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Dirac operators on Lorentzian manifolds
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zitterbewegung@gmail.com1
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Joined: 24 May 2006
Posts: 8

PostPosted: Sat Jul 15, 2006 10:54 am    Post subject: PDF version of Idea Reply with quote

Here is a pdf of the idea with minor modifications.
http://1946875.tripod.com/brane-hybrid-manifold.pdf

[Moderator's note: the newsgroup does not endorse this document. LM]
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zitterbewegung@gmail.com1
science forum beginner


Joined: 24 May 2006
Posts: 8

PostPosted: Thu Jun 08, 2006 2:25 am    Post subject: Rewrite of my assumptions. Was: Mistakes (again) Reply with quote

[Moderator's note: no responsibility for the content. LM]

This is an attempt to define a bulk which corresponds to both a string
cosmological and brane cosmological formalism which disallows other
branes that exist outside this universe.

Our bulk is homogenous manifold similar to a Joyce manifold but this
then glued at every point to an AdS4 which forms a surface to the
manifold . This manifold forms the background of everything that
exists. This bulk exists

Definition of the Hybrid Manifold

We define our bulk as continuous map of all points on a AdS4 (which may
or may be periodic) manifold and the Joyce manifold.

We state that the observation of the propagation of point particles is
actually a reflection of the current shape of the Joyce manifold. On
our manifold when the Joyce manifold bends or is warped then the AdS4
manifold's characteristics change as a result of the continuous
mapping. In other words the AdS4 manifold shows the shape of the
underlying substructure of the Joyce manifold. The Joyce manifold is
bound to a certain set of shapes and transformations which must follow
the laws of physics in the AdS4.

But now we must extend the hybrid manifold to have properties that
reflect current physical observations. We must have a manifold that is
both restricted to a subset of shapes that can exist on the manifold.
Also we must deal with the fact that energy is quantized.

A quantization of energy using hybrid manifolds

As a string moves through the part of our manifold that consists of the
G2 manifold it bends the manifold slightly so that we can detect the
point particle on the AdS4 space. The surface of our new manifold is
glued and because of this it can reflect properties of its bending The
reason we can detect different messenger particles is because as a
string moves through our hybrid manifold it is bound at certain parts
of the hybrid manifold due to the structure of the manifold. There are
only a specific set of path's that the string can move through and that
determines how the surface is bent.

Another property is that due to quantization the vibrations of the
string follow an ascending or descending chain of vibrational patters
as the strings gain or lose energy. Therefore each string follows a set
of vibrations given a specific amount of energy that a string
possesses.

Disallowing Violations of the Laws of Physics

Certain properties of physics can be disallowed by the Joyce Manifold
being unable to transform to that state ; because to do so would change
the topology of the space so greatly that the energy required would be
enormous.

Gravity Representation

There are two ways to represent gravity using this hybrid space. One
way is to give it a formal messenger particle and have a given energy
value for it. Another method is to say that gravity is a result of the
specific shape of the Joyce manifold and requires no messenger particle
whatsoever. This representation will be chosen once we have direct
evidence for either outcome and may remain independent of the theory.
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zitterbewegung@gmail.com1
science forum beginner


Joined: 24 May 2006
Posts: 8

PostPosted: Thu Jun 08, 2006 2:24 am    Post subject: Hybrid Manifold Reply with quote

Abstract: [Moderator's note: no responsibility for content. LM]

This is an attempt to define a bulk which corresponds to both a string
cosmological and brane cosmological formalism which disallows other
branes that exist outside this universe.

Our bulk is homogenous manifold similar to a Joyce manifold but this
then glued at every point to an AdS4 which forms a surface to the
manifold . This manifold forms the background of everything that
exists.

Definition of the Hybrid Manifold

We define our bulk as continuous map of all points on a AdS4 (which may
or may be periodic) manifold and the Joyce manifold.

We state that the observation of the propagation of point particles is
actually a reflection of the current shape of the Joyce manifold. On
our manifold when the Joyce manifold bends or is warped then the AdS4
manifold's characteristics change as a result of the continuous
mapping. In other words the AdS4 manifold shows the shape of the
underlying substructure of the Joyce manifold. The Joyce manifold is
bound to a certain set of shapes and transformations which must follow
the laws of physics in the AdS4.

But now we must extend the hybrid manifold to have properties that
reflect current physical observations. We must have a manifold that is
both restricted to a subset of shapes that can exist on the manifold.
Also we must deal with the fact that energy is quantized.

A quantization of energy using hybrid manifolds

As a string moves through the part of our manifold that consists of the
G2 manifold it bends the manifold slightly so that we can detect the
point particle on the AdS4 space. The surface of our new manifold is
glued and because of this it can reflect properties of its bending The
reason we can detect different messenger particles is because as a
string moves through our hybrid manifold it is bound at certain parts
of the hybrid manifold due to the structure of the manifold. There are
only a specific set of path's that the string can move through and that
determines how the surface is bent.

Another property is that due to quantization the vibrations of the
string follow an ascending or descending chain of vibrational patters
as the strings gain or lose energy. Therefore each string follows a set
of vibrations given a specific amount of energy that a string
possesses.

Disallowing Violations of the Laws of Physics

Certain properties of physics can be disallowed by the Joyce Manifold
being unable to transform to that state because it is impossible.
Certain holes that may occur may require a great deal of energy which
are prohibitive to accomplish because they require a great deal of mass
, energy, or mass/energy density. Therefore they would be difficult or
maybe even impossible to accomplish.

Gravity Representation

There are two ways to represent gravity using this hybrid space. One
way is to give it a formal messenger particle and have a given energy
value for it. Another method is to say that gravity is a result of the
specific shape of the Joyce manifold and requires no messenger particle
whatsoever. This representation will be chosen once we have direct
evidence for either outcome and may remain independent of the theory.


Conjectures

An interesting conclusion that we can get from this formalism is that
we don't need a set of interacting branes to start the process of the
formation of the universe. Instead some event must create a difference
inside of the space and make it form into a stable state. Therefore we
can confine the universe formation to just this brane.

Another conclusion we might be able to draw is that the underlieing
substructure allows for the space to be improbablly flat and bypasses
all horizons by being glued to a singular space that influences all
spacetime. That it is merely a propery of the substructure and
independant of an alternate form of matter.
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Puppet_Sock
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 124

PostPosted: Mon Jun 06, 2005 9:41 pm    Post subject: Re: From Magnetic Monopole to Magnetic Strings Reply with quote

Golden Boar wrote:
Quote:
I made a huge mistake here, a positive magnetic monopole would
annihilate with a negatice magnetic monopole, so no magnetic strings
would be formed.

Not necessarily. Monopoles can have electric and topological
charge. There's a name for it objects with both, but I don't
recall it just off. Both values need to be coserved.

You should get a review article or two before you continue.
Socks
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mihai cartoaje
science forum beginner


Joined: 29 May 2005
Posts: 10

PostPosted: Sun May 29, 2005 6:03 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

i have written:

Quote:
j = Psi^h gamma^0 gamma^a Psi eta_ab w^b
which has *d*j = 0.

This is not always true from the construction.
Actually,

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

has the divergence *d*j = j^b w^a(v_a, v_b).

FAIK, it might work to find a coordinate
system in which w = sqrt( eta g ) has the
property v_a . w^a = 0. At first order,
a solution exists iff R = 0 + O(h^2).
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pirillo
science forum beginner


Joined: 02 May 2005
Posts: 5

PostPosted: Tue May 17, 2005 6:11 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

Quote:
you will want to take the time average over an interval which is
large enough compared to the fluctuations but small enough to be local
in time with respect to the long-term behaviour.

Yeah, but you said integrate over the "whole" world sheet
not over a "short time band" on the worldsheet. So you're averaging
over the "whole" infinite history. I'm just saying what you seemed to
say --- not what you meant. Which I now think is to integrate over a small
time band.

Quote:
To be frank, I feel that the discussion of this point is getting a
little off-topic for sci.physics.strings.

And questions about Gerbes, Calabi Yau manifolds and all these objects
which are purely mathematical are not! Ha Smile
I think this "is" very stringy!
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Thu May 12, 2005 6:33 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

"pirillo" <ultraman2002@hotmail.com> schrieb im Newsbeitrag
news:1115858009.327557.225680@z14g2000cwz.googlegroups.com...
Quote:
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.

Oh, so you do sort of a " average size for the whole history"
operator which say first computes the average distance from cm at each
time and then averages this over the whole history.

Hmmm?

I am sorry, but I have a hard time understanding what you are confused
about.

The idea we are talking about is not particular to string theory at
all but seems to be just a matter of common sense: You have some fluctuating
something and want to get an idea of its rough size. So you average the
distance of all its points from its center of mass and, since its
fluctuating, average these distances over some period of time. If that piece
of something is systematically growing or shrinking on larger time scales
you will want to take the time average over an interval which is large
enough compared to the fluctuations but small enough to be local in time
with respect to the long-term behaviour.

To be frank, I feel that the discussion of this point is getting a little
off-topic for sci.physics.strings.
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pirillo
science forum beginner


Joined: 02 May 2005
Posts: 5

PostPosted: Thu May 12, 2005 5:46 am    Post subject: Re: Size of strings compared to size of elementary particles Reply with quote

Quote:
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.

Oh, so you do sort of a " average size for the whole history"
operator which say first computes the average distance from cm at each
time and then averages this over the whole history.

Hmmm?

I thought in curved backgrounds you did this at one instant
and found that the string became larger as time passed.
Or, were you saying more like there's an external parameter
which labels a family of spacetimes (the string lives in)
and as you vary this parameter, the quantity you described
above varies, although --if one changes the target space,
then one changes the states.
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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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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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pirillo
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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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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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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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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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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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