Antiunitary operator

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an antiunitary transformation is a <a href="/facts/Bijective/j4gTuTmW">bijective</a> <a href="/facts/Antilinear_map/GTXzWMKc">antilinear map</a>

U
        :
        
          H
          
            1
          
        
        →
        
          H
          
            2
          
        
        
      
    
    {\displaystyle U:H_{1}\to H_{2}\,}

between two <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a> such that

⟨
        U
        x
        ,
        U
        y
        ⟩
        =
        
          
            
              ⟨
              x
              ,
              y
              ⟩
            
            ¯
          
        
      
    
    {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}}

for all 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 in 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H_{1}}
 
, where the horizontal bar represents the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a>. If additionally one has 
 
 
 
 
 H
 
 1
 
 
 =
 
 H
 
 2
 
 
 
 
 {\displaystyle H_{1}=H_{2}}
 
 then 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is called an antiunitary operator.
Antiunitary operators are important in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> because they are used to represent certain <a href="/facts/Symmetry_(physics)/gJmgsooI">symmetries</a>, such as <a href="/facts/T-symmetry/Tgf2LJMc">time reversal</a>. Their fundamental importance in quantum physics is further demonstrated by <a href="/facts/Wigner%27s_theorem/IHmtm9bF">Wigner's theorem</a>.

Antiunitary operator open-in-new

Antiunitary operator