Path-ordering

In <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, path-ordering is the procedure (or a <a href="/facts/Meta-operator/v97nj54w">meta-operator</a> 
 
 
 
 
 
 P
 
 
 
 
 {\displaystyle {\mathcal {P}}}
 
) that orders a product of operators according to the value of a chosen <a href="/facts/Parameter/zFvVFMv9">parameter</a>:

P
          
        
        
          {
          
            
              O
              
                1
              
            
            (
            
              σ
              
                1
              
            
            )
            
              O
              
                2
              
            
            (
            
              σ
              
                2
              
            
            )
            ⋯
            
              O
              
                N
              
            
            (
            
              σ
              
                N
              
            
            )
          
          }
        
        ≡
        
          O
          
            
              p
              
                1
              
            
          
        
        (
        
          σ
          
            
              p
              
                1
              
            
          
        
        )
        
          O
          
            
              p
              
                2
              
            
          
        
        (
        
          σ
          
            
              p
              
                2
              
            
          
        
        )
        ⋯
        
          O
          
            
              p
              
                N
              
            
          
        
        (
        
          σ
          
            
              p
              
                N
              
            
          
        
        )
        .
      
    
    {\displaystyle {\mathcal {P}}\left\{O_{1}(\sigma _{1})O_{2}(\sigma _{2})\cdots O_{N}(\sigma _{N})\right\}\equiv O_{p_{1}}(\sigma _{p_{1}})O_{p_{2}}(\sigma _{p_{2}})\cdots O_{p_{N}}(\sigma _{p_{N}}).}

Here p is a <a href="/facts/Permutation/JSw9ukmv">permutation</a> that orders the parameters by value:

p
        :
        {
        1
        ,
        2
        ,
        …
        ,
        N
        }
        →
        {
        1
        ,
        2
        ,
        …
        ,
        N
        }
      
    
    {\displaystyle p:\{1,2,\dots ,N\}\to \{1,2,\dots ,N\}}

σ
          
            
              p
              
                1
              
            
          
        
        ≤
        
          σ
          
            
              p
              
                2
              
            
          
        
        ≤
        ⋯
        ≤
        
          σ
          
            
              p
              
                N
              
            
          
        
        .
      
    
    {\displaystyle \sigma _{p_{1}}\leq \sigma _{p_{2}}\leq \cdots \leq \sigma _{p_{N}}.}

For example:

P
          
        
        
          {
          
            
              O
              
                1
              
            
            (
            4
            )
            
              O
              
                2
              
            
            (
            2
            )
            
              O
              
                3
              
            
            (
            3
            )
            
              O
              
                4
              
            
            (
            1
            )
          
          }
        
        =
        
          O
          
            4
          
        
        (
        1
        )
        
          O
          
            2
          
        
        (
        2
        )
        
          O
          
            3
          
        
        (
        3
        )
        
          O
          
            1
          
        
        (
        4
        )
        .
      
    
    {\displaystyle {\mathcal {P}}\left\{O_{1}(4)O_{2}(2)O_{3}(3)O_{4}(1)\right\}=O_{4}(1)O_{2}(2)O_{3}(3)O_{1}(4).}

In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below.

Path-ordering open-in-new

Path-ordering