---
title: "Supersymmetry"
chunk: 2/6
source: "https://en.wikipedia.org/wiki/Supersymmetry"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:40:45.384774+00:00"
instance: "kb-cron"
---

  
    
      
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                ¯
              
            
          
          
            
              
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        =
        2
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            μ
          
        
        
          )
          
            α
            
              
                
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    {\displaystyle \{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}=2(\sigma ^{\mu })_{\alpha {\dot {\beta }}}P_{\mu }}
  

and all other anti-commutation relations between the 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
s and commutation relations between the 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
s and 
  
    
      
        P
      
    
    {\displaystyle P}
  
s vanish. In the above expression 
  
    
      
        
          P
          
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        =
        −
        i
        
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    {\displaystyle P_{\mu }=-i\partial _{\mu }}
  
 are the generators of translation and 
  
    
      
        
          σ
          
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    {\displaystyle \sigma ^{\mu }}
  
 are the Pauli matrices.
There are representations of a Lie superalgebra that are analogous to representations of a Lie algebra. Each Lie algebra has an associated Lie group and a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.

=== Supersymmetric quantum mechanics ===

Supersymmetric quantum mechanics adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. Supersymmetric quantum mechanics often becomes relevant when studying the dynamics of supersymmetric solitons, and due to the simplified nature of having fields which are only functions of time (rather than space-time), a great deal of progress has been made in this subject and it is now studied in its own right.
SUSY quantum mechanics involves pairs of Hamiltonians which share a particular mathematical relationship, which are called partner Hamiltonians. (The potential energy terms which occur in the Hamiltonians are then known as partner potentials.) An introductory theorem shows that for every eigenstate of one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy. This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy.

=== Supersymmetry in quantum field theory ===
In quantum field theory, supersymmetry is motivated by solutions to several theoretical problems, for generally providing many desirable mathematical properties, and for ensuring sensible behavior at high energies. Supersymmetric quantum field theory is often much easier to analyze, as many more problems become mathematically tractable. When supersymmetry is imposed as a local symmetry, Einstein's theory of general relativity is included automatically, and the result is said to be a theory of supergravity. Another theoretically appealing property of supersymmetry is that it offers the only "loophole" to the Coleman–Mandula theorem, which prohibits spacetime and internal symmetries from being combined in any nontrivial way, for quantum field theories with very general assumptions. The Haag–Łopuszański–Sohnius theorem demonstrates that supersymmetry is the only way spacetime and internal symmetries can be combined consistently.
While supersymmetry has not been discovered at high energy, see Section Supersymmetry in particle physics, supersymmetry was found to be effectively realized at the intermediate energy of hadronic physics where baryons and mesons are superpartners. An exception is the pion that appears as a zero mode in the mass spectrum and thus protected by the supersymmetry: It has no baryonic partner. The realization of this effective supersymmetry is readily explained in quark–diquark models: Because two different color charges close together (e.g., blue and red) appear under coarse resolution as the corresponding anti-color (e.g. anti-green), a diquark cluster viewed with coarse resolution (i.e., at the energy-momentum scale used to study hadron structure) effectively appears as an antiquark. Therefore, a baryon containing 3 valence quarks, of which two tend to cluster together as a diquark, behaves like a meson.

=== Supersymmetry in condensed matter physics ===
SUSY concepts have provided useful extensions to the WKB approximation. Additionally, SUSY has been applied to disorder averaged systems both quantum and non-quantum (through statistical mechanics), the Fokker–Planck equation being an example of a non-quantum theory. The 'supersymmetry' in all these systems arises from the fact that one is modelling one particle and as such the 'statistics' do not matter. The use of the supersymmetry method provides a mathematical rigorous alternative to the replica trick, but only in non-interacting systems, which attempts to address the so-called 'problem of the denominator' under disorder averaging. For more on the applications of supersymmetry in condensed matter physics see Efetov (1997).
In 2021, a group of researchers showed that, in theory, 
  
    
      
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        =
        (
        0
        ,
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    {\displaystyle N=(0,1)}
  
 SUSY could be realised at the edge of a Moore–Read quantum Hall state. However, to date, no experiments have been done yet to realise it at an edge of a Moore–Read state. In 2022, a different group of researchers created a computer simulation of atoms in 1 dimensions that had supersymmetric topological quasiparticles.

=== Supersymmetry in optics ===
In 2013, integrated optics was found to provide a fertile ground on which certain ramifications of SUSY can be explored in readily-accessible laboratory settings. Making use of the analogous mathematical structure of the quantum-mechanical Schrödinger equation and the wave equation governing the evolution of light in one-dimensional settings, one may interpret the refractive index distribution of a structure as a potential landscape in which optical wave packets propagate. In this manner, a new class of functional optical structures with possible applications in phase matching, mode conversion and space-division multiplexing becomes possible. SUSY transformations have been also proposed as a way to address inverse scattering problems in optics and as a one-dimensional transformation optics.

=== Supersymmetry in dynamical systems ===