C-theorem

In <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> the C-theorem states that there exists a positive real function, 
 
 
 
 C
 (
 
 g
 
 i

,
 μ
 )
 
 
 {\displaystyle C(g_{i}^{},\mu )}
 
, depending on the <a href="/facts/Coupling_constant/Queop1TR">coupling constants</a> of the quantum field theory considered, 
 
 
 
 
 g
 
 i

{\displaystyle g_{i}^{}}
  
, and on the energy scale, 
  
    
      
        
          μ

{\displaystyle \mu _{}^{}}
 
, which has the following properties:

<ul><li>
 
 
 
 C
 (
 
 g
 
 i

,
 μ
 )
 
 
 {\displaystyle C(g_{i}^{},\mu )}
 
 decreases monotonically under the <a href="/facts/Renormalization_group/Ugy0I1IR">renormalization group</a> (RG) flow.</li>
<li>At fixed points of the <a href="/facts/RG_flow/Ugy0I1IR">RG flow</a>, which are specified by a set of fixed-point couplings 
 
 
 
 
 g
 
 i
 
 
 ∗
 
 
 
 
 {\displaystyle g_{i}^{*}}
 
, the function 
 
 
 
 C
 (
 
 g
 
 i
 
 
 ∗
 
 
 ,
 μ
 )
 =
 
 C
 
 ∗
 
 
 
 
 {\displaystyle C(g_{i}^{*},\mu )=C_{*}}
 
 is a constant, independent of energy scale.</li></ul>
The theorem formalizes the notion that theories at high energies have more degrees of freedom than theories at low energies and that information is lost as we flow from the former to the latter.

C-theorem open-in-new

C-theorem