Pseudounitary

Pseudounitary refers to a generalization of unitary operators in the context of indefinite inner product spaces. In a standard Hilbert space, a unitary operator $U$ satisfies $U^\dagger U = UU^\dagger = I$, where $U^\dagger$ is the adjoint of $U$ and $I$ is the identity operator. This means that unitary operators preserve the norm of vectors.

In an indefinite inner product space, the inner product is not necessarily positive-definite. Instead, the inner product is defined by a non-degenerate bilinear form, often represented by a Gram operator $G$. A pseudounitary operator $U$ in such a space is defined by the condition $U^\dagger G U = G$, where the adjoint $U^\dagger$ is taken with respect to the indefinite inner product. This condition implies that pseudounitary operators preserve the structure of the indefinite inner product, but not necessarily the "magnitude" of vectors in the same way as unitary operators in Hilbert spaces.

The concept of pseudounitary operators arises in various areas of physics, particularly in quantum field theory and in the study of relativistic quantum mechanics. For instance, in quantum field theories formulated on Minkowski spacetime, operators that preserve the Lorentz metric (which is an indefinite form) are pseudounitary. These operators play a crucial role in maintaining the physical consistency of theories that involve transformations like Lorentz boosts and rotations. The study of pseudounitary operators is essential for understanding the symmetries and dynamics of physical systems described by indefinite inner products.