Somehow I am confused about a sign in the D1-brane action:

The DBI term for vanishing dilaton and gauge field is

sqrt( -det(G+B) ) = sqrt( -det(G) - (B_01)^2 ) .

But since det(G) 0 and (B_01)^2 = 0 the term under the square root
can become negative.

This would be avoided if the _temporal_ part of the B-field pullback
were multiplied by i as in

sqrt( -det(G + iB) ) = sqrt( -det(G) + (B_01)^2 ) .

This form is also what one would get from Wick-rotating the Euclidean
DBI action.

Am I making an elementary mistake? (Well possible, I am tired...)

On Tue, 13 Jan 2004, Urs Schreiber wrote:

Somehow I am confused about a sign in the D1-brane action:
The DBI term for vanishing dilaton and gauge field is

sqrt( -det(G+B) ) = sqrt( -det(G) - (B_01)^2 ) .

But since det(G) 0 and (B_01)^2 = 0 the term under the square root
can become negative.

That's correct, because the sign must be such that it *can* become
negative. I will explain it below.

This would be avoided if the _temporal_ part of the B-field pullback
were multiplied by i as in

sqrt( -det(G + iB) ) = sqrt( -det(G) + (B_01)^2 ) .

If you only redefine the coefficient of the temporal part, you obviously
break Lorentz invariance. If you redefine all coefficients, then you will
face the same problems with the space-space (magnetic) components of the
B-field.

But at the end, it is not important what *you* decide to write down. The
important question is what the action is in reality, and it can be
calculated, and there is indeed sqrt( -det(G) + (B_01)^2 ... ) which, on
mathematical grounds, can grow sqrt(negative).

I am sure that you've heard of it, but let me remind you why this is an
important physical effect that you must know. In the action, you really
find not only "B", but the sum "B+F" with the right coefficients - this is
the only combination that is invariant under the 2-form gauge invariance.

Consequently, a nonzero value of B_{01} is, up to a gauge transformation,
equivalent to a nonzero value of F_{01} which is the electric field living
on the brane. This means that you are really asking whether there is an
upper bound on the value of the electric field. Be sure that there is one!

Imagine an open string inside the D-brane. Its ends are oppositely (+-)
charged under the U(1) gauge field of the brane (or fundamental vs.
antifundamental of U(N), if you consider a stack of D-branes). In other
words, the endpoints behave as "electrons" or "quarks" with opposite
charges. If the separation of the endpoints is "R" (a spatial vector),
then the open string acts as a dipole p = Q.R.

The energy of a dipole in the electric field is the sum of the
electrostatic energies of both endpoints, namely p.E (the dot product with
the electric field) which is Q.R.E. Note that it is proportional to "R",
and it can have both signs, depending on the orientation of the dipole.
However there is also a contribution to the energy from the tension,
namely T.|R|. Note that for large enough electric field "E", the energy of
the dipole will be more relevant than the tension, and if you orient the
dipole in the "negative" direction, the full energy of the open string,
including the tension, will be negative, proportional to R, and - in fact
- unbounded from below.

In other words, the open string will attempt to grow, and it will become
unstable. This situation can never really occur physically because in
large electric fields, open strings will be spontaneously created which
will reduce the value of the electric field.

OK, the value of the electric field where the tension can precisely
compensate the energy of the E-directed dipoles, is called the "critical
electric field" and it is precisely the point where you reach the zero
under the square root. Before you reach this point of the critical
electric field, there is an interesting scaling limit. A new
non-gravitational decoupled theory can be separated in this regime, and it
is called NCOS - non-commutative open string theory.

All the best
Lubos
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On Tue, 13 Jan 2004 21:17:59 +0000 (UTC), Urs Schreiber wrote:

The DBI term for vanishing dilaton and gauge field is

sqrt( -det(G+B) ) = sqrt( -det(G) - (B_01)^2 ) .
But since det(G) 0 and (B_01)^2 = 0 the term under the square root
can become negative.

So what, that was the original motivation for the DBI action: There is
a limiting electric field so the self energy of the electron is
regulated. Compare this to the relativistic gamma factor that can
become imaginary for superluminal velocities. Some years ago, there
was some discussion of this issue, look up references around

SPACE-TIME NONCOMMUTATIVE FIELD THEORIES AND UNITARITY.
By Jaume Gomis, Thomas Mehen (Caltech & CIT-USC),. CALT-68-2272, CITUSC-00-023, May 2000. 15pp.
Published in Nucl.Phys.B591:265-276,2000
e-Print Archive: hep-th/0005129

Robert


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Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling

"Robert C. Helling" wrote in message ...
On Tue, 13 Jan 2004 21:17:59 +0000 (UTC), Urs Schreiber wrote:

The DBI term for vanishing dilaton and gauge field is

sqrt( -det(G+B) ) = sqrt( -det(G) - (B_01)^2 ) .
But since det(G) 0 and (B_01)^2 = 0 the term under the square root
can become negative.

So what, that was the original motivation for the DBI action: There is
a limiting electric field so the self energy of the electron is
regulated.

I see, didn't know this.

Compare this to the relativistic gamma factor that can
become imaginary for superluminal velocities. Some years ago, there
was some discussion of this issue, look up references around

SPACE-TIME NONCOMMUTATIVE FIELD THEORIES AND UNITARITY.
By Jaume Gomis, Thomas Mehen (Caltech & CIT-USC),. CALT-68-2272,

Thanks! I'll look it up.

BTW, there was a certain reason that I began to worry about this sign.
But this is a story that is hard to tell in ASCII. If anyone is
interested he or she should have a look at the String Coffee Table
entry http://golem.ph.utexas.edu/string/archives/000288.html which I
have just written, and where the formulas can be viewed in
pretty-printed style (when the necessary fonts are downloaded).

In that entry I discuss the canonical analysis of the D-string action
a little bit and show why I am interested in a similar action but with
that factor of "i" included. I also mention a paper where such a
factor actually appears, even though Lubos warns me that this paper
may have problems.