*Post by m***@yahoo.com**Post by John Baez*h_{ij} = g_{ij}

h_{i5} = A_i

h_{55} = 1

HOWEVER, I don't find that the velocity of the particle around

the circle is constant!

You got stumped by this problem?

That's easy. You've got the wrong metric. It's

h_{ij} = g_{ij} + K A_i A_j

h_{i5} = h_{5i} = K A_i

h_{55} = K

L = 1/2 (g_{ij} u^i u^j) + K/2 (A_i u^i + u^5)^2

[etc]

This can be worked out in a transparent fashion (at least when not

reduced to ASCII form!) in the more general case, within the principal

bundle formalism, using the *left-quotient* operator. The derivative of

the quotient completely captures in a much more general and clearer

context the notion of a connection, making everything easier to work

with.

So, some background and notation first. Let P be acted on by a group

G. This gives you a product operation with signature

P.G -> P

and properties

pe = p (e = group identity);

p(gh) = (pg)h,

with cosets

pG = { pg: g in G }

One can then define the set

p\q = { g in G: pg = q }.

The stablizer of point p is just p\p. The group acts transitively on P

if all the p\q are non-empty. It acts freely if p\q has at most one

member -- in which case one may consider the quotient as a partial

operation

p\q: defined for all q in pG

with the properties:

p p\q = q

p\p = e

(p\q)^{-1} = q\p

p\q q\r = p\q

(pg)\(qh) = g^{-1} p\q h.

When these spaces are also manifolds, one can consider the actions of

these operators under derivatives. Recalling that the key property of

tangent vectors on a manifold M is that for a curve r(t)

r'(t) is in T_{r(t)}(M),

we'll use the notation denoting by m' (with subscripts) a general

element of T_m(M).

With the coset notation, a local SECTION on the space P is defined as a

map

s: P/G = M -> P

such that

s(m)G = m.

A principal bundle is thus a manifold P acted on by a symmetry group G

in such a way that its cosets M = P/G can be arranged into a manifold,

which can be coordinatized by differentiable local sections whose

domains cover M.

In the following, for a function f: M -> N, I'll use the notation Df:

TM -> TN to denote its deriative. So, for f: R^m -> R^n, Df is just

the mxn Jacobian matrix of f. Its product with a vector v in TM is

denoted Df[v] in TN.

The product P.G -> P, naturally induces one with signatures:

p.T_g(G), T_p(P).g -> T_{pq}(G)

with the property

(pg)' = p'g + pg'.

Differentiating the relation,

s(m)gG = s(m)G = m

one finds:

ds_m[m']G = ds_m[m']gG + s(m)g'G = ds_m[m']G = m'.

Thus

ds_m[m']G = m'; s(m)g' = 0.

The corresponding action for quotients

p\T_q(P), T_p(P)\q -> T_{p\q}(G)

is not given at the outset, but must be postulated; and is assumed to

have the property

(p\q)' = p'\q + p\q'.

The connection one-form is just

omega_p[p'] = p\p'.

The horizontal lift of a vector m', m = pG, to a point p is

L_p[m'] = s(m) ds_m[m']\p + ds_m[m'] s(m)\p

or more briefly

L = s ds\p + ds s\p

The section-relative connection is

B = s\ds

or more precisely:

B_m[m'] = s(m)\ds_m[m'].

Its action under a global transformation s -> sg is

B -> (sg)\d(sg) = g^{-1} s\ds g = g^{-1} B g

and its action under a local transformation

s(m) -> s(m) x(m)

is

B -> (sx)\d(sx) = x^{-1} s\(ds x + s x')

= x^{-1} s\ds x + x^{-1} x'

= x^{-1} B x + x^{-1} x'.

In a similar way, one may define natural actions of the group on the

cotangent space through the properties:

g.T*_p(P).h -> T*_{gph}(P)

(gwh)[p'] = w[g^{-1} p' h^{-1}]

as well as a quotient

p\T*_q(P) -> T*_{p\q}(P)

p\w[q'] = w[p q'].

This time, it's the product that requires the additional structure of

the connection, with

p.T*_g(G) -> T*_{pg}(P)

p.a[q'] = a[p\q'].

(One can also define operations for the tangent and cotangent spaces:

T_p(P)\T*_q(P) -> R

p'\w = w[p']

).

So with the background out of the way...

Given a metric k on G that is invariant under left and right products

k_{hgf}(h g'_1 f, h g'_2 f) = k_g(g'_1, g'_2)

and a metric g on the base space M, define the bundle metric on P by

h = g + k(omega, omega)

or more precisely:

h_p(p'_1, p'_2) = g_{pG}(p'_1G, p'_2G) + k_{pG}(p\p'_1, p\p'_2).

In the general case, the scaling of k may be positionally dependent on

the points in the base space M.

Given a section s, one may decompose P and TP as follows:

p = s(m) g; where m = pG; g = s(pG)\p

with

p' = ds[m'] g + s g'.

One finds that

p'G = ds[m']gG + s g'G = m' + 0 = m',

and

p\p' = g^{-1} s\(ds g + s g') = g^{-1} B_m[m'] g + g^{-1} g'

or

p\p' = g^{-1} u g

introducing the horizontal velocity

u = B_m[m'] + v

and gauge velocity

v = g' g^{-1}.

In terms of the decomposition, the metric can be written as:

h_p(p'_1, p'_2) = g_m(m'_1, m'_2) + k_m(u'_1, u'_2).

In component form, this becomes:

h_{AB}p'_1^A p'_2^B = g_{mn} m'_1^m m'_2^n + k_{ab} u'_1^a u'_2^b.

using the summation convention on the indices.

The components of the total metric are thus:

h_{mn} = g_{mn} + k_{ab} B^a_m B^b_n

h_{mb} = k_{ab} B^a_m

h_{an} = k_{ab} B^b_n

h_{ab} = k_{ab}.

So, from this you can readily find the geodesic equations of motion.

These arise from the Lagrangian:

L(p,p') = 1/2 h_p(p',p').

Variation of L gives you the desired results.

Working within the decomposition, p = s(m) g as above, define the

variation of g by

Delta(g) = D g.

then

Delta(g^{-1}) = -g^{-1} D.

and the variation of v will be:

Delta(v) = Delta(g' g^{-1})

= (Delta g)' g^{-1} - g' g^{-1} D

= (Dg)' g^{-1} - v D

= D' g g^{-1} + D v - v D

= D' + [D,v].

The variations with respect to D give you the charge and precession

equation (Wong's equation) for the charge. The variation with respect

to m' gives you the ordinary momentum and the Lorentz force law coupled

with the charge.

First, employing the derivative notation alluded to before:

Delta(u) = Delta(B_m[m'] + v)

= DB_m[Delta(m)][m'] + B_m[Delta(m')] + D' + [D,v].

Thus

Delta(L) = Delta(1/2 g_m(m',m') + 1/2 k_m(u,u))

= 1/2 Dg_m[Delta(m)](m',m') + 1/2 Dk_m[Delta(m)](u,u)

+ g_m(m',Delta(m')) + k_m(u,Delta(u)).

Picking out the variation with respect to D', one finds the

corresponding momentum

p_D = k_m(u, ())

which is just u with its (Lie) index lowered.

Picking out the variation with respect to D, one finds the force term

F_D:

F_D = k_m(u, [(),v]) = p_D[[(),v]].

Since p_D is a cotangent vector in T*G, it's natural to extend the Lie

bracket to the cotangent space. So, let's consider first how the Lie

bracket is defined. For a Lie vector e' in the tangent space T_e(G) =

L, one has

g e' g^{-1} in T_{g e g^{-1})(G) in L.

Therefore, L is closed under the adjoint operator

ad_g(e') = g e' g^{-1},

and derivatives of this operator make sense. The Lie bracket is then

just

[g',e'] = (g e' g^{-1})'.

Explicitly:

[g'(s), e'(t)] = d^2(g(s) e(t) g(s)^{-1})/ds dt.

So, over the cotangent space T_e*(G) = L*, one has the coadjoint action

defined in the same way

co_g(w) = g w g^{-1}

and Lie bracket

[g', w] = (g w g^{-1})'.

Applying this to a Lie vector e', one finds

[e_1',w][e_2'] = (e_1 w e_1^{-1})'[e_2']

= (e_1 w e_1^{-1}[e_2'])'

= (w[e_1^{-1} e_2' e_1])'

= w[(e_1^{-1} e_2' e_1)'].

Since

e_1^{-1}'(t) = -e_1^{-1}(t) e_1'(t) e_1^{-1}(t)

which evaluated at the given t where e_1(t) = e is just

e_1^{-1}'(t) = -e_1'(t),

then the Lie bracket for e_1^{-1} is the negative of that for e_1.

Thus

[e_1',w][e_2'] = -w[[e_1',e_2']]

= w[[e_2',e_1']].

Or, in more prosaic notation:

[a,w][b] = w[[b,a]], for lie vectors a, b in L.

Since the metric k is assumed to be invariant under action of G to the

left and right, one has:

d/ds (k_{m}(g(s) h_1' g(s)^{-1}, g(s) h_2' g(s)^{-1}))

= d/ds (k_h(h_1', h_2') = 0.

Applying this to the individual vectors using the Leibnitz property,

one gets

d/ds (k(g h_1' g^{-1}, g h_2' g^{-1}))

= k(d(g h_1' g^{-1})/ds, g h_2' g^{-1})

+ k(g h_1' g^{-1}, d(g h_2' g^{-1})/ds)

= k([g', h_1'], h_2') + k(h_1', [g', h_2']).

Thus for Lie vectors, a, b, c in L:

k([a,b],c) + k(b,[a,c]) = 0.

In particular, since the metric is symmetric, then for b = c,

k([a,b],b) = 0.

Therefore, the force can be rewritten as:

F_D = k_m(u, [(),v])

= k_m(u, [(),u - B])

= k_m(u, [(),u]) - k_m(u, [(),B])

= -k_m(u, [(),B])

= -p_D([(),B])

= -[B,p_D] = [p_D, B].

The Wong equation for the precession of the charge is thus

d(p_D)/dt = [p_D, B[dm/dt]].

The variation with respect to m' yields the momentum

p = g_m(m', ()) + k_m(u, B_m())

= g_m(m', ()) + p_D(B_m())

= g_m(m', ()) + p_D.B_m.

In component form

p_m = g_{mn} m'^n + p_D_a B^a_m,

which is the usual expression for the canonical momentum in the

presence of a Lorentz force given by the potential Lie vector B coupled

to the charge Lie vector p_D.

The corresponding force is the variation with respect to m:

F = 1/2 Dg_m[](m',m') + 1/2 Dk_m[](u,u) + k_m(u,DB_m()[m']).

= 1/2 Dg_m[](m',m') + 1/2 Dk_m[](u,u) + p_D(DB_m()[m']).

The corresponding equation of motion is dp/ds = F. Differentiating p,

we find

dp/ds = Dg_m[m'](m',()) + g_m(m'',()) + p_D'[B_m()] + p_D[dB_m[m'][]]

= Dg_m[m'](m',()) + g_m(m'',()) + p_D[[B[m'],B[]]] +

p_D[dB_m[m'][]].

Thus

Dg_m[m'](m',()) + g_m(m'',()) + p_D[[B[m'],B[]]] + p_D[dB_m[m'][]]

= 1/2 Dg_m[](m',m') + 1/2 Dk_m[](u,u) + p_D(DB_m[][m']).

Defining the Yang-Mills force

G(v, w) = DB_m[v][w] - DB_m[w][v] + [B_m[v], B_m[w]]

and Christoffel coefficients

Gamma(u,v,w) = 1/2 (Dg[u](v,w) + Dg[v](u,w) - Dg[w](u,v))

the equations of motion become

g_m(m'', ()) + Gamma(m', m', ()) = G((), m') + 1/2 Dk_m[](u,u).

Applying this to the vector m', and noting that

(g(m', m'))' = 2 g(m'', m') + 2 Gamma(m', m', m')

we get

(g(m', m'))' = Dk_m[m'](u, u).

If the gauge metric has an inverse, k^{-1}, one may define the charge

magnitude

|p_D|^2 = k^{-1}(p_D, p_D).

= p_D[u]

= k(u, u)

and

Dk_m[m'](u, u) = -D{k^{-1}_m)(p_D, p_D).

Adjoint invariance applies to this metric as well

k^{-1}([a,b], c) + k^{-1}(b, [a,c]) = 0.

Therefore, under the derivative, its action becomes

(|p_D|^2)' = D(k^{-1})[m'](p_D, p_D) + 2 k^{-1}(p_D, [p_D, B])

= D(k^{-1})[m'](p_D, p_D)

= -Dk_m[m'](u, u).

Thus, one arrives at the constant of motion:

g(m', m') + |p_D|^2.

If the gauge metric is constant, then each part is a constant of motion

separately.