The theory of alternating
forms describes wholly anti-symmetric tensor fields; those of rank n are
called n-forms. On a smooth manifold of dimension m, the m-forms constitute a
one-dimensional tensor rank; and provide the means to specify integration. Thus
areas are specified by 2-forms; by area element field
I simply mean a
2-form.
In general, on a smooth manifold, a Leibniz operator of gradient rank yields
a Riemann tensor whose rank is a 2-form tensored with a linear map from
gradients to gradients; contracting the latter with a covariant rank 2 tensor
annihilated by the Leibniz operator reduces the Riemann tensor to a sum of
symmetric products of 2-forms. In General Relativity, the metric of space-time
is annihilated by the covariant differential operator, a Leibniz operator of
gradient rank, so the latter's Riemann tensor, contracted with the metric, is a
sum of symmetric products of 2-forms. Einstein's field equation also includes a
term, the energy, momentum, stress
tensor, which includes a term derived
from the Maxwell field (electric and magnetic fields, unified), which is also a
2-form.
On an even-dimensional smooth manifold, the measure (the thing which
mediates integration) can be expressed non-uniquely as a square
of a
half-dimension-form; in the case of a 4-manifold (such as space-time appears to
be), this is a 2-form again. Thus I am highly interested in 2-forms on
4-manifolds.
Maximal rank alternating forms are what mediates integration; you can think of the familiar dx.dy.dz as in fact dx^dy^dz (and so on, for more than three dimensions), wherein dx, dy and dz are formally gradient fields of our co-ordinates, forming a basis of gradients; integration cuts space-time up into tiny boxes, with vector sides, and the three-form dx^dy^dz contracts with the vectors along the sides to get the volume of each box, by which to scale a function of position that you can then sum over boxes to get the integral. Doing this with arbitrary co-ordinates (rather than presuming x, y and z are orthonormal co-ordinates, where changing one of them by L moves a distance L) requires pulling in a geometric factor derived from the metric, which combines a pair of vectors to give an inner product; when given the same vector as both of its pair, this is the square of that vector's length. The determinant of this metric is then the square of the tensor that replaces dx^dx^dz more generally.
In quantum mechanics, orthodoxy deals with a complex scalar function, ψ, of position for which the integral of ψ*ψ is one while the integral of ψ*dψ yields, with a suitable imaginary scaling, the momentum of the particle described by the wave function ψ. That's all conducted at one moment in time with respect to orthonormal spatial co-ordinates and presumes a flat space-time with a canonical integration, which makes it untidy to describe compatibly with general relativity, for which space-time is a smooth manifold with variable curvature and no special preferance among co-ordinate systems or measures (a measure is the thing that mediates integration). Furthermore, it depends on complex scalars, rather than real ones – though complex can often be eliminated by recognising it as a cipher for rotations.
What I'm going to describe below is a substitute for that complex scalar field, ψ, namely an anti-symmetric tensor of rank 2; in place of conjugation, which gave us ψ*, I'll be introducing a non-trivial tensor field derived from the metric of space-time, which consumes ψ and yields a substitute for ψ*; when this is combined with ψ, it yields a tensor which can mediate integration and thus be used to integrate scalar fields on space-time (or vector fields on space-like submanifolds of codimension 1).
To do integration on a smooth manifold of dimension m, one needs a wholly
antisymmetric covariant tensor field of rank m on the manifold – that can
eat an m-sided box
on the manifold to produce a scalar; sub-dividing the
manifold into many small boxes and summing the resulting scalars is then the
natural equivalent of how we obtain integration on a vector space. This kind of
tensor field is known as an m-form; the space of m-forms on a smooth manifold of
dimension m is one-dimensional, so the different ways we can specify integration
can be arrived at by scaling one of them using scalar fields; it suffices to use
a never-zero one as the unit thus scaled. A manifold is described as orientable
if there does exist a nowhere-zero m-form on the whole manifold; and one special
case where this is guaranteed is where there is a nowhere-singular metric
(which, incidentally, thus has constant signature) – take the metric's
determinant, apply a non-zero constant scaling that makes it positive (in an
appropriate sense) and it'll have square roots which will be non-zero m-forms.
Thus integration over our four-dimensional space-time uses a 4-form, μ = √−det(g), to which we can apply scalings. The space of 2-forms on space-time can potentially provide us with a square root, ψ, for μ, specified by: the wholly antisymmetric part of ψ×ψ is μ. For example, using orthonormal co-ordinates [t,x,y,z] with g = dt×dt −dx×dx −dy×dy −dz×dz, yielding dt^dx^dy^dz as one √−det(g);
since each −sign([…]) is of an odd permutation, hence yields −(−1), and each +sign([…]) is of an even permutation, hence yields +(+1); so we sum eight 1 terms to get 8 and divide it by 8.
In general, we can take any non-zero covariant 2-form, divide it by μ to get a contra-variant 2-form and apply g×g to the result to obtain a 2-form whose antisymmetric product with the original is a non-zero 4-form. If we integrate this over space-time, we'll get a scalar result; or we can contract it with a vector field (e.g. a time-like unit vector) and integrate over a 3-sub-manifold (e.g. the a space-like snapshot at a particular moment in time) to obtain a scalar. If we have a time co-ordinate and perform the latter on the space-like snapshot at each moment of time, using dt/g as vector field, the scalar result will depend only on our time co-ordinate; as long as our original 2-form varyies smoothly, so will this scalar result; provided it is positive at all times, we can take its square root (again smoothly varying) and divide our 2-form at all positions on each space-like snapshot by the snapshot's time's value; the result should be a smoothly varying 2-form we can use in place of the original; for this replacement, the dt/g integral is 1 over each space-like snapshot at a moment of time.
Thus our (g×g)(ψ/μ) ← ψ operation replaces complex
conjugation in the orthodox quantum mechanical approach, leaving us with a real
tensor rank in place of complex numbers; and with a mechanism for describing
the quantum field
of a particle as a tensor field on the smooth manifold
inhabited by the particle. We can apply all usual tensor operators of our
smooth manifold to it without needing special strange rules for the
consequences; and we can explore what co-ordinate independent form the
differential equations of quantum mechanics must take to work with it.
We can get away with the same basic trick on any even-dimensional smooth
manifold; if its dimension is 2.i, then i-forms on it will do the job; the
metric, g, is a covariant rank 2 tensor whose determinant is a linear map from
contravariant 2.i-forms to covariant 2.i-forms; provided the metric is
nowhere-singular with constant signature, some constant multiple of the metric's
determinant will be positive so have square roots, one of which we can select as
μ (the other will then be −μ); we can then use bulk(×,
({g}:|i))∘(: x/μ ←x :) as a linear isomorphism on (covariant)
i-forms to use as our conjugation
. The space of i-forms has dimension
(2.i)!/i!/i!, a.k.a. chose(2.i, i); for successive values of i we get chose(4,2)
= 6, chose(6,3) = 20, chose(8,4) = 70, chose(10,5) = 252, chose(12,6) = 924
(note: chose(2.i, i)/chose(2.(i−1), i−1) =
2.(2.i−1)/i = 4.(1−1/2/i) for each i, so chose(2.i, i) grows
asymptotically a little slower than power(i, 4), a.k.a. 4i; in
fact chose(2.i, i) tends to power(i, 4)/√(π.i)
asymptotically).
It's also worth noting that the Riemann tensor, once contracted with the metric, is a sum of symmetric products of 2-forms, and the Maxwell field tensor is a 2-form; so 4-dimensional space-time (where 2-forms and half-dimension-forms are the same thing) is particularly suited to √μ fields and their kin. Totally antisymmetrising the Riemann tensor will annihilate it (because antisymmetrising on its first three factors does so), so it doesn't relate usefully to μ – however, if we tensor two copies of it together, permute the first two factors of each copy to the left and the last two to the right, then antisymmetrise separately over the first four and last four factors of the result, we'll get some multiple of det(g); determining what multiple may be of interest (though it may be zero again).
The electromagnetic field is, in Special and General Relativity, described
by an antisymmetric second rank tensor field over space-time; via the metric and
measure, it's associated with an antisymmetric (dim −2)-rank tensor. In
four dimensions, that's another second rank tensor, but I'll abstain from
assuming dim is 4. Maxwell's equations tell us the antisymmetrised derivatives
of these two tensors; that of the second-rank tensor is (a third-rank) zero;
that of the (dim −2)-rank tensor is a (dim −1)-rank tensor that
describes the charge/current density – which is zero in free space. So
let's pause to wrap our heads round, maybe even build some foundations to
eventually make some intuitive sense of, what such tensor fields look
like
and, indeed, how to picture them at all.
As it happens, there's one class of antisymmetric tensor fields that we can
visualise: area or volume elements and line lengths; of these, area elements are
second-rank. The measure (a square root of ± the determinant of the
metric) puts a numeric value on the 4-volume
of a region of space-time;
except that, when dim isn't 4, that's actually a dim-volume. This measure is an
antisymmetric tensor field whose rank of rank Alt(dim, T), where T is the
tangent bundle of space-time and dim is its dimension; Alt(dim, T) is a
one-dimensional space whose vectors are all, at each point of the manifold,
simple scalar multiples of the antisymmetric product of any gradient basis. Our
antisymmetric tensors generally are,
The Alt(n, T) are known as n-forms; the electromagnetic field is a
2-form and its Hodge dual is a dim−2-form. Aside from the
antisymmetrisation, a 2-form has the same tensor rank as the metric, G⊗G.
(This means we can also take its determinant, getting a tensor of the same rank
as det(g); in 4-D, this tensor is just ±det(g) times the square of the
(3-vector) inner product, E·B, of the electric and magnetic fields that
we can read out of the electromagnetic field tensor. Its square root is then
E·B times the measure. Note that, both for this and det(g) we could
consider a square root
of rank Alt(n, T)⊗Alt(m, T) with n+m = dim;
the square
being taken by combining each factor's n-term with the other's
m-term, antisymmetrically, (u×v)^(v×u) = ±(u^v)×(u^v),
potentially supplying deg(g)'s negation.)
Written by Eddy.