##
**Dynamical breaking of supersymmetry.**
*(English)*
Zbl 1258.81046

It is an honour for me to be the reviewer of this seminal and renowned paper. I start mentioning the works of Gaston Darboux who solved some differential equations that appeared in quantum mechanics some years afterwards.

In 1882 G. Darboux published the seminal paper “Sur une proposition relative aux équations linéaires” [C. R. XCIV, 1456–1459 (1882; JFM 14.0264.01)], where he presented a quite general result that today in a particular case is a very important theorem that involves the so-called Darboux transformation. Darboux proved that if one knows how to integrate the equation \[ {d^2y \over dx^2} = (\Phi(x) + m)y, \] for all values of the constant \(m\), then one can obtain an infinite set of equations, displaying the variable parameter in the same way, which are integrable for any value of the parameter. This proposition can also be found in his book [Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II. Paris. Gauthier-Villars and Fils (1889; JFM 21.0744.02), p. 210].

Strangely, Darboux’s transformation was forgotten for a long time until it was recovered as an exercise in 1927 by E. Ince in his book [Ordinary differential equations. Longmans, Green & Co, London (1927; JFM 53.0399.07), p. 132, Exercises 5, 6 and 7], where he followed the same formulation as Darboux.

In 1930, P. A. M. Dirac published [The principles of quantum mechanics. London: H. Milford (1930; JFM 56.0745.05)], where he gave a mathematically rigorous formulation of quantum mechanics. Later, in 1955, M. M. Crum, inspired by Liouville’s work on Sturm-Liouville systems, developed a kind of iterative generalisation of the Darboux transformation. Crum did not refer to Darboux.

Almost one hundred years after Darboux’s proposition, Edward Witten, in the paper under review, gave birth to a “toy model” of quantum field theory, the “supersymmetric quantum mechanics” (SUSY QM), where he discussed general conditions for dynamical supersymmetry breaking. The paper contains eight sections, plus introduction and conclusions, in particular, in Section 6 he considers two models which involve systems with dimensions less than four where dynamical breaking of supersymmetry is plausible. These models have been used by a lot of researchers of supersymmetric quantum mechanics for the two-dimensional case.

Following the material under review, a supersymmetric quantum mechanical system is one in which there are operators \(Q_i\) that commute with the Hamiltonian \(\mathcal H\), \[ [Q_i,\mathcal H]=0,\quad i=1,\ldots,n,\tag{wit1} \] and satisfy the conditions \[ \{Q_i,Q_j\}=\delta_{ij}\mathcal H,\quad \{Q_i,Q_j\}=Q_iQ_j+Q_jQ_i.\tag{wit2} \] Witten’s theory points out that the simplest example of a supersymmetric quantum mechanical system corresponds to the case \(n=2\), which is considered in this paper. The wave function of \(\mathcal H\Phi=E\Phi\) is therefore a two-component Pauli spinor, \[ \Phi(x)= \left( \begin{matrix} \Psi_+(x) \\ \Psi_-(x) \end{matrix} \right). \] The supercharges \(Q_i\) are defined as \[ Q_\pm=\frac{\sigma_1p\pm\sigma_2W(x)}2,\quad Q_+=Q_1,\, Q_-= Q_2,\tag{wit3} \] where the superpotential \(W\) is an arbitrary function of \(x\), \(p = -i{d\over dx},\) and \(\sigma_i\) are the usual Pauli spin matrices. Using the expressions above we obtain \(\mathcal H\): \[ \mathcal{H}=2Q_-^2=2Q_+^2=\frac{I_2p^2+I_2W^2(x)+\sigma_3{d\over dx}W(x)}2,\tag{wit4} \] where \(I_2\) is the identity matrix of size \(2\times 2\).

The supersymmetric partner Hamiltonians \(H_\pm\) are given by \[ H_\pm=-{1\over 2}{d^2\over dx^2}+V_\pm,\quad 2V_\pm=W^2\pm{d\over dx}W. \] The potentials \(V_\pm\) are called supersymmetric partner potentials and are linked with the superpotential \(W\) through a Riccati equation. Thus, \({\mathcal H}\) can be written as \({\mathcal H}=H_+\oplus H_-\) (the block matrix \(\text{diag}(\mathcal{H})=(H_+,H_-)\)), which leads to the Schrödinger equations \(H_+\Psi_+=E\Psi_+\) and \(H_-\Psi_-=E\Psi_-\), and for instance, to solve \({\mathcal H}\Phi=E\Phi\) is equivalent to solve simultaneously \(H_+\Psi_+=E\Psi_+\) and \(H_-\Psi_-=E\Psi_-\).

V. B. Matveev and M. Salle in their book [Darboux transformations and solitons. Berlin etc.: Springer (1991; Zbl 0744.35045)] interpret the Darboux theorem as Darboux covariance of a Sturm-Liouville problem and proved the following result:

The case \(n=2\) in Witten’s Supersymmetric Quantum Mechanics is equivalent to a single Darboux transformation.

Since this work of Witten, thousands of papers on SUSY QM have been written. One of these papers was written in 1983 by L. Gendenshteïn [Sov.Phys., JETP 38, 356–359 (1983; Zbl 1398.81106)], where the concept of “shape invariance” arises, which has been referred to by a lot of authors. Several techniques and approaches have been applied to SUSY QM. Recently, a Galoisian approach has been developed by different researchers, including myself, for Witten’s supersymmetric quantum mechanics (see for example the monograph [the reviewer, Galoisian approach to supersymmetric quantum mechanics: the integrability analysis of the Schrödinger equation by means of differential Galois theory. Saarbrücken: VDM Verlag Dr Müller (2010)]; and see also the paper [the reviewer, {J. J. Morales-Ruiz} and J. A. Weil, Rep.Math.Phys.67, No. 3, 305–374 (2011; Zbl 1238.81090)]. However, one question stated in the paper under review has not been solved yet, the general study of the case \(n\geq 4\) for SUSY QM, in particular, a complete analysis for supersymmetric quantum mechanical systems of dimension \(n=4\) has not been achieved yet.

In 1882 G. Darboux published the seminal paper “Sur une proposition relative aux équations linéaires” [C. R. XCIV, 1456–1459 (1882; JFM 14.0264.01)], where he presented a quite general result that today in a particular case is a very important theorem that involves the so-called Darboux transformation. Darboux proved that if one knows how to integrate the equation \[ {d^2y \over dx^2} = (\Phi(x) + m)y, \] for all values of the constant \(m\), then one can obtain an infinite set of equations, displaying the variable parameter in the same way, which are integrable for any value of the parameter. This proposition can also be found in his book [Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II. Paris. Gauthier-Villars and Fils (1889; JFM 21.0744.02), p. 210].

Strangely, Darboux’s transformation was forgotten for a long time until it was recovered as an exercise in 1927 by E. Ince in his book [Ordinary differential equations. Longmans, Green & Co, London (1927; JFM 53.0399.07), p. 132, Exercises 5, 6 and 7], where he followed the same formulation as Darboux.

In 1930, P. A. M. Dirac published [The principles of quantum mechanics. London: H. Milford (1930; JFM 56.0745.05)], where he gave a mathematically rigorous formulation of quantum mechanics. Later, in 1955, M. M. Crum, inspired by Liouville’s work on Sturm-Liouville systems, developed a kind of iterative generalisation of the Darboux transformation. Crum did not refer to Darboux.

Almost one hundred years after Darboux’s proposition, Edward Witten, in the paper under review, gave birth to a “toy model” of quantum field theory, the “supersymmetric quantum mechanics” (SUSY QM), where he discussed general conditions for dynamical supersymmetry breaking. The paper contains eight sections, plus introduction and conclusions, in particular, in Section 6 he considers two models which involve systems with dimensions less than four where dynamical breaking of supersymmetry is plausible. These models have been used by a lot of researchers of supersymmetric quantum mechanics for the two-dimensional case.

Following the material under review, a supersymmetric quantum mechanical system is one in which there are operators \(Q_i\) that commute with the Hamiltonian \(\mathcal H\), \[ [Q_i,\mathcal H]=0,\quad i=1,\ldots,n,\tag{wit1} \] and satisfy the conditions \[ \{Q_i,Q_j\}=\delta_{ij}\mathcal H,\quad \{Q_i,Q_j\}=Q_iQ_j+Q_jQ_i.\tag{wit2} \] Witten’s theory points out that the simplest example of a supersymmetric quantum mechanical system corresponds to the case \(n=2\), which is considered in this paper. The wave function of \(\mathcal H\Phi=E\Phi\) is therefore a two-component Pauli spinor, \[ \Phi(x)= \left( \begin{matrix} \Psi_+(x) \\ \Psi_-(x) \end{matrix} \right). \] The supercharges \(Q_i\) are defined as \[ Q_\pm=\frac{\sigma_1p\pm\sigma_2W(x)}2,\quad Q_+=Q_1,\, Q_-= Q_2,\tag{wit3} \] where the superpotential \(W\) is an arbitrary function of \(x\), \(p = -i{d\over dx},\) and \(\sigma_i\) are the usual Pauli spin matrices. Using the expressions above we obtain \(\mathcal H\): \[ \mathcal{H}=2Q_-^2=2Q_+^2=\frac{I_2p^2+I_2W^2(x)+\sigma_3{d\over dx}W(x)}2,\tag{wit4} \] where \(I_2\) is the identity matrix of size \(2\times 2\).

The supersymmetric partner Hamiltonians \(H_\pm\) are given by \[ H_\pm=-{1\over 2}{d^2\over dx^2}+V_\pm,\quad 2V_\pm=W^2\pm{d\over dx}W. \] The potentials \(V_\pm\) are called supersymmetric partner potentials and are linked with the superpotential \(W\) through a Riccati equation. Thus, \({\mathcal H}\) can be written as \({\mathcal H}=H_+\oplus H_-\) (the block matrix \(\text{diag}(\mathcal{H})=(H_+,H_-)\)), which leads to the Schrödinger equations \(H_+\Psi_+=E\Psi_+\) and \(H_-\Psi_-=E\Psi_-\), and for instance, to solve \({\mathcal H}\Phi=E\Phi\) is equivalent to solve simultaneously \(H_+\Psi_+=E\Psi_+\) and \(H_-\Psi_-=E\Psi_-\).

V. B. Matveev and M. Salle in their book [Darboux transformations and solitons. Berlin etc.: Springer (1991; Zbl 0744.35045)] interpret the Darboux theorem as Darboux covariance of a Sturm-Liouville problem and proved the following result:

The case \(n=2\) in Witten’s Supersymmetric Quantum Mechanics is equivalent to a single Darboux transformation.

Since this work of Witten, thousands of papers on SUSY QM have been written. One of these papers was written in 1983 by L. Gendenshteïn [Sov.Phys., JETP 38, 356–359 (1983; Zbl 1398.81106)], where the concept of “shape invariance” arises, which has been referred to by a lot of authors. Several techniques and approaches have been applied to SUSY QM. Recently, a Galoisian approach has been developed by different researchers, including myself, for Witten’s supersymmetric quantum mechanics (see for example the monograph [the reviewer, Galoisian approach to supersymmetric quantum mechanics: the integrability analysis of the Schrödinger equation by means of differential Galois theory. Saarbrücken: VDM Verlag Dr Müller (2010)]; and see also the paper [the reviewer, {J. J. Morales-Ruiz} and J. A. Weil, Rep.Math.Phys.67, No. 3, 305–374 (2011; Zbl 1238.81090)]. However, one question stated in the paper under review has not been solved yet, the general study of the case \(n\geq 4\) for SUSY QM, in particular, a complete analysis for supersymmetric quantum mechanical systems of dimension \(n=4\) has not been achieved yet.

### MSC:

81Q60 | Supersymmetry and quantum mechanics |

81T60 | Supersymmetric field theories in quantum mechanics |

### Citations:

Zbl 0744.35045; Zbl 1238.81090; JFM 14.0264.01; JFM 21.0744.02; JFM 53.0399.07; JFM 56.0745.05; Zbl 1398.81106
Full Text:
DOI

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