Dear Maryam,
In Physics the unitary transformations are summarized in continuous groups of symmetry U(n) which are formed by antisymmetric matrices M of dimension nxn, such that they product M*M=I, where I is the identity matrix of dimension nxn. One example is the isospin group U(2) formed by the 2x2 Pauli's matrices and which follow a Clifford algebra.
If these groups of transformations want to work on the quantum states, then a subgroup is taken SU(n) which adds the condition that they determinants need to be just 1. This is necessary for preserving the statistical intrepretation of the wave function.
Dear Kai,
All the continous groups of transformotion in Physics conserve the inner product, because it provides a complex or real number. The unitary transfomations needs to do more things as I have told you and which are even more details to put. There are very good books on the subject as the Hammermish.
Sorry, I have written badly the name of the author using just memory. The correct reference is:
Hamermesh, M. (2005) Group Theory, in Mathematical Tools for Physicists (ed G. L. Trigg), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, FRG. doi: 10.1002/3527607773.ch7
Briefly, a bounded transformation T: H → H, where H is a Hilbert space, which is isometric, that is ||T|| = 1, and surjective (in other words, an isomorphism whose range and domain coincide) is called unitary.
I take the opportunity and warmly recommend to the interested the truly wonderful book Unitary Transformations in Solid State Physics, by Max Wagner (North-Holland, Amsterdam, 1986).
Dear Christian,
It is interesting your application of the symplectic structures in Physics for the unitary transformation, but I think that this only happens when the rank of the matrices is pair, i.e.
2n x 2n. In such a case you can generalize the complex numbers to quaternions, biquaternions etc as parts of Clifford algebras, isn't?
Dear Christian,
Thank you very much for reference and explanation.
The application that I know for the symplectic manifolds M is in classical mechanics to present the Liouville theorem in phase space of fluids ,where Legendre transformations are allowed. I remember to have been reading papers of Elie Cartan showing the 2-differential closed form defined on the cotangent space for a vector which has to be nondegenerate.
What I wonted to say is that, since in odd dimensions, antisymmetric matrices are not invertible and the sumplectic differential two-form, the skew-symmetric condition implies that M has even dimension. Thus the matrices representation must be also 2nx2n and the interpretaiton of numbers is just an extra that is not needed.
Unitary matrix is a n x n complex square matrix. It obeys the condition U†U= I , where dagger denotes transpose-conjugate and I denotes the unit matrix of dimension n.
Consider non-relativistic quantum mechanics of a single particle. If a wave function undergoes a transformation such as ψ→ U ψ, then ψ†ψ will transform to ψ†U†Uψ=ψ†ψ, which means that probability density P(x) remains invariant if wave function undergoes a unitary transformation.
Another mathematical property of unitary matrices can be noted. All rows and columns of a unitary matrix are of length one. All rows are orthogonal to each other whereas all columns are also orthogonal to each other. In other words rows form a unitary basis, and similarly columns also form a unitary basis.
- Badges
- Science topic
- Similar topics
- Physics