On May 25, 7:16 am, Lax <Lax.Cla...@gmail.com> wrote:

On May 25, 6:12 am, Hatto von Aquitanien <a...@AugiaDives.hre> wrote:

What does a mathematician call the thing a physicist calls a scalar
field? I am of the impression that "scalar field" to a mathematician is
an algebraic structure devoid of geometric meaning.

A function from a subset of F^n into F, where F is a field, (thus F^n
is can be considered vector space).

A field (to a mathematician) is a section on a bundle whose base space
is 4-dimensional (for field theory and dynamics), 3-dimensional (for
statics or equilibrium physics in field theory) or 1-dimensional (for
mechanics) or even 0-dimensional (for statics or equilibrium physics
in mechanics). This is the modern mathematical treatment of a field
and includes the notion of particles and systems with finite numbers
of degrees of freedom, also, as special cases.

If the base space X is locally coordinatized by X: (x^0,...,x^{n-1})
(where n = 1, 3 or 4 here), the bundle Q will have local coordinates
Y: (x^0, ..., x^{n-1}, q^0, ..., q^{p-1}). The "fibre" Q_x at each
point (x) in X is a space locally coordinatized by Q_x: (q^0, ...,
q^{p-1}). The usual definition of a fibre bundle also requires that
the fibres Q_x, Q_x' for two different points (x), (x') in X have the
same structure, so that one can talk about a "typical fibre" Q
irrespective of the point x.

For a scalar field, n = 4 and p = 1; the fiber is 1-dimensional and is
either the real line or a circle.

For n = 1, the coordinate X: (t) is just time. For an ordinary
particle (without internal structure) n = 1, p = 3. The field
coordinates are Q: (x, y, z) (i.e. the coordinates of the particle),
and a section is then a map r: t in (a,b) |-> (x(t), y(t), z(t)),
which describes a trajectory for a given time interval (a,b).

The bundle is trivial if it is topologically equivalent to a product
space
Y = X x Q,
with the "fibre" space Q possessing the coordinates (q^0,...,q^{p-1}).
Not all bundles are trivial. Therefore, this is NOT the same thing as
saying that the space is
Total Configuration Space = (Spacetime) x (Field Configuration
Space).

All bundles over the 1-dimensional base space, however, are trivial if
the base space is the real line. If, on the other hand, the base space
is a circle, then there are plenty of non-trivial bundles (the Moebius
strip, in fact, provides the basis for the example of one).

A local section q: X -> Y is a map q: (x) |-> (x, q(x)) that picks out
a particular configuration on the fibre at each point x.

If the domain of the section dom(q) = { x in X: q(x) is defined } is
all of the base space X, then the section is a global section and can
then be thought of as a function. But more generally, sections are not
functions!

More to the point: a field is NOT regarded as merely a function in the
standard mathematical treatment of field theory -- which is what the
original question relates to.

The first derivatives of the field reside in the jet bundle J1(Y),
whose coordinates locally are
J1(Y): (x^m: m=0,...,n-1; q^a: a=0,...,p-1; v^a_m: a=0,...,p-1,
m=0,...,n-1).
The section q: X -> Y representing a field then extends to its "first
derivative" section (which is called the first jet extension of q),
j1(q): (x) |-> (x, q(x), dq(x)/dx); i.e. v^a_m = dq^a/dx^m.

For mechanics, n = 1, and v^a_m reduces to the single component v^a =
dq^a/dt -- the "generalized velocity". Therefore, the extra
coordinates v^a_m are sometimes called the fields "velocity"
components.

The configuration space for a field has n+p diimensions. Counting the
velocity components, you get (n+1)(p+1)-1 dimensions for the total
space J1(Y). For a scalar field (n = 4, p = 1), this gives you
respectively 5 and 9 for the dimensions of Y and J1(Y).

Lagrangians, for instance, are defined in reference to these objects.
So, the action S is actually a function of a region Omega in the base
space X, and a section q: X -> Y (the space of sections is sometimes
denoted Gamma(Y) or explicitly Gamma(X->Y)),
S(Omega, q) = integral (L(j1(q)) omega)
where omega = dx^0 dx^1 ... dx^{n-1} and the integral is taken over
the region Omega and the Lagrangian L: J1(Q) -> R.