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\textbf{Homework 1.}
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Consider a time dependent one body operator
$$
H(t)=V_{ij}(t)C_{i}^{\dagger }C_{j}
$$
Define 
$$
B(t,t^{\prime })=T\exp \left[ -\int_{t^{\prime }}^{t}V(t^{\prime \prime
})dt^{\prime \prime }\right]
$$
$$
U(t,t^{\prime })=T\exp \left[ -\int_{t^{\prime }}^{t}H(t^{\prime \prime
})dt^{\prime \prime }\right] 
$$

and 
$$
{\cal{U}}(t,t^{\prime })_{ij}=\left\langle 0\left| C_{i}\,U(t,t^{\prime
})\,C_{j}^{\dagger }\right| 0\right\rangle
$$

a) Prove that 
$$
B(t,t^{\prime })_{ij}=
{\cal{U}}(t,t^{\prime })_{ij}
$$

b) Define ``Heisenberg operators''

$$
C_{Hi}(\tau )\equiv U(0,\tau )C_{i}U(\tau ,0)
$$
$\,\,\,$and prove that
$$
C_{Hi}(\tau )=\sum_{j}B_{ij}(\tau ,\tau ^{\prime })C_{j}(\tau ^{\prime })
$$

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