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Intheoretical physics, thesuperpotential is a function insupersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in theSchrödinger equation. The partner potentials have the samespectrum, apart from a possibleeigenvalue of zero, meaning that the physical systems represented by the two potentials have the same characteristic energies, apart from a possible zero-energy ground state.

Consider aone-dimensional, non-relativistic particle with a two state internal degree of freedom called "spin". (This is not quite the usual notion of spin encountered in nonrelativistic quantum mechanics, because "real" spin applies only to particles inthree-dimensional space.) Letb and itsHermitian adjointb^† signifyoperators which transform a "spin up" particle into a "spin down" particle and vice versa, respectively. Furthermore, takeb andb^† to be normalized such that theanticommutator {b,b^†} equals 1, and take thatb² equals 0. Letp represent themomentum of the particle andx represent itsposition with [x,p]=i, where we usenatural units so that${\displaystyle \hslash =1}$. LetW (the superpotential) represent an arbitrarydifferentiable function ofx and define the supersymmetric operatorsQ₁ andQ₂ as

${\displaystyle Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW)b^{†}\right]}$

${\displaystyle Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW)b^{†}\right]}$

The operatorsQ₁ andQ₂ are self-adjoint. Let theHamiltonian be

${\displaystyle H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W^{\prime }}{2}(b{b}^{†}-b^{†}b)}$

whereW' signifies the derivative ofW. Also note that {Q₁,Q₂}=0. Under these circumstances, the above system is atoy model ofN=2 supersymmetry. The spin down and spin up states are often referred to as the "bosonic" and "fermionic" states, respectively, in an analogy toquantum field theory. With these definitions,Q₁ andQ₂ map "bosonic" states into "fermionic" states and vice versa. Restricting to the bosonic or fermionic sectors gives twopartner potentials determined by

${\displaystyle H=\frac{p^2}{2}+\frac{W^2}{2}\pm \frac{W^{\prime }}{2}}$

Insupersymmetricquantum field theories with fourspacetime dimensions, which might have some connection to nature, it turns out thatscalar fields arise as the lowest component of achiral superfield, which tends to automatically be complex valued. We may identify the complex conjugate of a chiral superfield as an anti-chiral superfield. There are two possible ways to obtain an action from a set of superfields:

• Integrate a superfield on the whole superspace spanned by${\displaystyle x_{0,1,2,3}}$ and${\displaystyle \theta ,\overline{\theta }}$,

or

• Integrate a chiral superfield on the chiral half of a superspace, spanned by${\displaystyle x_{0,1,2,3}}$ and${\displaystyle \theta }$, not on${\displaystyle \overline{\theta }}$.

The second option tells us that an arbitraryholomorphic function of a set of chiral superfields can show up as a term in a Lagrangian which is invariant under supersymmetry. In this context, holomorphic means that the function can only depend on the chiral superfields, not their complex conjugates. We may call such a functionW, the superpotential. The fact thatW is holomorphic in the chiral superfields helps explain why supersymmetric theories are relatively tractable, as it allows one to use powerful mathematical tools fromcomplex analysis. Indeed, it is known thatW receives no perturbative corrections, a result referred to as theperturbative non-renormalization theorem. Note that non-perturbative processes may correct this, for example through contributions to thebeta functions due toinstantons.

• Komar superpotential

• Stephen P. Martin,A Supersymmetry Primer.arXiv:hep-ph/9709356.
• B. Mielnik and O. Rosas-Ortiz, "Factorization: Little or great algorithm?",J. Phys. A: Math. Gen. 37: 10007-10035, 2004
• Cooper, Fred; Khare, Avinash; Sukhatme, Uday (1995). "Supersymmetric quantum mechanics".Physics Reports.251 (5):267–385.arXiv:hep-th/9405029.Bibcode:1995PhR...251..267C.doi:10.1016/0370-1573(94)00080-M.

• Supersymmetry
• Supersymmetric quantum field theory
• Potentials

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