What is the square root of the Dirac Delta Function? Is it defined for functional integrals? Can it be used to describe quantum wave functions?
∫∞−∞f(x)√δ(x−a)dx
Answer
You can think of the Dirac distribution as being an element in some operator valued Banach algebra. One can then use the Riesz calculus (http://en.wikipedia.org/wiki/Holomorphic_functional_calculus) to define an arbitrary holomorphic function of this Dirac delta (or more general distributions).
In particular (much like the Cauchy-Integral Formula) we have, for a Banach algebra A and holomorphic function f, for a∈A:
f(a)=12πi∮f(z)z−adz ;
This can in principle be evaluated. For example, you can represent the Dirac delta as a matrix (if you are thinking about finite dimensions). The integration is then fairly straightforward (for the square-root you would set f(z)=z1/2=e12log(z)). You will have to be careful about branch cuts for this particular case.
As for quantum wave functions, I could not say. However, this tool is very useful to define Dirac operators. For example, you could consider the algebra A to consist of differential operators of some dimension. Taking the Klein-Gordon operator ∇2+m2 one can attempt to define square-roots of this operator by evaluating the above integral.
No comments:
Post a Comment