Natural Science Forum / Physics / Research / March 2010

In many cases, yes. But it depends on the properties of the operators.
There is no simple answer. Infinite-dimensional Lie-group theory is full
of similarities to the finite-dimensional case, but also full of
counterexamples to these. Read books by Hilgert if you really want to
study these questions.

Arnold Neumaier

The issue of matrix algebra is a red herring, since this actually has
nothing to do with matrix algebra at all.

It's a close relative of the Campbell-Baker-Hausdorff formula and also
closely relative to the "logarithmic coordinates" representation. You
have results of the following form. First define the logarithmic form
of the product function:
  A (+) B def= log(exp(A) exp(B))
Then, in a coordinate representation you have
  A (+) B = A + B + 1/2 [A,B] + O({A,B}^3).
Two things come out of this:
(1) In the limit (tA (+) tB)/t as t -> 0, you get A + B
(2) In the limit (tA (+) tB - t(A + B))/t^2 as t -> 0, you get 1/2 [A,
B].

An interesting exercise for you is the following: find the MOST
general analytic function A (+) B of A and B such that
(1) A (+) 0 = A = 0 (+) A
(2) sA (+) tA = (s + t)A
(3) (A (+) B) (+) C = A (+) (B (+) C)
where A, B and C are vectors in a given vector space.

On the larger question you posed: there doesn't seem to be universal
consensus on exactly what actually DOES define a Lie algebra or Lie
group when the underlying space is infinite dimensional. At the very
least, the issue is clouded. For instance, the diffeomorphism group
doesn't appear to be considered a Lie group, though many of the
results and techniques applicable to Lie algebras and Lie groups apply
to it. Nor, therefore, would the "gauge" and "authmorphism" groups for
principle bundles be considered Lie groups, though the usual results
of Lie algebra/group theory are just as applicable to this as they are
to the diffeomorphism group.

You may wonder, therefore, what would NOT be applicable. One answer is
easy to see: the Noether Theorem. It applies in classical theory, to
be sure. You end up getting an infinite-dimensional "current" -- i.e.,
a current J_X parametrized not by a discrete index, but by a vector
field X (the integral representation would then be J_X = integral_W
(T^m_n X^n (d^3x)_n) where the density T^m_n represents the rank (1,1)
densitized form of the corresponding stress tensor.

But a problem emerges with its application. Strictly speaking the
Noether theorem only works for compact regions W. You have to wing it
for non-compact regions by stipulation one or another idealization
(none of which are physically relevant). This idealization, in
reality, serves as nothing more than an expedient way to represent
compact regions without horizons or boundaries having to be taken into
consideration.

The reason this works find for ordinary Lie group symmetries is
because these symmetries leave points fixed. But that's PRECISELY the
situation that does NOT apply when you bring in any non-trivial member
of the "automorphism group" for a principal bundle, or (as a special
case) any diffeomorphism.

Diffeomorphisms move points.

Classically this is fine. The effect of the motion is captured by what
goes on at the boundary.

Quantum theoretically it is not fine! When a boundary moves
(particularly, a horizon), you do not have an equivalence of the
corresponding state spaces. Instead, the transformation between the
state spaces is an "incomplete Bogoliubov transformation". As a
result, therefore, when carrying out any non-trivial principal bundle
automorphism, or a diffeomorphism on the base space, you end up
breaking Noether's theorem -- unless the compact region in question
remains fixed on the boundary.

The result is that diffeomorphism symmetry must be spoiled at the
quantum level. An anomalous contribution to the corresponding Noether
current must therefore arise.

My conjecture is that this contribution is none other than the
"Einstein-Hilbert 3-current", i.e., the current
  G_X = integral (G^m_n root(-g) X^n (d^3x)_m).

In other words, it may very well be the case that gravity is ALREADY
implied by quantum field theory; and is implied in such a way that one
needs nothing more than quantum field theory on a classical background
manifold to start with. Instead of a "backreaction" equation, one gets
a "stress tensor anomaly".

I haven't yet proven this conjecture, but I have a good idea how the
proof will work, if it can be proven. The key element is already
contained in the observations made in Jacobson's 1995 paper, in which
he derived semi-classical gravity from the laws of thermodynamics, by
adopting the conventions:
Law 0: Temperature T taken as the Unruh temperature
Law 1: Conservation law taken as the integral equation for the stress
tensor (thus leading to an expression for delta(Q))
Law 2: The formula T dS = delta(Q)
Law 3: Bekenstein Bound for entropy vs. horizon area (thus leading to
an expression for dS).

His construction used "approximate causal horizons" and brought in a
gravitational lensing effect and employed the Raychaudhuri equation to
get the key result. If the conjecture pans out, then it should be
possible to generalize this argument to one that generalizes the
Noether Theorem to a form applicable to either of the (infinite-
dimensional) "Lie" groups Aut(P) (automorphism group for a principal
bundle P) or Diff(M) (diffeomorphism group for a manifold M).

His construction is replaced my mine, where a compact region is
represented as a compact foliation generated by a compactly-supported
vector field D_t (that may or may not be time-like). The region W is
thus partitioned into 3-surfaces (W_t: t = a to b) and each surface
shares a common boundary H = Boundary(W_t) with all the others. That
boundary is the horizon, the boundary of the region W, itself, being
Boundary(W) = W_b - W_a.

The flow would decompose into 3 "modes": the transverse "mode" that
moves along a given surface W_t (while leaving the boundary H fixed),
the longitudinal "mode" which moves from one W_t to another (again,
leaving H fixed), and the third component: the most important one --
the one that moves H.

The last "mode" is the one that would generate the anomalous current
(i..e. the Einstein-Hilbert 3-current).