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# 物理代写|量子力学代写Quantum mechanics代考|PHYSICS332 The thermal propagator

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## 物理代写|量子力学代写Quantum mechanics代考|The thermal propagator

The thermal propagator is the time-ordered propagator in imaginary time. It is equivalent to the Euclidean correlation function in cylindrical geometry. Denote the thermal propagator by:
$$G_{T}^{(0)}(x, \tau)=\langle\phi(x, \tau) \phi(\mathbf{0}, 0)\rangle_{T}$$
It has the Fourier expansion
$$\langle\phi(\boldsymbol{x}, \tau) \phi(\mathbf{0}, 0)\rangle_{T}=\frac{1}{\beta} \sum_{n=-\infty}^{\infty} \int \frac{d^{d} p}{(2 \pi)^{d}} \frac{e^{i \omega_{n} \tau+i p \cdot x}}{\omega_{n}^{2}+\boldsymbol{p}^{2}+m^{2}}$$
where, once again, $\omega_{n}=2 \pi T n$ are the Matsubara frequencies.
We will now obtain two useful expressions for the thermal propagator. The expressions follow from doing the momentum integrals first. The Matsubara frequencies act as mass terms of a field in one dimension lower. This observation allows us to identify the integrals in eq. (5.210) with the Euclidean propagators of an infinite number of fields, each labeled by an integer $n$, in $d$ Euclidean dimensions with mass squared equal to
$$m_{n}^{2}=m^{2}+\omega_{n}^{2}$$

## 物理代写|量子力学代写Quantum mechanics代考|Second quantization and the many-body problem

Let us consider now the problem of a system of $N$ identical nonrelativistic particles. For the sake of simplicity, assume that the physical state of each particle $j$ is described by its position $x_{j}$ relative to some reference frame. This case is easy to generalize.

The wave function for this system is $\Psi\left(x_{1}, \ldots, x_{N}\right)$. If the particles are identical, then the probability density, $\left|\Psi\left(x_{1}, \ldots, x_{N}\right)\right|^{2}$, must be invariant under arbitrary exchanges of the labels that we use to identify (or designate) the particles. In quantum mechanics, particles do not have well-defined trajectories. Only the states of a physical system are well defined. Thus, even though at some initial time $t_{0}$ the $N$ particles may be localized to a set of well-defined positions $x_{1}, \ldots, x_{N}$, they will become delocalized as the system evolves. Furthermore, the Hamiltonian itself is invariant under a permutation of the particle labels. Hence, permutations constitute a symmetry of a many-particle quantum mechanical system. In other words, in quantum mechanics, identical particles are indistinguishable. In particular, the probability density of any eigenstate must remain invariant if the labels of any pair of particles are exchanged.

If we denote by $P_{j k}$ the operator that exchanges the labels of particles $j$ and $k$, the wave functions must satisfy
$$P_{j k} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)=e^{i \phi} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)$$
Under a further exchange operation, the particles return to their initial labels, and we recover the original state. This sample argument then requires that $\phi=0, \pi$, since $2 \phi$ must not be an observable phase. We then conclude that there are two possibilities: either $\Psi$ is even under permutation and $P \Psi=\Psi$, or $\Psi$ is odd under permutation and $P \Psi=-\Psi$.

## 物理代写|量子力学代写Quantum mechanics代考|The thermal propagator

$$G_{T}^{(0)}(x, \tau)=\langle\phi(x, \tau) \phi(\mathbf{0}, 0)\rangle_{T}$$

$$\langle\phi(\boldsymbol{x}, \tau) \phi(\mathbf{0}, 0)\rangle_{T}=\frac{1}{\beta} \sum_{n=-\infty}^{\infty} \int \frac{d^{d} p}{(2 \pi)^{d}} \frac{e^{i \omega_{n} \tau+i p-x}}{\omega_{n}^{2}+\boldsymbol{p}^{2}+m^{2}}$$

$$m_{n}^{2}=m^{2}+\omega_{n}^{2}$$

## 物理代写|量子力学代写Quantum mechanics代考|Second quantization and the many-body problem

$$P_{j k} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)=e^{i \phi} \Psi\left(x_{1}, \ldots, x_{j}, \ldots, x_{k}, \ldots, x_{N}\right)$$

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