Unitary transformation

More precisely, a unitary transformation is an isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function

$${\displaystyle U:H\to H_{2}\,}$$

between two inner product spaces, ${\displaystyle H}$ and ${\displaystyle H_{2},}$ such that

$${\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H}\quad {\text{ for all }}x,y\in H.}$$

A unitary transformation is an isometry, as one can see by setting ${\displaystyle x=y}$ in this formula.

In the case when ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

A closely related notion is that of antiunitary transformation, which is a bijective function

${\displaystyle U:H_{1}\to H_{2}\,}$

between two complex Hilbert spaces such that

${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }$

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_{1}}$, where the horizontal bar represents the complex conjugate.

• Antiunitary
• Orthogonal transformation
• Time reversal
• Unitary group
• Unitary operator
• Unitary matrix
• Wigner's theorem
• Unitary transformations in quantum mechanics