Hilbert space of direct products

[Wiemster]

How come if all states in the representation space (of say rotations) have the same energy, Hilbert space can be written as a direct product space of these representation spaces?

 

[dextercioby]

Direct product ? More like a direct sum.

Daniel.

 

[denian]

The concept of Hilbert space is a fundamental tool in mathematical physics, particularly in the study of quantum mechanics. It is a mathematical structure that allows us to describe the states of a physical system and the evolution of those states over time. In the case of rotations, the representation space refers to the set of all possible states of a system under rotation. These states can be described by a set of quantum numbers, such as spin, angular momentum, and energy.

Now, if all states in the representation space have the same energy, it means that they are degenerate states. This implies that they can be transformed into each other without changing the energy of the system. In other words, they are equivalent states in terms of energy. This degeneracy can occur due to symmetries of the system, such as rotational symmetry.

In such a case, the Hilbert space can be written as a direct product space of these representation spaces. This means that the states in the Hilbert space can be written as a combination of states from each representation space. This is because in a direct product space, every state is a unique combination of states from different spaces. In this case, the different representation spaces correspond to different quantum numbers, and the direct product space allows us to describe all possible combinations of these quantum numbers.

In summary, if the states in the representation space have the same energy, it implies a degeneracy in the system, which allows us to write the Hilbert space as a direct product space. This is a powerful mathematical tool that helps us understand and study the symmetries and degeneracies in physical systems, such as rotations.