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John Baez
#6
Nov4-06, 03:20 PM
P: n/a
In article <437BBA7A.5020804@xs4all.nl>,
Gerard Westendorp <westy31@xs4all.nl> wrote:

>John Baez wrote:


>> The idea of p-form electromagnetism is to replace point particles
>> by strings or higher-dimensional membranes. To see how this goes,
>> it's enough to look at 2-form electromagnetism.
>>
>> In 2-form electromagnetism, the star of the show is a 2-form, A.
>> As already mentioned, a 2-form is a gadget you can integrate over a
>> surface and get a number. In 2-form electromagnetism, this number
>> describes the change in phase that a charged string acquires as it
>> moves along, tracing out a surface in spacetime.


>Could string theory really be that easy?


No. But, the *bosonic* string is quite easy.

I've described the action that a string gets by interacting with
the 2-form analogue of the electromagnetic field - string theorists
call this the B field, or sometimes the "Kalb-Ramond" field. There's
also another term in the action, which is just proportional to the
*area* of the string worldsheet. So, for a string of tension m and
charge q,

string action = m (area of worldsheet)
+ q (integral of B field over worldsheet)

which is just like the action for a point particle in general relativity:

particle action = m (length of worldline)
+ q (integral of A field over worldline)

where now m is the particle's mass.

The fun starts when you try to quantize this theory, and see that
nasty stuff happens except in 26 dimensions - and that even then,
the string has tachyonic vibrational modes.

And, the fun *really* gets going when you consider the supersymmetric
version of the string.

But, the basic idea is pretty simple. Anyone who wants to learn
this stuff should read Zweibach's book "A First Course in String Theory".
It's aimed at smart undergraduates. I'm not sure undergrads should be
spending time on a theory for which there's no evidence, but it's a
very good book.

>My understanding till now has been:
>
>scalar field
><-> spin 0 particles
><-> potential (phi) defined on vertices
><-> field defined on edges, d(phi)
>
>vector field
><-> spin 1 particles
><-> potential (A) defined on edges
><-> field (F) defined on faces (loops of edges), F= dA
>
>The next step up, in which the potential is defined on faces,
>and the field is defined on "solids", ( "loops" of faces), did
>seem natural to me, but I thought this would just be a field of
>spin 2 particle wave functions.


This is the right story for "discretized p-form electromagnetism",
which you can read about here:

http://math.ucr.edu/~derek/pform/

You are talking about

0-form electromagnetism (the scalar field phi),

then

1-form electromagnetism (the vector potential A),

then

2-form electromagnetism (the Kalb-Ramond field B),

and so on...

and you're getting fooled into thinking there's a certain
pattern that's not really there.

In 4d spacetime a 0-form acts like a spin-0 particle since it
transforms in the spin-0 representation of the rotation group
(i.e. a scalar doesn't change when you rotate it).

and

In 4d spacetime a 1-form acts like a spin-1 particle since it
transforms in the spin-1 representation of the rotation group
(i.e. a 1-form transforms like a vector when you rotate it).

but you don't get spin-2 particles this way, since:

In 4d spacetime a 2-form acts like a spin-1 particle since it
transforms in the spin-1 representation of the rotation group
(i.e. a 2-form transforms like a vector when you rotate it).

So, gravity is very different than the B field. String theory
involves both. Certain string theorists love to proclaim that
"string theory predicts gravity", which is true in a sense -
though the word "postdict" is more appropriate. But, string
theory also predicts the B field... which you don't tend to hear
string theorists boast about, for some mysterious reason.