Finite Dimensional Representation of SU(2)

In summary: Fotescu, an undergraduate student at the University of British Columbia, has provided a proof that the representations are unitary.
  • #1
kakarukeys
190
0
Does anybody know whether the following irreducible representations of SU(2) are unitary?

g belongs to SU(2)
[tex][U_j(g) f](v) = f(g^{-1} v)[/tex]

f is an order-2j homogeneous complex polynomial of two complex variables v = (x, y)

e.g. for [tex]j = 1[/tex], [tex]f = 2x^2 + 3xy + 4y^2[/tex]
 
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  • #2
kakarukeys said:
Does anybody know whether the following irreducible representations of SU(2) are unitary?
g belongs to SU(2)
[tex][U_j(g) f](v) = f(g^{-1} v)[/tex]
f is an order-2j homogeneous complex polynomial of two complex variables v = (x, y)
e.g. for [tex]j = 1[/tex], [tex]f = 2x^2 + 3xy + 4y^2[/tex]
Hi, they are. For example: let's look at the complex vector space V_2 of polynomials which are homogeneous of degree 2. This vectorspace is isomorphic to the vectorspace of 2 times 2 complex symmetric matrices through f <- -> A, f(v) = v^T A v. The action of the group corresponds then to A --> \bar{g} A g* where v^T is the transpose, g* g = 1 and \bar{g} is the complex conjugate of the matrix g. The scalar product on this space is < A,B> = tr(A*B) where tr is trace. Under the action of g, this becomes:
<U(g)A, U(g)B> = tr(g A* g^T \bar{g} B g* ) = tr(A*g^T \bar{g} B) = tr(A*B)
For higher j you can use the isomorphism with the symmetric tensors and generalized traces.

Cheers,

Careful
 
  • #3
I don't see how the trace corresponds to the inner product. Is this the map between the column vector f and the symmetric matrix A?
f=(f1,f2,f3)^T --> A = ( (f1, 1/2 f2), (1/2 f2, f3) )
since (x y) . A . (x y)^T = f1 x^2 + f2 xy + f3 y^2
If this is true then for two vectors f -> A and g -> B the diagonal elements of the product A*B are
A*B = ( (f1*g1 +f2*g2/4 , blah ) , (blah, f2*g2/4 + f3*g3) )
The trace of this matrix is Tr(A*B) = f1*g1 + f2*g2/2 + f3*g3
but this is not the inner product of f and g:
<f,g> = f1*g1 + f2*g2 + f3*g3

I'm sure your result is correct (that the representation is unitary) so how does one fix the proof?
Thanks
Alex
 

1. What is the definition of a finite dimensional representation of SU(2)?

A finite dimensional representation of SU(2) is a mathematical way of describing how the special unitary group SU(2) acts on a finite-dimensional vector space. This means that the elements of SU(2) can be represented as matrices that transform the vectors in the vector space in a certain way.

2. How are finite dimensional representations of SU(2) used in physics?

Finite dimensional representations of SU(2) are used in physics to describe the symmetries of physical systems. For example, in quantum mechanics, the spin of a particle can be represented by a finite dimensional representation of SU(2).

3. What is the significance of the dimension of a finite dimensional representation of SU(2)?

The dimension of a finite dimensional representation of SU(2) is related to the number of possible states or degrees of freedom in a physical system. This is important in understanding the behavior and properties of the system.

4. How are finite dimensional representations of SU(2) related to other mathematical concepts?

Finite dimensional representations of SU(2) are closely related to other mathematical concepts such as Lie groups, Lie algebras, and representation theory. They provide a mathematical framework for understanding the symmetries and transformations of physical systems.

5. Can finite dimensional representations of SU(2) be generalized to other groups?

Yes, the concept of finite dimensional representations can be extended to other groups, such as SU(N) and SO(N). These groups have similar properties to SU(2) and their finite dimensional representations are used in various areas of physics and mathematics.

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