations has an explicit first
integral of the corresponding Euler-Lagrange equations. There is no
reciprocal statement. Third, this theorem is not confined to continous
fields (see Hamiltonian mechanics).
--
-drl
Pmb
"Danny Ross Lunsford" <antima...@yahoo.com> wrote in message
news:LyTHb.1062$se6...@newssvr24.news.prodigy.com...
> Pmb wrote:
> > Noether's theorem states -
> > (1) For every continuous symmetry of the laws of physics, there must
exist
> > a conservation law.
> > (2) For every conservation law, there must exist a continuous symmetry.
>
> Noether's theorem states nothing of the kind.
That is not to my understanding. In fact this is what I found on it
>From -- http://thesaurus.maths.org/dictionary/map/word/2300
---------------------------------------------
There is a one-to-one correspondence between continuous symmetries of the
laws of physics, and conservation laws in physics.
---------------------------------------------
>From -
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html
---------------------------------------------
Emmy Noether's first piece of work when she arrived in Göttingen in 1915 is
a result in theoretical physics sometimes referred to as Noether's Theorem,
which proves a relationship between symmetries in physics and conservation
principles.
---------------------------------------------
>From - http://www.emmynoether.com/noeth.htm
---------------------------------------------
For every continuous symmetry of the laws of physics, there must exist a
conservation law.
For every conservation law, there must exist a continuous symmetry.
---------------------------------------------
etc.
However shortly after I posted that question (which was last week on Dec.
23) I found the answer I was looking for. It is found here
http://www.physics.ucla.edu/~cwp/pubs/noether.trans/english/mort186.html
> First, it is math, not physics. Second, it states that a variational
integral that is invariant
> under a one parameter fami
In her paper the Noether does speak physics, i.e. Lagrangians, GR etc.
However I have not gotten around to reading it yet.
More later
Thank you for your response.
Pmb
Michael Varney
news:1vVFb.34$1o...@nwrdny01.gnilink.net...
http://www.physics.ucla.edu/~cwp/articles/noether.trans/english/mort186.html
http://www.physics.ucla.edu/~cwp/articles/noether.trans/german/emmy235.html
http://www.physics.ucla.edu/~cwp/Phase2/Noether,_Amali...@861234567.html
Uncle Al
Here is my pile of stuff,
http://www.physics.ucla.edu/~cwp/articles/noether.trans/english/mort186.html
http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html
http://www.emmynoether.com/
http://arXiv.org/abs/physics/9807044
http://www.physics.buffalo.edu/phy511/section3.pdf
Olver, PJ, Applications of Lie Groups to Differential Equations, 2nd
Edition (Graduate Texts in Mathematics, vol.107) (Springer-Verlag, New
York: 1993); Noether's Theorm, p. 272ff.
Noether, E Gesammelte Abhandlungen (Collected Publications), edited by
Nathan Jacobson (Springer Verlag: New
York, 1983)
Good luck and do better.
--
Uncle Al
http://www.mazepath.com/uncleal/qz.pdf
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)
Pmb
"Uncle Al" <Uncl...@hate.spam.net> wrote
> http://www.physics.buffalo.edu/phy511/section3.pdf
I tried this and it doesn't work. Anyone know why?
Pmb
Uncle Al
It looks like the course syllabus was updated, with the exclusion of
that topic.
http://www.physics.buffalo.edu/phy511/
Phil
> (1) For every continuous symmetry of the laws of physics, there must exist
a
> conservation law.
> (2) For every conservation law, there must exist a continuous symmetry.
>
Of course you have to add some conditions to make these statements true, but
even then there are far more subtle questions about what is a "meaningful"
or
"useful" or "non-trivial" or "vakid" conservation law. That is still
disputed.
For example, a Lorentz boost in SR seems to be a resonable example
of a continuous symmetry, but is the conservation law it gives really
useful?
If it is then why is it not given the same status as conservation of energy,
momentum and angular mometnum which are related to the other generators
of the Lorentz transforms? If not, how do we distiguish it from the others?
Noether actually developed another theorem sometimes called her second
theorem. It applies to local symmetries and was targeted at GR. Ever since
people have argued about whether or not this second theorem implies that
energy and momentum conservation laws in GR are trivial. The argument
started with Noether, Hilbert, Klein, Weyl and Einstein but more recently it
is John Wheeler who leads the group of relativitists that think they are
trivial.
He likes to think of energy conservation as the same thing as saying that a
boundary of a boundary is zero. Many experts follow his lead, yet the same
arguments are easily applied to other gauge theories, so is conservation of
electric charge trivial in the same sense?
People here recently argued here that special relativity is generally
covariant
so do the same results from GR apply there too? if not why not? Does a
meaningful conservation law have to be a quantity expressed in terms of
first
derivatives only? Does the formulation have to be covariant? Would it matter
if a conservation law of energy could give a non-zero energy density on
empty
flat space-time? (Many formulations of energy conservation in GR break at
least some of these conditions but you can keep any one) How do we make
these questions precise anyway? What would we mean by a "local"
conservation law and how important is that? Different physicists and
philosophers have different answers to these questions, and do you suppose
they always state them up-front? Of course not, hence great confusion
prevails.
Even the relativity FAQ gives a rather inadequate answer on the subject of
energy conservation in GR, but that is excusable on the grounds that the
answer depends on the questions above and perhaps a few more, so a
complete answer would be very long and still disputed.
BTW there is also a "third theorem" sometimes called the "Bondary theorem"
due to Klein which followed Noether's work in 1918. You will want to look
at that one too if you want to form an informed opinion on this. Search in
Google using the names and key phrases I have used here to find some recent
papers.
Hendrik van Hees
Pmb wrote:
> Here is my question: Does anyone know of an online English version and
> the original publiucation date and how Noether herself stated her
> theorem?
>
> John Baez has an online explaination of it here
> http://math.ucr.edu/home/baez/noether.html
>
> He gives a date of 1915. At the bottom of this page there is this link
> http://www.emmynoether.com/noeth.htm
>
> It gives a publication date of 1905
I know of this one:
E. Noether, Invariante Variationsprobleme, Nachr. Ges. Wiss. Gottingen,
Math.-Phys. Kl. (1918) 235
It's very general and a very clear paper.
--
Hendrik van Hees Fakultät für Physik
Phone: +49 521/106-6221 Universität Bielefeld
Fax: +49 521/106-2961 Universitätsstraße 25
http://theory.gsi.de/~vanhees/ D-33615 Bielefeld
Danny Ross Lunsford
Pmb wrote:
> "Danny Ross Lunsford" <antima...@yahoo.com> wrote in message
> news:LyTHb.1062$se6...@newssvr24.news.prodigy.com...
>> Pmb wrote:
>> > Noether's theorem states -
>> > (1) For every continuous symmetry of the laws of physics, there must
> exist
>> > a conservation law.
>> > (2) For every conservation law, there must exist a continuous symmetry.
>>
>> Noether's theorem states nothing of the kind.
>
> That is not to my understanding. In fact this is what I found on it
>
>>From -- http://thesaurus.maths.org/dictionary/map/word/2300
Don't believe everything you read on the Internet. :)
I'd suggest reading a book on the calculus of variations (good ones:
Caratheodory, Gelfand-Fomin). If the one-parameter family involves time
translations, then the first integral can be *interpreted* as a
conservation law. Noether's theorem itself is a generalization of the ideas
behind the method of "Routhian cyclic coordinates" in Hamiltonian
mechanics. You might also want to read some of the works of Sophus Lie (and
not derived texts, unless they are by Klein).
-drl
Lubos Motl
On 29 Dec 2003, Danny Ross Lunsford wrote:
> Noether's theorem states nothing of the kind. First, it is math, not
> physics.
Pmb's explanation of the theorem was absolutely OK. Noether's theorem is a
central result in theoretical physics. If you want to see resources that
claim exactly the same thing, i.e. that Noether's theorem is about
*physics* and it is *one-to-one*, see e.g.
http://en.wikipedia.org/wiki/Noether's_theorem
http://thesaurus.maths.org/dictionary/map/word/2300
Emmy Noether was a mathematician, but it does not change the fact that she
rocked the world of theoretical physics, not so much the world of pure
mathematics.
> There is no reciprocal statement.
Of course that there is a reciprocal statement, and let me repeat this
reciprocal statement. For every conserved quantity, there is a symmetry.
Well, the symmetry is easy to construct. Using the language of quantum
mechanics, it is generated by the conserved charge. The infinitesimal
transformation of an operator is given by the commutator with the charge -
which has an obvious classical counterpart involving the Poisson brackets.
> Third, this theorem is not confined to continous fields (see
> Hamiltonian mechanics).
The relation between conserved quantities and symmetries holds for
discrete symmetries, too - e.g. if a theory has the parity symmetry, it
conserves a charge whose value is +1 or -1 - but Noether's classical
derivation only applies to continuous symmetries acting on continuous
fields.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
Hendrik van Hees
> Emmy Noether was a mathematician, but it does not change the fact that
> she rocked the world of theoretical physics, not so much the world of
> pure mathematics.
I'd not go so far. She was one of the founders of what is known as
"modern algebra". The classical books by van der Waerden contain a lot
of material, if not the most of it, which was taught by Noether in her
lectures, which contained much material found by herself and her
scholars.
It is funny to know that she herself didn't like the results of her
early work on "theory of invariants", which she had undertaken as her
PhD thesis. Among these early works is also the celebrated theorem
about symmetries and conservation laws which is one of the most
important guide lines for theory building in modern physics.
Indeed, Noether's theorem says that for any one-parameter Lie group,
which leaves the action, from which the equations of motion are derived
by making it stationary in the sense of variational calculus, there
exists a conserved quantity. That holds true for classical point
mechanics as well as for classical field theory.
In its hamiltonian formulation the conserved quantity is the generator
of the symmetry in the sense of symplectomorphisms where the symplectic
structure applies to phase space with the Lie algebra given by the
Poisson brackets, defined on differentiable phase-space functions. Also
the opposite holds true: If there is a conserved quantity, this
quantity generates a symmetry of the equations of motion.
It is also interesting to know that Noether's theorem also applies to
quantum theory. One has only to be careful with anomalies, i.e., one
has to keep in mind the fact that the quantised version of a classical
system might lose a symmetry, which applies to the classical theory.
If the symmetry is not anomalously broken, in the operator formalism the
conserved quantity is represented by an selfadjoint operator, which
generates a unitary symmetry operator by exponentiation, which leads to
the representation of a one-parameter Lie symmetry. Here, the Lie
algebra is defined by the selfadjoint operators on Hilbert space with
the commutator (multiplied by 1/i) as Lie bracket. Lie symmetries are
represented by unitary operators (only discrete symmetries can be
represented by antiunitary operators; time reversal symmetry must be
represented by an antiunitary operator in order to stay within the
Hilbert space with a Hamiltonian, which is bounded from below).
An important example for an anomaly is dilatation symmetry in massless
field theories, e.g., massless QCD: The classical action has the
symmetry, and there is no dimensionful parameter in the theory, in
other words, it is scale invariant.
The reason, why this is not a symmetry of the quantised theory, is that
the path integral measure is not invariant under the symmetry, which
leads to additional terms in the effective action that break dilatation
symmetry. While in the classical symmetry, there is no parameter of the
dimension of energy, especially the trace of the energy-momentum tensor
vanishes, this is not the case in the quantised theory.
Technically the same comes out in perturbation theory, where one cannot
find a regularisation, that keeps the dilatation symmetry intact, and
any renormalisation procedure introduces a renormalisation scale. In
the \overline{MS} scheme of dimensional regularisation, usually used in
perturbative QCD (see, e.g., the review of particle physics), this
introduces the scale \Lambda_{QCD}.
>> There is no reciprocal statement.
>=20
> Of course that there is a reciprocal statement, and let me repeat this
> reciprocal statement. For every conserved quantity, there is a
> symmetry. Well, the symmetry is easy to construct. Using the language
> of quantum mechanics, it is generated by the conserved charge. The
> infinitesimal transformation of an operator is given by the commutator
> with the charge - which has an obvious classical counterpart involving
> the Poisson brackets.
>=20
>> Third, this theorem is not confined to continous fields (see
>> Hamiltonian mechanics).
>=20
> The relation between conserved quantities and symmetries holds for
> discrete symmetries, too - e.g. if a theory has the parity symmetry,
> it conserves a charge whose value is +1 or -1 - but Noether's
> classical derivation only applies to continuous symmetries acting on
> continuous fields.
The reason might be that, for classical theories, the discrete
symmetries are less important, because they do not imply conservation
laws, while they do in quantum theory, as you write above, or have I
overseen something? Is there an interesting physical implication from
discrete symmetries in classical theory?
--=20
Hendrik van Hees Fakult=E4t f=FCr Physik=20
Phone: +49 521/106-6221 Universit=E4t Bielefeld=20
Fax: +49 521/106-2961 Universit=E4tsstra=DFe 25=20
http://theory.gsi.de/~vanhees/ D-33615 Bielefeld=20
Danny Ross Lunsford
Phil wrote:
> For example, a Lorentz boost in SR seems to be a resonable example
> of a continuous symmetry, but is the conservation law it gives really
> useful?
> If it is then why is it not given the same status as conservation of
> energy, momentum and angular mometnum which are related to the other
> generators of the Lorentz transforms? If not, how do we distiguish it from
> the others?
Conservation laws are really time-parametrized flows. I can't see how to do
that naturally for a boost.
> Even the relativity FAQ gives a rather inadequate answer on the subject of
> energy conservation in GR, but that is excusable on the grounds that the
> answer depends on the questions above and perhaps a few more, so a
> complete answer would be very long and still disputed.
This seems to me fairly straightforward - if you can convert a covariant
divergence to an ordinary one by some property of the differentiated
quantity, then you can have a conservation law. GR has no conservation law
as such because of the incomplete locality (direction is localized but not
length). In a Weyl geometry where both are localized, there is a real
conservation law. Indeed it has always struck me as curious that this
extremely important point is more or less ignored.
> BTW there is also a "third theorem" sometimes called the "Bondary theorem"
> due to Klein which followed Noether's work in 1918. You will want to look
> at that one too if you want to form an informed opinion on this. Search in
> Google using the names and key phrases I have used here to find some
> recent papers.
Could you point to Klein's work? Thanks in advance. I had thought that Klein
had retired to teaching by 1918.
--
-drl
Pmb
>
> Pmb wrote:
>
>
> > "Danny Ross Lunsford" <antima...@yahoo.com> wrote in message
> > news:LyTHb.1062$se6...@newssvr24.news.prodigy.com...
> >> Pmb wrote:
> >> > Noether's theorem states -
> >> > (1) For every continuous symmetry of the laws of physics, there must
> > exist
> >> > a conservation law.
> >> > (2) For every conservation law, there must exist a continuous
symmetry.
> >>
> >> Noether's theorem states nothing of the kind.
> >
> > That is not to my understanding. In fact this is what I found on it
> >
> >>From -- http://thesaurus.maths.org/dictionary/map/word/2300
>
> Don't believe everything you read on the Internet. :)
Hence the reason for this post.
However I'd rather read Noether's original paper rather than some author
telling me what Noether proved
> I'd suggest reading a book on the calculus of variations (good ones:
> Caratheodory, Gelfand-Fomin). If the one-parameter family involves time
> translations, then the first integral can be *interpreted* as a
> conservation law. Noether's theorem itself is a generalization of the
ideas
> behind the method of "Routhian cyclic coordinates" in Hamiltonian
> mechanics.
Yes. I'm quite aware of that. In faqt I explained that to bilge who had it
confused with something else. In fact Goldstein explains all this very well.
bilge applied it only to conserved currents. Seems that he didn't realize
how it was applied to discrete systems. In fact Goldstein uses cyclic
coordinates to obtained conserved quantities as an application of Noether's
theorem.
Thanks
Pmb
Hendrik van Hees
> Conservation laws are really time-parametrized flows. I can't see how
> to do that naturally for a boost.
The question with the Lorentz boost also puzzles me a little bit. I'll
think about it over the weekend.
Concerning quantum mechanics, it seems quite clear, what is the point.
Noether's theorem applies to quantum fields, with the caveat I posted
before, concerning anomalies.
Noether's theorem needs the invariance of the local quantum fields under
symmetry transformations. The local fields do not transform with
unitary matrices for Lorentz boosts. That must be true, because there
are no finite-dimensional unitary representations of the proper
orthochronous Lorentz group, but only for the rotation group (which is
a compact subgroup of the Lorentz goup). So there are selfadjoint
operators which generate rotations. These are the operators of total
angular momentum.
For boosts, the generators are not unitary. So these symmetries do not
define observables in the sense of quantum mechanics.
For a complete treatment of the Poincare group in quantum mechanics, you
can read my script about qft on my home page, but stay tuned, it will
be revised in the next time, especially concerning this group
theoretical stuff (Appendix B) and the notion of the local field
concept (Chapt. 4).
http://theory.gsi.de/~vanhees/publ/lect.pdf
Danny Ross Lunsford
> However I'd rather read Noether's original paper rather than some author
> telling me what Noether proved
Why? Did you learn gravity from the Principia? Often the book presentation
has been refined and boiled down to the essential elements. It is very rare
that an original paper has the best presentation of an issue. Try, for
example, to learn about electrodynamics from Maxwell's 1865 paper. Again I
urge you to look up the Gelfand-Fomin book, because the presentation is
mathematically accurate while accessible to the non-mathematician (that is,
engineers and physicists).
There is a general point I'm trying to make. Instead of "learning Noether's
theorem", you can take the opportunity to learn how Lie invented a lot of
extremely interesting math in the study of systems of differential
equations. You will then be allowed to see the Noether result in historical
context.
--
-drl
Lubos Motl
> The question with the Lorentz boost also puzzles me a little bit. I'll
> think about it over the weekend.
The boosts are symmetries of the action - of the full physical system in
spacetime - but they don't commute with the Hamiltonian: the commutator of
a boost and the Hamiltonian is proportional to the momentum in the
direction of the boost.
Even though the boosts don't commute with the Hamiltonian, these
symmetries *do* imply something analogous as the conservation laws,
following Noether's procedures. But because the commutator is nonzero, the
law associated with the boost symmetry is - in fact - that the center of
mass moves with a constant (time-independent) velocity that can be
nonzero.
> Noether's theorem needs the invariance of the local quantum fields under
> symmetry transformations. The local fields do not transform with
> unitary matrices for Lorentz boosts.
Of course that they do, otherwise your theory would not be unitary. If you
apply a finite boost on a state |psi> that is normalized to <psi|psi>=1
(i.e. the total probability that an unboosted system exists is one), then
- of course - the boosted state B|psi> must also satisfy
<psi|Bdagger.B|psi>=1 because the probability that the boosted system
exists must still be equal to one.
> That must be true, ...
Well, maybe it must be true. The only problem of this statement is that it
is apparently *not* true. ;-)
> because there are no finite-dimensional unitary representations of the
> proper orthochronous Lorentz group, ...
That's right. It is why this representation is guaranteed to be
infinite-dimensional. Boosting an object with a nonzero energy-momentum
vector - and we don't want to consider vanishing four-momentum objects
because the vacuum is the only guy of this sort - changes its momentum,
and therefore - of course - the full representation space must include
states with all possible momenta, and therefore it is guaranteed to be
infinite-dimensional. This differs from the rotational group where it is
meaningful to consider operators or objects with vanishing momenta that
transform under the rotational group, and therefore finite-dimensional
unitary representations of the rotational group are important.
> For boosts, the generators are not unitary. So these symmetries do not
> define observables in the sense of quantum mechanics.
Once again, they *are* unitary.
Hendrik van Hees
> Even though the boosts don't commute with the Hamiltonian, these
> symmetries *do* imply something analogous as the conservation laws,
> following Noether's procedures. But because the commutator is nonzero,
> the law associated with the boost symmetry is - in fact - that the
> center of mass moves with a constant (time-independent) velocity that
> can be nonzero.
Yes, it's exactly the same as in the case of Galilei transformations
("Galilei boosts"): The conserved quantity is explicitly dependent on
time. Thus, we have
D_t K={K,H}+\partial_t K,
where K is the conserved quantitity and the generator of boosts (for
Lorentz or Galilei boosts). We find
K=X-t P,
where X is position vector of the centre of energy (mass in the
Galileian case) and P the conserved total momentum of the system.
I do not repeat the calculation here, because it's given (for the
quantum case) by Weinberg in Vol. I of his books about qft (Sect. 7.4,
pp 316).
>
>> Noether's theorem needs the invariance of the local quantum fields
>> under symmetry transformations. The local fields do not transform
>> with unitary matrices for Lorentz boosts.
>
> Of course that they do, otherwise your theory would not be unitary. If
> you apply a finite boost on a state |psi> that is normalized to
> <psi|psi>=1 (i.e. the total probability that an unboosted system
> exists is one), then - of course - the boosted state B|psi> must also
> satisfy <psi|Bdagger.B|psi>=1 because the probability that the booste=
d
> system exists must still be equal to one.
>
>> That must be true, ...
>
> Well, maybe it must be true. The only problem of this statement is
> that it is apparently *not* true. ;-)
>
>> because there are no finite-dimensional unitary representations of
>> the proper orthochronous Lorentz group, ...
>
> That's right. It is why this representation is guaranteed to be
> infinite-dimensional.
> Boosting an object with a nonzero
> energy-momentum vector - and we don't want to consider vanishing
> four-momentum objects because the vacuum is the only guy of this sort
> - changes its momentum, and therefore - of course - the full
> representation space must include states with all possible momenta,
> and therefore it is guaranteed to be infinite-dimensional. This
> differs from the rotational group where it is meaningful to consider
> operators or objects with vanishing momenta that transform under the
> rotational group, and therefore finite-dimensional unitary
> representations of the rotational group are important.
>
>> For boosts, the generators are not unitary. So these symmetries do
>> not define observables in the sense of quantum mechanics.
>
> Once again, they *are* unitary.
You are right, that was my mistake. Sorry.
Gauge
> Pmb wrote:
>
> > However I'd rather read Noether's original paper rather than some author
> > telling me what Noether proved
>
> Why? Did you learn gravity from the Principia?
By the way - there is a big difference between learing gravity from
the Principia and reading the Principia. One act teaches you the
subject and the other act teaches you if what you taught was accurate.
Pmb
Lubos Motl
On 2 Jan 2004, Hendrik van Hees wrote:
> I'd not go so far. She was one of the founders of what is known as
> "modern algebra".
I apologize. What I wrote was terribly insensitive to this loving and
intelligent genius of mathematics whom I admire so much. What I should
have written is that her PhD thesis on the theory of invariants rocked the
world of physics, not so much of mathematics. ;-) Thanks for your history
lecture.
> The reason might be that, for classical theories, the discrete
> symmetries are less important, because they do not imply conservation
> laws, while they do in quantum theory, as you write above, or have I
> overseen something? Is there an interesting physical implication from
> discrete symmetries in classical theory?
The implication is that if you know one solution, you can get another one
by a symmetry transformation. But this is true both for discrete as well
as continuous symmetries.
Urs Schreiber
"Danny Ross Lunsford" <antima...@yahoo.com> schrieb im Newsbeitrag
news:jarIb.3749$mh3....@newssvr22.news.prodigy.com...
>
> Phil wrote:
>
> > For example, a Lorentz boost in SR seems to be a resonable example
> > of a continuous symmetry, but is the conservation law it gives really
> > useful?
>
> > If it is then why is it not given the same status as conservation of
> > energy, momentum and angular mometnum which are related to the other
> > generators of the Lorentz transforms? If not, how do we distiguish it
from
> > the others?
>
> Conservation laws are really time-parametrized flows. I can't see how to
do
> that naturally for a boost.
Generellay, if k is a Killing vector field and t the tangent to a geodesic,
then
C = k.t
is a conserved quantity along the geodesic (in order to check, take the
covariant derivative of C along the geodesic and use Killing's equation.)
On Minkowski space, choose k to be the Killing vector associated with a
boost, e.g.
k_mu = [x^1, -x^0,0,0] .
Then the conserved quantity is
C = gamma x^1 - gamma v^1 x^0
To see the meaning of this formula more clearly devide by gamma and
rearrange:
<=> x^1 = const + v^1 x^0 .
The conserved quantity is hence proportional to the position of the particle
at the time origin. In general, for many-particle systems, the "conserved
quantity" is the center of energy at t=0.
For more answers to this question see
http://math.ucr.edu/home/baez/boosts.html .
> > Even the relativity FAQ gives a rather inadequate answer on the subject
of
> > energy conservation in GR, but that is excusable on the grounds that the
> > answer depends on the questions above and perhaps a few more, so a
> > complete answer would be very long and still disputed.
One should note that the above depends crucially on the existence of a
certain symmetry, manifested by the existence of Killing vector fields. Even
in GR there is energy conservation IF a timelike Killing vector field
exists, i.e. if the gravitational field is stationary.
Michael Varney
"Pmb" <some...@somewhere.com> wrote in message
news:ia6Jb.11325$x34....@nwrdny02.gnilink.net...
>
> However I'd rather read Noether's original paper rather than some author
> telling me what Noether proved
This is not the best way to learn physics. Rarely will the original paper
explain things as well as later authors. Even though Einstein invented
Relativity, he was not the end all final word in it. Progress does happen,
and has been happening for the last 100 years.
John Baez
In article <Pine.LNX.4.31.04010...@feynman.harvard.edu>,
Lubos Motl <mo...@feynman.harvard.edu> wrote:
>On 3 Jan 2004, Hendrik van Hees wrote:
>> The question with the Lorentz boost also puzzles me a little bit. I'll
>> think about it over the weekend.
>The boosts are symmetries of the action - of the full physical system in
>spacetime - but they don't commute with the Hamiltonian: the commutator of
>a boost and the Hamiltonian is proportional to the momentum in the
>direction of the boost.
>
>Even though the boosts don't commute with the Hamiltonian, these
>symmetries *do* imply something analogous as the conservation laws,
>following Noether's procedures. But because the commutator is nonzero, the
>law associated with the boost symmetry is - in fact - that the center of
>mass moves with a constant (time-independent) velocity that can be
>nonzero.
This issue keeps coming up on sci.physics.research, since most textbooks
don't deal with it. So, I made a little webpage about this subject:
Symmetry under boosts gives what conserved quantity?
http://math.ucr.edu/home/baez/boosts.html
It starts out like this:
........................................................................
In article <8fc591$eu4$1...@nnrp1.deja.com>, fi...@my-deja.com wrote:
>The invariance of the Lagrangian under boosts (either Lorentzian or
>Galilean) gives conservation of what?
Heh. People keep asking this question, so by now I just pull out my
handy little file of one of the previous discussions we've had about
this... see below.
>At least I searched in many books and they didn't even mention that.
Right. That's why everybody keeps asking this question....
>Is Noether's theorem not valid in such cases or what?
It's valid, all right.
>and if Noether's theorem still
>works in these cases, why the heck do books not talk about the conserved
>quantities corresponding to such symmetries?
Either 1) the textbook writers are too stupid to have thought about this
issue, or 2) they have decided it's better to let everybody figure this
out for themselves, or 3) they feel the answer is not sufficiently important
to waste a precious paragraph on it. I don't know. When I become king of
the universe, I will make all books on mechanics mention this issue.
But for now, read these old articles [...]
Ashok Prasad
only to continuous symmetries. Can someone direct me to a reference
about its extension to discrete symmetries that Lucus and some others
have mentioned?
Thanks in advance.
Ashok
[Moderator's note: 1. Quoted text deleted. Please don't quote entire
articles. Also, don't top-post. In other words, put your reply
after, not before, what you're replying to.
2. The quoted post, which I deleted, was about applying Noether's
theorem to boosts, which are a continuous, not discrete, family of
symmetries. -TB]
Uncle Al
Ashok Prasad wrote:
>
> I thought, before I read this thread, that Noethers theorem applies
> only to continuous symmetries. Can someone direct me to a reference
> about its extension to discrete symmetries that Lucus and some others
> have mentioned?
> Thanks in advance.
Noether's theorem requires a continuous symmetry or a symmetry that
can be approximated by a Taylor series or other sum of
infinitesimals. Noether's theorem with its dependence upon smooth Lie
groups is inappropriate for truly discrete symmetries.
Example: Parity is the reversal in sign of all coordinates. There is
no way to sneak up on it. Chirality plus 180 degree rotation is *not*
parity, certainly not if rotation is directionally quantized or
quantized more coarsely than lattice positions.
If anybody can explicity demonstrate or give a literature citation as
to how parity is validly treated by Noether's theorem, I will recant.
Danny Ross Lunsford
Uncle Al wrote:
> Noether's theorem requires a continuous symmetry or a symmetry that
> can be approximated by a Taylor series or other sum of
> infinitesimals. Noether's theorem with its dependence upon smooth Lie
> groups is inappropriate for truly discrete symmetries.
Yes.
> Example: Parity is the reversal in sign of all coordinates. There is
> no way to sneak up on it. Chirality plus 180 degree rotation is *not*
> parity, certainly not if rotation is directionally quantized or
> quantized more coarsely than lattice positions.
>
> If anybody can explicity demonstrate or give a literature citation as
> to how parity is validly treated by Noether's theorem, I will recant.
The way to do it is - enlarge the transformation group, restrict oneself
ot proper transformations in the larger group, focus on the ones that
cause a reversal in the subspace of the smaller group, and then hack off
the unnecessary terms generated in the context of the larger group.
E.g. start with some (homogeneous, source-free) theory in 4d, expand to
6d with extra timelike dimensions, consider rotations of the "timespace"
about the 3rd axis by pi, which reverses the 1st and 2nd - now you have
a continuous transformation of 6d that produces a time reversal in 4d.
Throw away (i.e. contract out) the extra stuff to get back to 4d (if you
want). The extra stuff will actually carry valuable information about
how interactions behave under time reversal in the inhomogeneous 4d
theory. The contraction (throwing out) should reveal the proper form of
the possible interactions in the inhomogeneous case.
-drl
Uncle Al
Time is continuous, homogeneous, and isotropic in the real world
(until Planck quantities). One can in principle slow down, stop, and
reverse. Notherian treatments of difficult time-dependent symmetries
are both elegant and commonplace.
Parity transformation of coordinates has no analog to "slow down,
stop, and reverse." It's an all or nothing operation along each
axis. Rotations may be very dangerous in this context, because
angular momenta are explicitly quantized - and at a coarse scale.
Quantitative parity divergence is normalized by the three moments of
inerita of the 3D chiral body in question (four in 4D, etc.)
We know that a 3D chiral object immersed in 4D (spatial only) can be
rotated into its opposite hand without inversion. It has been said
that chirality only exists in 2N+1 dimensions, not in 2N ones. This
statement is falsified by the following 2D case,
______ ______
| |
___ | | ___
| | | | | |
| _| | | |_ |
|______| |______|
In 3D the chiral bodies are trivially interconverted by rotation
only. I'm not at all clear how Noether swallows this if the rotation
is quantized and at least one necessary click is, at least in
principle, forbidden by local transition rules.
Gravitation, despite classical field theory, will yield to
quantization. Matter diddles about angstroms, muclei are
femtometers. What is the characteristic interaction scale of
gravitons? I'm not convinced that continuous rotation (continuous
angular momentum) is a valid model of a quantized world.
Can you remove low-dimension quantization by going to a superset in
higher dimensions?
Danny Ross Lunsford
> In 3D the chiral bodies are trivially interconverted by rotation
> only. I'm not at all clear how Noether swallows this if the rotation
> is quantized and at least one necessary click is, at least in
> principle, forbidden by local transition rules.
>
> Can you remove low-dimension quantization by going to a superset in
> higher dimensions?
This is a very interesting point and I have thought a lot about it
without reaching a definite conclusion. My scheme for seeing parity as a
contraction of rotation clearly requires introducing two new dimensions
in order to do the rotation, and happily this also preserves the
projective topology (even and odd dimensionsal projective spaces are
topologically distinct). Is it a coincidence that 1->3 this also brings
in the natural quantization associated with angular momentum in 3d?
Probably not.
-drl