Differential geometry in Gauge Fields, Knots and Gravity?
alleg...@uchicago.edu
technical for sci.math or sci.physics. I'm trying to learn
differential geometry from a book by John Baez and Javier Muniain
called Gauge Fields, Knots and Gravity, but I'm having trouble
understanding the authors' definitions of vector fields and 1-forms.
I think most people start by defining a tangent vector at a point $p$
of a manifold $M$ to be a function from $C^infty(M)$ to the real
numbers which is linear and satisfies Leibniz's product rule. Then you
can define a vector field to be a function which associates, to every
point of the manifold, a tangent vector at that point.
Baez and Muniain take a different approach which I don't understand.
They first define a vector field to be a function from $C^infty(M)$ to
$C^infty(M)$ which is linear over the real numbers and satisfies
Leibniz's product rule. Now this definition of a vector field is
different from the one I just gave; Baez and Muniain are saying that a
vector field is a function on the set of smooth functions, not a
function on the manifold itself. This seems to cause difficulties
later when the authors state that a vector field is a section of the
tangent bundle. A section is a function defined on the manifold, not
the set $C^infty(M)$ of smooth functions.
I'm also confused when Baez and Muniain talk about 1-forms. These
authors define a 1-form to be a linear function that sends vector
fields to smooth functions. Later the authors state that a 1-form is a
section of the cotangent bundle, but this is impossible since a
section is a function defined on the manifold itself, not the set of
vector fields.
So I have several questions: What is a vector field? What is a 1-form?
Do these functions act on the points of the manifold, or other sets?
Sorry to ask such an elementary question. It's been bothering me for a
long time.
Maarten Bergvelt
On 2008-09-11, alleg...@uchicago.edu <alleg...@uchicago.edu> wrote:
> I think most people start by defining a tangent vector at a point $p$
> of a manifold $M$ to be a function from $C^infty(M)$ to the real
> numbers which is linear and satisfies Leibniz's product rule.
Right, a tangent vector is a "directional derivative" at p, acting on
functions defined near p.
> Then you
> can define a vector field to be a function which associates, to every
> point of the manifold, a tangent vector at that point.
Right, and the association should depend smoothly on p. But this means
that if you have a function an a vector field, you get for each p a
number depending smoothly on p, i.e., you get another C^\infty
function. So a vector field is not only a function on M giving for
each p a tangent vector, but also a machine that takes a function and
produces another function (by differentiation).
This is really the same as in calculus: you can define the derivative
of a function f(x) at x_0, and get a number, or take the derivative
of f everywhere, and get a new function f^\prime(x).
I think you should just work through all the concepts in differential
geometry in the case where M is one dimensional, e/g. R, the real line.
That should clarify a lot.
> Baez and Muniain take a different approach which I don't understand.
> They first define a vector field to be a function from $C^infty(M)$ to
> $C^infty(M)$ which is linear over the real numbers and satisfies
> Leibniz's product rule. Now this definition of a vector field is
> different from the one I just gave; Baez and Muniain are saying that a
> vector field is a function on the set of smooth functions, not a
> function on the manifold itself. This seems to cause difficulties
> later when the authors state that a vector field is a section of the
> tangent bundle. A section is a function defined on the manifold, not
> the set $C^infty(M)$ of smooth functions.
>
> I'm also confused when Baez and Muniain talk about 1-forms. These
> authors define a 1-form to be a linear function that sends vector
> fields to smooth functions. Later the authors state that a 1-form is a
> section of the cotangent bundle, but this is impossible since a
> section is a function defined on the manifold itself, not the set of
> vector fields.
>
> So I have several questions: What is a vector field? What is a 1-form?
> Do these functions act on the points of the manifold, or other sets?
Think about a differential operator like L= d/dx + a(x) on the real
line. This is a function of x taking values in operators. So it both
"acts" on the base manifold in this case the real line, >>>and<<< also
acts on functions on the manifold. There is no contradiction.
Vector fields are just slightly more complicated objects than you have
seen before.
Hope this helps.
--
Maarten Bergvelt
Theo Wollenleben
> I'm trying to learn differential geometry from a book by John Baez
> and Javier Muniain called Gauge Fields, Knots and Gravity, but I'm
> having trouble understanding the authors' definitions of vector
> fields and 1-forms.
>
> I think most people start by defining a tangent vector at a point $p$
> of a manifold $M$ to be a function from $C^infty(M)$ to the real
> numbers which is linear and satisfies Leibniz's product rule. Then
> you can define a vector field to be a function which associates, to
> every point of the manifold, a tangent vector at that point.
Baez and Muniain actually use the same definition. In the section
"Tangent Vectors" they give this definition of a tangent vector
independently of their previously defined concept of vector fields
(using the different approach you describe below):
"Henceforth, we will simply /define/ a tangent vector at p in M to be
a function from C^infinity(M) to R satisfying these three properties."
Later in the section "Vector Bundles" they define sections in the
standard way, calling the set of all sections of the tangent bundle
Gamma(TM).
> Baez and Muniain take a different approach which I don't understand.
> They first define a vector field to be a function from $C^infty(M)$
> to $C^infty(M)$ which is linear over the real numbers and satisfies
> Leibniz's product rule.
And that's an element of what they call the set of all vector fields
Vect(M).
> Now this definition of a vector field is different from the one I
> just gave; Baez and Muniain are saying that a vector field is a
> function on the set of smooth functions, not a function on the
> manifold itself. This seems to cause difficulties later when the
> authors state that a vector field is a section of the tangent bundle.
> A section is a function defined on the manifold, not the set
> $C^infty(M)$ of smooth functions.
I think the confusion comes from the sloppiness of the authors when they
formulate:
"Exercise 64. Show that a section of the tangent bundle is a vector
field."
Perhaps they mean:
Show that F: Gamma(TM) -> Vect(M), F(s)(f)(p):=s(p)(f) is an isomorphism
of modules.
> I'm also confused when Baez and Muniain talk about 1-forms.
I guess the confusion with 1-forms corresponds to the vector field
confusion.
Rock Brentwood
this is as follows. The manifold M has a transformation group Diff(M)
comprising the active transformations of M into itself -- i.e.
diffeomorphisms.
The Lie algebra of Diff(M) consists of all infinitesimal flows. These
are the vector fields. At each point, x of the manifold M, the flow
delta takes on the value delta^{mu}(x) d/dx^{mu} of a tangent vector
at x.
The Lie bracket of this Lie algebra is the Lie bracket of the vector
fields, themselves. The Lie algebra is infinite-dimensional
(uncountably infinite dimenions, but with a countable-dimensional
dense subspace).
The tangent vectors at a point x are the first derivatives of curves
passing through x. Thus, if r(s) is a curve (over some interval s in
(a,b)) and s(c) = x for some c in (a,b), then r'(c) is a tangent
vector at x. In general, r'(s) is in the tangent space T_{r(s)}(M).
The tangent vectors are infinitesimal line segments. The vector fields
connect up these segments at adjacent points and produce the above-
mentioned flows.
In a more algebraic approach, you end up pulling the rug out from
underneath everything and forget where the function space EM =
C^{infinity}(M) comes from. Instead, you treat it as a regular algebra
(with products, sums of functions defined point-wise).
Then you can talk about the Leibnitz operators of this algebra D: EM -
> EM (or "derivations"). The "inner derivations" are those that
correspond to differential operators
D = delta^{mu} d/dx^{mu}
while the outer derivations are those that do not. For a finite-
dimensional C^{infinity} manifold, all derivations are inner. This is
a consequence (and generalized restatement) of Taylor's Theorem.
Taylor's Theorem, however, does not generally holds for functions of
an infinite number of variables, so there need not be a Taylor's
Theorem for infinite-dimensional manifolds. (An example of an infinite-
dimensional manifold is Diff(M). But I don't know if it has outer
derivations).
The correct approach is the one mathematicians take: define the
vectors and vector fields the same way as they had always been
historically conceived: as arrows and flows. THEN characterize them in
relation to derivations. But they are not actually defined as
derivations.
So, the Baez treatment is a circumlocution (similarly for the way the
geometry underlying gauge fields is handled).
If you want a more cohesive formulation, the best place to look for is
a monograph that works in terms of principal bundles for gauge fields,
and associated bundles for "matter" fields that exhibit gauge
invariance. This is the closest that I've seen to what can be called a
standard treatment of these two applications.
Both the Baez book, and the older book by David Bleecker (I forget the
title, it's from 1980 or 1981) evade associated bundles -- at the cost
of having it come around and bite from the rear. Bleecker, in fact,
avoids them to "avoid the 'complication' of their formulation", but
then defeats the point by bringing them in anyway, later on.
If you want to handle this from a purely algebraic point of view,
there is in fact a large algebraic underpinning to both formalisms
which contains 90% of the respective formalisms.
A Lie group G has operations
(a,b) in G |-> ab in G;
a in G |-> a^{-1} in G;
e in G.
Since it's also a differentiable manifold, then the tangent spaces
(and even co-tangent spaces) inherit a many-sorted algebra from this.
a in G, v in T_b(G), c in G |-> avc in T_{abc}(G)
such that
d/ds (a b(s) c) = a b'(s) c.
Then you can factor the tangent bundle into the gauge gruop G and the
tangent space L = T_e(G) at the identity by
T_g(G) = g L = L g.
The left-factoring produces left-invariant fields, g |-> gv (v in L);
the right factoring produces right-invariant fields g |-> vg (v in L)
(or maybe I got the terms backwards, I forget). The inverse operators
g\(_): v in T_g(G) |-> g\v in T_e(G) = L
(_)/g: v in T_g(G) |-> v/g in T_e(G) = L
are the "Cartan-Maurer forms".
A princpal bundle P over G is a many-sorted algebra that introduces
the following additional operators:
a product p in P, g in G |-> pg in PG
and a partially-defined quotient
p in P, q in P/G |-> p\g in G.
The space partitions into cosets P/G = {pG: p in P} where each coset
or "orbit" is defined by
pG = { pg: g in G}
and basically describes a point pG = m of the base space M = P/G. This
makes a principal bundle a copy of the base space M, in which each
point m in M is replaced by a copy pG of the group G.
The operations are inherited by the tangent spaces:
v in T_p(P), g in G |-> vg in T_{pg}(P)
p in P, v in T_g(G) |-> pv in T_{pg}(G).
The vertical vectors reside are those of the form pv, for v in T_g(G).
It can always be reduced to a product in PL by
pv = pg(g\v).
The quotient, however, does NOT generalize to tangent spaces. To
define the operations
v in T_p(P), q in pG |-> v\q in T_{p\q}(G)
and
p in qG, v in T_q(P) |-> p\v in T_{p\q}(G)
requires additional structure.
It requires a connection. In fact, the quotient IS the connection. The
connection 1-form is just
omega_p(v) = p\v, for v in T_p(P).
Conversely, the quotient can be expressed in terms of the connection.
Finally, an associated bundle X extends the many-sorted algebra by
introducing a vector space V (which carries the degrees of freedom of
a prospective matter fields), and an operation
g in G, v in V |-> gv in V
(which represents the gauge transformation behavior on the matter
fields).
The extra structure amounts to new operations of the form
p in P, v in V |-> p.v in X
p in P, x in X |-> p\x in V
which satisfies the properties
p\(q.x) = (p\q) x
p.p\x = x
The space X is a bundle with a projection operator
pi(p.v) = pG.
So operation p\x, like the quotient for P, is restricted only to those
x where pi(x) = pG.
All these generalize to operations on the tangent spaces. The
derivative of the quotient operator, as in the case of P, again both
requires (and yields) a definition for a connection. This connection
is none other than the covariant derivative operator on V.
For the space P, you can use the connection to define "horizontal"
lifts. This is done with the aid of a local section S: M -> P. A
section is defined by the property
S(m)G = m
where m is identified with the corresponding orbit in P. The
horizontal vectors H_p(P) in T_p(P) are those given by
H_p(P) = { v in T_p(P): p\v = 0 }.
Then you can use S to convert place a vector v in T_m(M) onto a
horizontal vector at in H_p(P) at some point p in the orbit
corresponding to m. The explicit formula is simply:
Lift_p(v) = S(m) (S_{*m}(v)\p) + S_{*m}(v) S(m)\p.
Or, more briefly,
Lift_p = S dS\p + dS S\p.
A similar construction can be done for an associated bundle X.
All the foregoing shows that there is a simplicity underlying both the
principal bundle and associated bundle formalisms that belie the
notion that they ought to be avoided. In fact, there is a great deal
of transparency to be gained by actually making use of them, instead
of the various obscuring circumlocutions employed in references like
Bleecker or Baez in their place.