What does it mean for a change of variables to be UNITARY?

[AxiomOfChoice]

So if I'm changing from variables $x,y$ to variables $\alpha = f(x,y), \beta = g(x,y)$, what exactly does it mean to stay this change of variables is unitary, and how can I tell if it is or if it isn't?

 

[Petr Mugver]

I never encountered it, but probably it means that the jacobian matrix of the change of variables is a unitary matrix. In other words: you differentiate f and g with respect to x and y, put this four functions into a matrix M, and verify that U multiplied by the hermitian conjugate of U is the identity 2 x 2 matrix. To find the hermitian conjugate of a matrix you transpose it and then you take the complex conjugate.

I'm not really sure that all this is true though.

 

[ulriksvensson]

A unitary transformation is one that preserves complex norm, ie complex numbers (or functions) of unit absolute value transforms to complex numbers (or functions) of unit absolute value. If you consider real valued functions of real variables this is the same as orthogonal (or more correctly orthonormal) transformations. For a transformation to be unitary the Jacobian must be a unitary matrix as stated above. This will preserve the volume $$V = \int_{V}dx_{1}...dx_{n}$$.