Re: phase and conservation of charge in QM
- From: Rock Brentwood <markwh04@xxxxxxxxx>
- Date: Mon, 7 Jan 2008 13:00:39 +0000 (UTC)
On Dec 29 2007, 5:48 am, 0o.M42...@xxxxxxxxx wrote:
I was looking through the Feynman lectures on physics, and he briefly
mentioned that the invariance with respect to overall phase of
wavefunctions is related to the conservation of charge. Is this true,
and if so, can someone explain the reasoning behind it?
Indeed, there is a structure underlying all this that is still
(mostly) unknown to the world (so far).
First, on your question: in classical electromagnetic theory, one
writes down the Lorentz law:
dp/dt =3d e (E + v x B); dT/dt =3d e (E.v)
for a particle with kinetic energy T, momentum p and velocity v acting
in an electric field E and magnetic field B related to the magnetic
potential A and electric potential phi by:
B =3d curl A, E =3d -grad phi - @A/@t.
Here, @ denotes partial derivative
When you substitute these reductions into the equations of motion to
find the motion in terms of the potentials, themselves, you get:
dp/dt =3d e (-grad phi - @A/@t + v x (curl A)), dT/dt =3d e (-v.grad
phi - v.@A/@t)
Now ... add to the momentum and energy the corresponding "potentials",
eA and e phi and use the chain rule to compute their total time
derivatives
dA/dt =3d @A/@t + grad (A.v); d(phi)/dt =3d @(phi)/@t + v.grad phi,
where in the "grad", v is kept constant. Note also that
v x (curl A) =3d grad(A.v) - v.grad A,
again, keeping v constant inside the "grad".
Then, you get
d(p + eA)/dt =3d -grad(e(phi - A.v)), d(T + e phi)/dt =3d @(e(phi -
A.v))/@t.
This expresses the motion of the charge in terms of its CANONICAL
momentum P =3d p + eA and total energy H =3d T + e phi.
For a quantum particle, under the standard Schroedinger
interpretation, the canonical momentum P and total energy H are
replaced by -i h-bar del and i h-bar @/@t. Then the potential A
becomes a correction on -i h-bar del, while phi becomes a correction
on i h-bar @/@t.
For a wave function psi, if phi is constant, for instance, then you
could rewrite
(i h-bar @/@t - e phi) psi =3d i h-bar (@/@t + i e phi/h-bar) psi =3d
exp(-i e phi/h-bar) i h-bar @/@t (psi exp(i e phi/h-bar)).
Thus, the potential corresponds the time-like gradient, @/@t, in the
shift of the phase of the wave function. A similar consideration
applies for A.
Usually, this overall setup is modelled in a geometry known as a
principal bundle. In effect, this takes seriously the equation above
involving d/dt (p + eA) representing the equation of motion for the
ACTUAL momentum of a charge. The eA part is then the shadow left
behind by the unseen part of the charge's motion -- a motion taking
place with respect to an extra degree of freedom. The "e" is the
component of the charge conjugate to this extra degree of freedom. The
"unseen" element is the phase.
The conservation of charge is the conservation of momentum applied to
this extra component of momentum.
However, this kid glove handling of everything, denying the thing
you're working with, is confusing. So, let's cut through the tangle
and tell it straight out, like it actually is.
Number one, the phase IS there. Indeed, each point in spacetime is now
to be regarded as having a copy of a circle S_1. A particle's
trajectory is, therefore, not only described by the chain of points in
space-time that its worldline lies on, but also the chain of points on
the respective copies of S_1 that the phase resides on.
The geometry is best described NOT by a principal bundle, but simply
by the product of the unit circle S_1 and the spacetime manifold M.
(For, unbeknownst to most people who work with the underlying geometry
at this technical level, a principal bundle IS a trivial bundle --
just not a trivial GROUP bundle. It's a trivial TORSOR bundle).
The "cannot be seen" aspect of the phase is completely analogous to
the notion that a "0" cannot be seen in an ordinary Euclidean
geometry. All points in Euclidean space are on an equal standing, so
none of the gets to be called "0". This distinguishes a Euclidean
geometry from what's known as a vector space, where a "0" would be
defined. Instead, a Euclidean space is an example of what's known as
an affine space.
Analogously, the circle S_1 is often confused with the group U(1),
which is the group that defines angle addition (i.e., addition modulo
360 degrees). However, there is no point on S_1 that distinguishes
itself as the "0 degree phase". For, the circle S_1 is NOT a group.
It's what we call a torsor.
Phase is perfectly well-defined and meaningful. But it's an element of
a torsor. Unlike the case of U(1), where you have the operation of
addition, for S_1, you have the operation of RELATIVE addition: (s - t
+ u). it satisfies the axioms:
s - t + t =3d s =3d t - t + s, s - t + (u - v + w) =3d (s - t + u) - v +
w
which define a non-Abelian torsor, and the property
s - t + u =3d u - t + s
which defines an Abelian torsor.
The difference of points in S_1 can be taken to represent a "phase
change" (s - t). Thus, one can actually DEFINE U(1) from S_1 as just
the set of formal differences
U(1) =3d { (s-t): s, t in S_1 }
where one needs to pose the following consitency relation
(s-t+u)-v =3d u-(t-s+v).
Then, it will actually follow from the axioms on S_1 that U(1) is a
group, with the operations defined by
(s-t) + (u-v) =3d (s-t+u) - v, and -(s-t) =3d t-s, and 0 =3d s-s =3d t-t.
This is the group of all relative phase differences U(1) =3d delta(S_1).
Another way of producing U(1) from S_1 is to pick a point o in S_1 and
arbitrarily deem it 0 degrees. This corresponds to "choosing a gauge".
The result is a "relativized" group (S_1)_o with operations given by
a + b =3d a - o + b; -a =3d o - a + o.
One can show that (S_1)_o and delta(S_1) are identical, as groups,
through the correspondences:
s in (S_1)_o |-> s-o in delta(S_1)
and
(s-t) in delta(S_1) |-> s-t+o in (S_1)_o.
Each relativization (S_1)_o may be considered a copy of U(1) attached
to the point o. This is analogous to the concept of a tangent space,
but here the tangent spaces are groups identical to U(1). Instead,
they're called fibers.
So, if M is the space-time manifold, one constructs the appropriate
geometry by simply affixing a copy of S_1 to each point of M,
resulting in the space
P =3d M x S_1.
One assumes, in addition, that
Gauge Axiom: There is an atlas of gauges; i.e., partial maps o: M -
S_1 whose domains cover M, such that(o-o'): m |-> o(m) - o'(m) is a differentiable map over the common
intersection of the domains of any two gauges o and o'.
The difference between the two maps is called a gauge transformation.
The resulting structure is mathematically equivalent to what's called
a principal bundle. Algebraically, it has 2 operations defined on it:
Phase change at a point: P x U(1) -> P, given by (m,s) + (t-u) =3d
(m,s-t+u)
Phase difference of two settings at a point: P x_M P -> U(1), given
by (m,s) - (m,s') =3d s-s'
where
P x_M P =3d M x S_1 x S_1
is known as a "bundle spliced" product of P with itself.
Associated with each point p =3d (m,s) in P is the COSET or ORBIT of the
point pU(1) =3d { pg: g in U(1) } =3d {m} x S_1, which is just the copy of
S_1 attached to the point m. Thus, the second operation, applies to
two points on the same coset.
The properties of the 2 operations are:
(p + g) - p =3d g
p + (p' - p) =3d p'.
I won't go too far beyond this point, other than to point out that the
potential is naturally identified as the derivative of the difference
operation (p - p').
If you picked out a gauge o: M -> S_1, with a domain dom(o) =3d U subset
of M, then you could define the electromagnetic 4-vector, up to a
proportionality, as
A(x).u =3d d/ds ((x(s), o(x(s)) - (x, o(x)))
evaluated at x(s) =3d x, assuming the curve (x(s)) crosses the point x
with tangent x'(s) =3d u at x. This choice of potential is gauge-
dependent. If you switch over to a different gauge o'(x), and compare
the resulting potential A' with this one, you'll get the usual gauge
transformation formula
(A'(x) - A(x)).u =3d d/dt ( (x(s),o'(x(s))) - (x(s),o(x(s))) ) =3d d/ds
(o'(x(s)) - o(x(s)))) =3d d/ds (o'-o)(x(s)) =3d (o'-o)'(x).
Applying the chain rule to the latter, you get
(o'-o)'(x) =3d u.d(o'-o)(x).
Thus, A' - A =3d d(o'-o).
Significantly, this all generalizes to NON-Abelian gauge groups. Thus,
for instance, if S_3 denotes the hypersphere, the corresponding gauge
group is SU(2), the group generated from the Pauli spin matrices. Only
here, the charge e becomes a vector and is no longer conserved.
Instead, it precesses.
That's because the non-Abelian version of the Maxwell relations for
the fields vs. potentials, are
B =3d curl A + A x A; E =3d -grad phi - @A/@t + phi A - A phi.
So, when you substitute in the equations of motion involving the
canonical momentum and total energy
d/dt (p + e A) =3d -grad (e (phi - v.A) ), d/dt (H + e phi) =3d @/@t (e
(phi - v.A))
you will only get the Lorentz force laws
dp/dt =3d e (E + v x B), dH/dt =3d e v.E
by postulating a non-zero expression for de/dt =3d e (A.v - phi) - (A.v
- phi) e.
That's Wong's equation.
.
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