Natural Science Forum / Physics / Research / May 2008

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This needs to be fixed and more carefully done. The combined n-form
algebra you're seeking takes place in Lambda(M) x L. This is what
needs to be more carefully parsed.

So, first things first. Given a vector space V of n dimensions, the
C(n,k) dimensional space Lambda^k(V) is the space formed by all k-
vectors. The 2-vectors in Lambda^2(V) are, for instance, linear
combinations of (u (x) v - v (x) u) for u, v in V, where (x) denotes
tensor product.

The space-time manifold M may be coordinatized locally by (x^m) where
the space-time index m ranges from 0 to n-1, and n is the dimension of
M. At each point m in M, the tangent space T_m(M) has local
coordinates (x-dot^m) and is equivalent, as a vector space, to R^n.
Together, these form the "fibres" of the tangent space TM whose local
coordinates are (x^m, x-dot^m).

The co-tangent space T_m*(M) is the dual of T_m(M) and has coordinates
(p_m). It, too, is equivalent to R^n. This forms the co-tangent bundle
T*(M) which has local coordinates (x^m, p_m).

Over each set of vector spaces one may define the spaces of anti-
symmetric k-vectors: Lambda^k(T_m(M)) and Lambda^k(T_m*(M)). They are
coordinatized locally by
  Lambda^k(T_m(M)): (v^{m1 ... mk})
and
  Lambda^k(T_m*(M)): (p_{m1 ... mk}),
respectively. These give you the spaces, respectively, of k-
multivectors and k-forms over the point m of M.

Together, these give you the spaces Lambda^k(T(M)) and
Lambda^k(T*(M)), with respective coordinates given by (x^m,
v^{m1...mk}) and (x^m,p_{m1...mk}).

The respective fields are smooth SECTIONS over these spaces. That is,
a k-multivector field v: M --> Lambda^k(T(M)), and a k-form field
omega: M --> Lambda^k(T*(M)) such that v(m) is in Lambda^k(T_m(M)) and
omega(m) is in Lambda^k(T_m*(M)) for each m in M; the mappings are as
C^{infinity} functions.

The spaces of k-form fields will be denoted as Lambda^k(M).

The space Lambda(M) will then be defined as the sum of the Lambda^k(M)
over all k = 0, 1, ..., n.

The 0-forms are scalar functions over M. This may be conveniently
taken as the space E(M), the space of C^{infinity} functions mapping M
to R. This is equivalent to Lambda^0(M). The potential 1-forms and
curvature 2-forms reside, componentwise in Lambda^1(M) and
Lambda^2(M). Since they are both Lie valued, then their spaces are the
tensor products
  Lambda^1(M) (x) L and Lambda^2(M) (x) L
where L is the Lie algebra of the underlying gauge group.

When L is treated as residing in a universal covering algebra, then
one can form products over Lambda(M) (x) L. This leads to a product
defined by
  (a (x) u) (b (x) v) = (ab) (x) (uv)
where a, b are in Lambda(M) and u, v are in L. The Lie bracket then
extends to a graded bracket over the combined algebra, given by
  [a(x)u, b(x)v] = (ab) (x) [u,v].
Expressed in terms of the product, this is just
  [a(x)u, b(x)v] = (a(x)u) (b(x)v) - (-1)^{AB} (b(x)v)(a(x)u)
where A, B are the degrees of the forms a and b, respectively.

That's the combined algebra you're looking for.

Note, in particular, for 1-forms a, b you have [a,b] = ab + ba, not ab-
ba. So, as a special case, for a 1-form a, you have [a,a] = 2a^2.

An author like Bleecker (in his 1980 tome on the math underlying gauge
theory and principal bundles) only defines the bracket. He doesn't use
the device of putting everuything inside a universal covering algebra,
so things get obscured. In place of the equation f = da + a^2, he can
only write f = da + 1/2 [a,a].

So, in n-dimensional spacetime, if A is the potential 1-form, and F
the field strength 2-form (using upper case now), their Lie components
are A^a, F^a which are, respectively, 1-form and 2-form fields
residing in Lambda^1(M) and Lambda^2(M). The field law becomes (dA +
A^2 = F), the Bianchi identity (dF + AF - FA = 0). Expressed in terms
of the graded Lie bracket, these become
   dA + 1/2 [A,A] = F; dF + [A,F] = 0.

The conjugate fields (corresponding to what Maxwell calls D and H) are
Lie co-vectors, and are captured by the (n-2) form which we'll call G.
Thus, for Minkowski spacetime, you have
  G = (D^x dy^dz + D^y dz^dx + D^z dx^dy)
  - (H_x dx + H_y dy + H_z dz) ^ dt.
These come out as derivatives of the field Lagrangian L. So if the
Lagrangian L is a n-form (the kernel of the action integral S =
integral(L)), then the total differential in terms of A and F would be
 dL = dA ^ J - dF ^ G.
The current (n-1) form is, in Minkowski space, the 3-form
  J = rho dx^dy^dz - (J_x dy^dz + J_y dz^dx + J_z dx^dy)^dt.

To write the field law and conservation law algebraically, you need to
also bring the dual Lie algebra L* within the universal covering
algebra. First, the bracket may be extend to an operation over L and
L* by
  [w,u].v = w.[u,v]
where w is in L*, and u,v in L; with ().() denoting the contraction.
Taking a basis (Y_a) for L, defining the structure coeffcients by
[Y_a,Y_b] = f^c_{ab} Y_c, this gives you a product [Y^c,Y_a] =
f^c_{ab} Y_b.

Assume this has been incorporated into the covering algebra in such a
way that [Y^c,Y_a] = Y^c Y_a - Y_a Y^c.

Then one can define a graded bracket that for (Lie-valued) and (Lie-
covalued) forms. That is, if u is in Lambda^A(M) x L and v is in
Lambda^B(M) x L*, then the bracket [u,v] will be in Lambda^{A+B}(M) x
L*, such that
  [u,v] = uv - (-1)^{AB} vu.

The field law then becomes
  dG + [A,G] = dG + AG - (-1)^n GA = J.

  dJ + [A,J] = [F,G] --> dJ + AJ + (-1)^n JA = FG - GF.