33.3 Supersymmetric quantum mechanics and diferential forms

33.3 Supersymmetric quantum mechanics and diferential forms

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · differentiable manifold

If one considers supersymmetric quantum mechanics in the case of $d$ degrees of freedom and in the Schr¨odinger representation, one has

$$\mathcal{H}=L^2({\mathbf{R}}^d)\otimes {\Lambda }^{\ast }({\mathbf{C}}^d)$$

the tensor product of complex-valued functions on ${\mathbf{R}}^d$ (acted on by the Weyl algebra $\mathrm{Weyl}(2d,\mathbf{C}))$ and anticommuting functions on ${\mathbf{C}}^d$ (acted on by the Cliford algebra $\mathrm{Cliff}(2d,\mathbf{C}))$ . There are two operators $Q_{+}$ and $Q_{-},$ adjoints of each other and of square zero. If one has studied diferential forms, this should look familiar. This space $\mathcal{H}$ is well known to mathematicians, as the complexvalued diferential forms on ${\mathbf{R}}^d.$ , often written ${\Omega }^{\ast }({\mathbf{R}}^d)$ , where here the ^∗ denotes an index taking values from 0 (the 0-forms, or functions) to d (the d-forms). In the theory of diferential forms, it is well known that one has an operator $d$ on ${\Omega }^{\ast }({\mathbf{R}}^d)$ with square zero, called the de Rham diferential. Using the inner product on ${\mathbf{R}}^d$ , a Hermitian inner product can be put on ${\Omega }^{\ast }({\mathbf{R}}^d)$ by integration, and then d has an adjoint $\delta ,$ also of square zero. The Laplacian operator on diferential forms is

$$\square =(d+\delta {)}^2$$

The supersymmetric quantum system we have been considering corresponds precisely to this, once one conjugates $d,\delta$ as follows

$$Q_{+}=e^{-W(q)}d{e}^{W(q)},Q_{-}=e^{W(q)}\delta e^{-W(q)}$$

In mathematics, the interest in diferential forms mainly comes from the fact that they can be constructed not just on ${\mathbf{R}}^d$ , but on a general diferentiable manifold $M.$ , with a corresponding construction of $d,\delta ,\square$ operators. In Hodge theory, one studies solutions of

$$\square \psi =0$$

(these are called “harmonic forms”) and finds that the dimension of the space of solutions $\psi \in {\Omega }^k(M)$ gives a topological invariant called the kth Betti number of the manifold $M$.

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原书 PDF · 印刷页 351、352、353、354、355、356、357

来源版本：2025-10-20

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