A. Zee, Quantum Field Theory in a Nutshell · Problem VIII.4.8

Field Theory Redux

Physics — QFT

Problem

If we can construct supersymmetric quantum field theory, surely we can construct supersymmetric quantum mechanics. Indeed, consider Q1≡12[σ1P+σ2W(x)]Q_1 \equiv \frac{1}{2}[\sigma_1 P + \sigma_2 W(x)] and Q2≡12[σ2P−σ1W(x)]Q_2 \equiv \frac{1}{2}[\sigma_2 P - \sigma_1 W(x)], where the momentum operator P=−i(d/dx)P = -i(d/dx) as usual. Define Q≡Q1+iQ2Q \equiv Q_1 + iQ_2. Study the properties of the Hamiltonian HH defined by {Q,Q†}=2H\{Q, Q^\dagger\} = 2H.

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