The term $\widehat p^{,2}/2$ is the target-space zero-mode kinetic energy. The constant $-1/12$ is the <zero-point energy> of the left- and right-moving oscillators: each chiral boson contributes $-1/24$. Finally,
$$
\sum_{n\geq1}\alpha_{-n}\alpha_n,
\qquad
\sum_{n\geq1}\widetilde\alpha_{-n}\widetilde\alpha_n
$$
count the energies of the two independent sets of <string oscillators>; one excitation of mode $n$ has energy $n$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 2 / i / Solution

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Zero-point energy is the nonzero ground-state energy left by quantum fluctuations. One harmonic oscillator contributes $\omega/2$ before any regularization or normal-ordering subtraction.


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