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Dirac operators on Lorentzian manifolds
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Volker Braun
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Joined: 26 Apr 2005
Posts: 6

PostPosted: Thu Apr 28, 2005 5:23 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

On Thu, 28 Apr 2005 08:15:57 -0400, Robert C. Helling wrote:
Quote:
For example (ignoring spinors), a Moebius strip (S^1 x R^1)/Z_2 [...]
But is that more complicated than scattering in other spaces that are
not asymptotically flat

Well, the Moebius strip is completely flat. The time-slices are
compact, but you could easily extend that by adding more flat,
noncompact space-direction(s). I don't see any problem with finding
asymptotic regions. Apart from being unable to decide whether they are
past or future, of course.

Volker
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Lubos Motl
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Joined: 02 May 2005
Posts: 38

PostPosted: Sun May 01, 2005 3:14 pm    Post subject: Re: Kaluza-Klein help needed Reply with quote

Dear John,

let me first say how the actual electromagnetism arises from the
five-dimensional KK Universe.

The charge of the particle "Q" is a quantized entity that is conserved,
and therefore it is identified with the total momentum along the extra
circle, not the total velocity. The momentum is conserved; the velocity is
not.

Second, the four-dimensional mass satisfies

m4^2 = m5^2 + p5^2

and it has an extra contribution from the momentum along the fifth
direction, i.e. from the charge. Note that for a fixed circumference of
x^5, "p5" is conserved, and therefore a constant value of m5 is the same
thing as the constant value of m4.

Third, your chosen metric

Quote:
h_{ij} = g_{ij}
h_{i5} = A_i
h_{55} = 1

is morally OK, but it's not the most natural "right" choice at the
non-linear level. Even if you decide to stabilize (and omit) the
KK-dilaton (g_{55}, which would appear as a pre-factor of the second
term in the equation below), it is better to consider the metric of the
form

ds^2 = dx_i^2 + (dx^5 + A_i.dx^i)^2 (**)

Note that *this* has the right gauge invariance inherited from the
diffeomorphisms. You can consider the periodicity of x^5 to be constant,
say 2.pi, and the gauge transformation is

x^5 -> x^5 + lambda (x^i) (##)

Note that if you transform your coordinates as in (##) and you
simultaneously transform

A_i -> A_i - d_i lambda (x^j),

then the metric (**) is gonna be invariant. Your metric does not have this
property if you're accurate: the x^5-rotating diffeomorphism does not act
as an electromagnetic U(1) transformation because it influences not only
h_{i5} but also your h_{ij}.

Note that "my" more accurate metric (**) diffes from yours - for example,
h_{ij} is no longer independent of A_{i} because it contains an extra
correction of the form A_{i}.A_{j}.

I am confident that if you redo the calculation with the "improved" metric
and with the assumption that the momentum_5 is the (conserved) charge, you
will obtain exactly the correct electromagnetic force.

All the best
Lubos
______________________________________________________________________________
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Lubos Motl
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Joined: 02 May 2005
Posts: 38

PostPosted: Sun May 01, 2005 3:34 pm    Post subject: Re: Kaluza-Klein help needed Reply with quote

I forgot to mention a trivial comment. The momentum "p" - in
electromagnetism - is what corresponds to "-i.hbar.derivative" with
respect to "x". But its action on the wavefunction is not gauge-invariant.
Of course, the U(1) gauge-invariant quantity is (p-eA), if you allow me to
use the non-relativistic formula, and this is the quantity that is equal,
in the nonrelativistic theory, to "mv" where "v" is velocity. In a
(special) relativistic theory, (p-eA) is not exactly velocity, because of
the gamma factors.

Once again about the identification of the charge and the 5-momentum. In
quantum theory, it is easy - the wavefunction depends on x^5 as
exp(i.Q.x^5) if the circumference of x^5 is taken to be 2.pi. Also, "Q"
defined in this way is conserved in the quantum theory. It's clear that in
the classical theory, this will correspond to a conserved feature of the
classical trajectory. The classical trajectory is a geodesic in 5
dimensions, obtained by a parallel transport of a small 5-vector p^M whose
(constant) invariant "p^M.p^N.g_{MN}" equals "m_5^2". And the conserved
momentum will come from the second term in the metric in (**) in the
previous message - this is the U(1) gauge-invariant quantity. But of
course dx^5/dx^0 has no reason to be conserved.
______________________________________________________________________________
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Arkadiusz Jadczyk
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Joined: 02 May 2005
Posts: 21

PostPosted: Sun May 01, 2005 7:47 pm    Post subject: Re: Kaluza-Klein help needed Reply with quote

On Sat, 30 Apr 2005 08:30:31 +0000 (UTC),
baez@math.removethis.ucr.andthis.edu (John Baez) wrote:

Quote:
h_{ij} = g_{ij}
h_{i5} = A_i
h_{55} = 1

You should take:

h_{ij} = g_{ij}+A_iA_i
h_{i5} = A_i
h_{55} = 1

because with your

h_{ij} = g_{ij}
h_{i5} = A_i
h_{55} = 1

h is not necessarily invertible and even when it is, d/dx^5 is not a
Killing vector.

ark

--

Arkadiusz Jadczyk
http://quantumfuture.net/quantum_future/jadpub.htm
--
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Arkadiusz Jadczyk
science forum beginner


Joined: 02 May 2005
Posts: 21

PostPosted: Sun May 01, 2005 7:48 pm    Post subject: Re: Kaluza-Klein help needed Reply with quote

On Sat, 30 Apr 2005 08:30:31 +0000 (UTC),
baez@math.removethis.ucr.andthis.edu (John Baez) wrote:

Quote:
For a course on a classical mechanics I decided to have my
students work out the geodesics on a 5-dimensional manifold
M x U(1) with the metric h given by

h_{ij} = g_{ij}
h_{i5} = A_i
h_{55} = 1

where i,j = 1,2,3,4, g is a metric on M and A is a 1-form
on M describing the electromagnetic vector potential.

As I wrote in my previous post, it should be

h_{ij} = g_{ij}+A_iA_j

Quote:
I was hoping to get the equation for the motion of
a charged particle in a electromagnetic field, namely

m (D^2q/Dt^2)^i = e F^i_j (dq/dt)^j

where:

dq/dt is the derivative of the path q(t),

D^2q/Dt^2 is its covariant 2nd derivative,

F_{ij} = d_i A_j - d_j A_i is the electromagnetic field,

Here one must be careful. We want to write geodesic equation in 5D.
It reads

D^2q/ds^2 = 0

PROVIDED "s" is the length parameter in 5D (or an affine variation of
it).

But "length" in 5D is not the same as "length in 4D - where we want to
have just standard Lorentz force and nothing more using 4D length.

The simplest way to get the desired result quickly is by noticing that the
problem is "gauge invariant" and that we can always choose the gauge (that
is x^5 coordinate) in such a way that the A_i vanish identically on the
trajectory (there are no obstruction for making A_i to vanish along
1-dimensional curve).

ark





--

Arkadiusz Jadczyk
http://quantumfuture.net/quantum_future/jadpub.htm
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JS2
science forum beginner


Joined: 01 May 2005
Posts: 1

PostPosted: Sun May 01, 2005 7:49 pm    Post subject: Re: Novice: Indivisibility of string Reply with quote

"Urs Schreiber" <Urs.Schreiber@uni-essen.de> wrote in message
news:Pine.LNX.4.62.0504270925220.17446@feynman.harvard.edu...

Quote:
...Well, if the string is a line, there are points on that line and the
distance between them may vary.

In essence this is not different from any continuum description of, say, a
violin string, which you may found in classical mechanics textbooks.
For the prupose of getting a good description of its vibrational dynamics
we can forget about the fact that the violin string consists of atoms and
model it as a 1-dimensional continuum.

So I guess the question comes down to the string being a one dimensional
string look like, and like your next post says, its really no stranger
than a mathematical point as an elementary particle. Okay, I can
live with that, seeing as how the only reason a point is easier to
take is that I'm used to the idea ... thanks for your help. Its a pretty
strange theory ... I suppose that's how all the people felt about relativity
and quantum mechanics at the start of the 20th century.
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pirillo
science forum beginner


Joined: 02 May 2005
Posts: 5

PostPosted: Mon May 02, 2005 11:15 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

I think ultimately (and even in the intermediate steps)]
the size of the string 1) does not matter 2) is Ill defined.
One first has to define what one means by string size.
1) Conceptually
2) Experimentally
To some people string size may just be the value
of a coupling constant.
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katerina
science forum beginner


Joined: 02 May 2005
Posts: 1

PostPosted: Mon May 02, 2005 11:15 am    Post subject: Re: FAQ's Reply with quote

Urs, I am looking forward to the answers for FAQ.
Until that time I would like to read something.
I have read The Greene's Elegant Universe (thank you, Lubos, for the
translation), and started to read hep-th/9702155v1.
I doubt that it was the best choice to start with that, but I don't know
how to choose. So I will continue reading with half understanding.

I suggest another question to FAQ -

What can I read about strings?
What mathematics is needed and what should I read to understand?

Thanks
Katerina
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PostPosted: Mon May 02, 2005 5:50 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

Jack Tremarco <jacktremarco@yahoo.com> wrote in message news:<1114017473.127196.51460@l41g2000cwc.googlegroups.com>...

Quote:
This is true if you ignore non-compactness issues, which can be quite
serious. In generic time-dependent backgrounds the honest answer is
closer to a "no", at least if you demand mathematical rigor.

I have an idea about how to generalize the Dirac equation to
4-dimensional curved space-times which have a property P such that all
harmonic charts which verify P are related by Lorentz tranformations.

In one harmonic chart which verifies P, set w: T(M) -> R^4 as,

w = sqrt( eta g )

With the help of vector fields,

w^a(v_b) = d_ab

the Dirac equation can be generalized to,

( gamma^a(i v_a - A(v_a)) - m ) Psi = 0

The current might be,

j = Psi^h gamma^0 gamma^a Psi eta_ab w^b

which has *d*j = 0.

mihai cartoaje
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Wed May 04, 2005 5:10 pm    Post subject: Re: FAQ's Reply with quote

"katerina" <bubenickova@plbohnice.cz> schrieb im Newsbeitrag
news:890522e8.0505012323.4d4148df@posting.google.com...


Quote:
What can I read about strings?
What mathematics is needed and what should I read to understand?

Have a look at this:

http://superstringtheory.com/

(But stay away from the discussion forum there.)
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Thu May 05, 2005 9:00 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

"pirillo" <ultraman2002@hotmail.com> schrieb im Newsbeitrag
news:1114877221.057878.148140@g14g2000cwa.googlegroups.com...
Quote:
I think ultimately (and even in the intermediate steps)]
the size of the string 1) does not matter 2) is Ill defined.
One first has to define what one means by string size.
1) Conceptually

This is another FAQ. The last time this came up was here:

http://groups.google.de/group/sci.physics.strings/msg/01c88b017e3ccc62?hl=de
The following was my reply at that time. (There is of course much room for
improving on that reply.)

Quote:


"mandro" <ultraman2...@hotmail.com> schrieb im Newsbeitrag
news:dec722c5.0407131057.2602b41b-100000@posting.google.com...



Quote:
Well, I already said, that I'd been informed that the average length of a
string is infinity.


Yes, but by regularizing (normal ordering) the observable which measures the
size of the string, one obtains a finite value which is physically very
interesting, since it can be related to black hole entropy considerations.

I recall that you, mandro, have asked these questions before, and I think I
had answered most of them, for instance in the thread


http://groups.google.de/groups?selm=dec722c5.0303061133.1bf83085%40po...


But maybe I wasn't pointing you to enough literature. Anybody interested in
these questions should have a look at the very nice paper


Thibault Damour, Gabriele Veneziano:
Self-gravitating fundamental strings and black-holes
hep-th/9907030


and references given there, where the observable measuring the rms size of a
string is given in equations (2.9)-(2.11).


The idea is quite simple: The mean squared diameter of the string is the
average of (X-X_0)^2, taken over the worldsheet, where X_0 is the center of
mass coordinate. Now expand X in terms of worldsheet Fourier modes as usual
and then integrate over the worldsheet coordinates in order to average. The
result is (2.11), which says that the rms size is proportional to


\sum_{n=1}^\infty \frac{1}{n^2} (\alpha_{-n} \cdot \alpha_n + \alpha_n
\cdot \alpha_{-n}).


Clearly, when you take the expectation value of this guy in any string state
you'll get an infinite contribution from pulling the annihilators \alpha_n
through the creators \alpha_{-n}. This is a common quantum effect and is
removed by normal ordering. It has been argued that this infinite
contribution to the string's length has a proper physical meaning - but the
point is that the remaining finite part has, too.


In particular, the finite part is related to string/black hole
correspondence, which I have tried to review here:


http://golem.ph.utexas.edu/string/archives/000379.html .


In Paris I had a chance to look at Barton Zwiebach's new textbook on string
theory (my own copy has not arribed yet) and I saw that there, too, a very
nice summary of the string/black hole correspondence along the lines
summarized at the above link is given. So maybe mandro and others will
benefit from having a look at that book.


<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<


Quote:
To some people string size may just be the value
of a coupling constant.

That's not quite right. The value of the coupling constant in 10D string
theory is related to the dilaton which again is related to the circumference
of an extra dimension.
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pirillo
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Joined: 02 May 2005
Posts: 5

PostPosted: Tue May 10, 2005 6:19 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

Quote:
To some people string size may just be the value
of a coupling constant.

That's not quite right. The value of the coupling constant in 10D
string
theory is related to the dilaton which again is related to the
circumference
of an extra dimension.

What if there's no extra compactified dimension, what then.
What if I visualize a bosonic string living in 4 spacetime dimensions
See, the word "string size" is nearly meaningless without adding a lot
of qualifiers. If it's the average size [as you prescibed wrt a given
cutoff
procedure] , wrt a string state then I expect this to wary as the state

varies, what state are you talking about?
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pirillo
science forum beginner


Joined: 02 May 2005
Posts: 5

PostPosted: Tue May 10, 2005 6:20 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

Do you integrate over the wholeworldsheet (X-X_0 )^2 ,
or do you integrate over the sigma coordinate only?
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Wed May 11, 2005 5:06 pm    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

On Tue, 10 May 2005, pirillo wrote:

Quote:
Do you integrate over the wholeworldsheet (X-X_0 )^2 ,
or do you integrate over the sigma coordinate only?


You want to average that over space _and_ time to get the rms size of the
string. I seem to recall that this is discussed in the references that I
provided.
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Wed May 11, 2005 5:10 pm    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

On Tue, 10 May 2005, pirillo wrote:

Quote:

To some people string size may just be the value
of a coupling constant.

That's not quite right. The value of the coupling constant in 10D
string
theory is related to the dilaton which again is related to the
circumference
of an extra dimension.

What if there's no extra compactified dimension, what then.


Then it's still not true that the string size is the value of a coupling
constant.


Quote:
What if I visualize a bosonic string living in 4 spacetime dimensions


The you have a noncritical string and are in pretty deep waters.


Quote:
See, the word "string size" is nearly meaningless without adding a lot
of qualifiers.


True. So go ahead and specify precisely which notion of "string size" you
are interested in.



Quote:
If it's the average size [as you prescibed wrt a given
cutoff
procedure] , wrt a string state then I expect this to wary as the state
varies, what state are you talking about?


Indeed. That procedure I mentioned gives you an operator and taking the
expectation value of that operator in a given state of the string gives
the rms size of the string in that state.
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