Projective Hilbert space

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and the foundations of <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the projective Hilbert space or ray space 
  
    
      
        
          P
        
        (
        H
        )
      
    
    {\displaystyle \mathbf {P} (H)}
  
 of a <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
  
    
      
        H
      
    
    {\displaystyle H}
  
 is the set of <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> 
  
    
      
        [
        v
        ]
      
    
    {\displaystyle [v]}
  
 of non-zero vectors 
  
    
      
        v
        ∈
        H
      
    
    {\displaystyle v\in H}
  
, for the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> 
  
    
      
        ∼
      
    
    {\displaystyle \sim }
  
 on 
  
    
      
        H
      
    
    {\displaystyle H}
  
 given by
</p>

w
        ∼
        v
      
    
    {\displaystyle w\sim v}
  
 if and only if 
  
    
      
        v
        =
        λ
        w
      
    
    {\displaystyle v=\lambda w}
  
 for some non-zero complex number 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
.
<p>This is the usual construction of <a href="/facts/Projectivization/KB2shv7I">projectivization</a>, applied to a complex Hilbert space. In quantum mechanics, the equivalence classes 
  
    
      
        [
        v
        ]
      
    
    {\displaystyle [v]}
  
 are also referred to as rays or projective rays. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed.
</p>

Projective Hilbert space open-in-new

Projective Hilbert space