Abstract
This chapter is a preliminary chapter for forthcoming discussions. First, we briefly review representation theories for boson and fermion. Second, we turn to consider partition function \(Z=\text {Tr}(\mathrm{e}^{-\beta \hat{H}})\). Third, we generalize it by turning on an insertion of \((-1)^{\hat{F}}\) into the trace: \(I=\mathrm{Tr}((-1)^{\hat{F}} e^{-\beta \hat{H}})\). This quantity is called Witten index, a prototype of the superconformal index in Chaps. 3–5. \(\hat{F}\) is fermion number operator which counts the number of fermionic excitations. In the last section, we generalize it and the generalized index gives the basis for Chap. 3.
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Notes
- 1.
The simplest way to derive this relation is to use the differential equation. For example,
$$\begin{aligned} \frac{\partial }{\partial x} \langle p | x \rangle&= \lim _{a \rightarrow 0} \frac{\langle p | x +a \rangle - \langle p | x \rangle }{a} \nonumber \\&= \lim _{a \rightarrow 0} \frac{\langle p | e^{- i \hat{p} a} | x \rangle - \langle p | x \rangle }{a} \nonumber \\&= \lim _{a \rightarrow 0} \frac{ e^{- i p a} \langle p | x \rangle - \langle p | x \rangle }{a} \nonumber \\&= - i p \langle p | x \rangle . \nonumber \end{aligned}$$ - 2.
Because of the fermionic natures in (2.13), we have to be careful with the order of \(\psi _+\) and \(\psi _-\).
- 3.
In order to derive this relation from the usual canonical quantization method, considering Poisson bracket is not enough. Instead of it, Dirac bracket is necessary.
- 4.
This is valid if there is no degeneracy.
- 5.
We have checked it only with \(\Psi _-\), but we can understand the case for \(\Psi _+\) in similar way.
References
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Tanaka, A. (2016). Preliminary—Quantum Mechanics. In: Superconformal Index on RP2 × S1 and 3D Mirror Symmetry. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-1398-0_2
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DOI: https://doi.org/10.1007/978-981-10-1398-0_2
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