- #1
muppet
- 602
- 0
Hi all,
It's a well-known result that any finite or compact group G admits a finite-dimensional, unitary representation. A standard proof of this claim involves defining a new inner product of two vectors by averaging the inner products of the images of the vectors under each element of the image of G under the representation. So given an inner product (a,b) on a vector space admitting some representation T(g) define
\langle a , b \rangle = \frac{1}{[g]} \sum_{g\in G} (T(g)a,T(g)b)
so that acting with an element of the image of G under the representation just reorders the sum, and hence the representation is unitary wrt this new inner product. This is for finite groups; for a compact group, I understand that we can replace the sum with an integral over group elements and divide by the "volume" of the group (e.g. 2pi for SO(2)) rather than the order [g].
Now, it seems to be part of the theoretical physics lore that the converse holds- that a noncompact group does not admit finite dimensional unitary representations. Can anyone direct me to a proof of this? I ask because the Lorentz group admits a faithful, four-dimensional representation that is "unitary" with respect to the Minkowski "inner product". The quotation marks are intended to indicate an awareness that the Minkowski inner product is really a symmetric bilinear form, as it's not positive definite. So it seems to me that a proof of this statement must hinge crucially on the positive definiteness of an inner product, even though our proof of the original result relies essentially upon the finiteness of the group and yet seemingly not at all upon the positive definiteness of the i.p.
Many thanks in advance.
It's a well-known result that any finite or compact group G admits a finite-dimensional, unitary representation. A standard proof of this claim involves defining a new inner product of two vectors by averaging the inner products of the images of the vectors under each element of the image of G under the representation. So given an inner product (a,b) on a vector space admitting some representation T(g) define
\langle a , b \rangle = \frac{1}{[g]} \sum_{g\in G} (T(g)a,T(g)b)
so that acting with an element of the image of G under the representation just reorders the sum, and hence the representation is unitary wrt this new inner product. This is for finite groups; for a compact group, I understand that we can replace the sum with an integral over group elements and divide by the "volume" of the group (e.g. 2pi for SO(2)) rather than the order [g].
Now, it seems to be part of the theoretical physics lore that the converse holds- that a noncompact group does not admit finite dimensional unitary representations. Can anyone direct me to a proof of this? I ask because the Lorentz group admits a faithful, four-dimensional representation that is "unitary" with respect to the Minkowski "inner product". The quotation marks are intended to indicate an awareness that the Minkowski inner product is really a symmetric bilinear form, as it's not positive definite. So it seems to me that a proof of this statement must hinge crucially on the positive definiteness of an inner product, even though our proof of the original result relies essentially upon the finiteness of the group and yet seemingly not at all upon the positive definiteness of the i.p.
Many thanks in advance.