Kleinert · 제16장 비평형

$$ P_{N}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\sqrt{\frac{2}{2 \pi L a}}^{2} e^{-\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2} / 2 L a} $$

$$ \varepsilon_{i j k} \varepsilon^{i m n}=\delta_{j}^{m} \delta_{k}^{n}-\delta_{j}^{n} \delta_{k}^{m}, \quad \varepsilon_{i j k} \varepsilon^{i j l}=2 \delta_{k}^{l} $$

$$ \Lambda_{L_{+}}(a, x)+\Lambda_{L_{-}}(a, x)=x\left[\Lambda_{L_{0}}(a, x)+\Lambda_{L_{\infty}}(a, x)\right] . $$

$$ A_{i}=\sum_{a} A_{i}^{a} T_{a} . $$

$$ \mathbf{j}=\rho e \mathbf{v}, $$

$$ \mathbf{j}=\rho e \mathbf{v} . $$

$$ P_{N}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\sum_{n=-\infty}^{\infty} P_{N}^{n}\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right) $$

$$ \int_{c-i \infty}^{c+i \infty} \frac{M d z}{2 \pi i} e^{z L} \tilde{f}(z) \tilde{g}(z)=\int_{0}^{L} d s f(s) g(L-S) $$

$$ \Lambda(a, x)=\frac{a+a^{-1}}{z}-1 . $$

$$ \left[T_{a}, T_{b}\right]=i f_{a b c} T_{c} . $$

$$ M \dot{\mathbf{v}}=e\left(\mathbf{E}+\frac{1}{c} \mathbf{v} \times \mathbf{B}\right) $$

$$ M \dot{\mathbf{v}}=e\left(\mathbf{E}+\frac{1}{c} \mathbf{v} \times \mathbf{B}\right) $$

$$ \varphi_{b}-\varphi_{a}=\int_{t_{a}}^{t_{b}} d t \dot{\varphi}(t)=\int_{t_{a}}^{t_{b}} d t \frac{x_{1} \dot{x}_{2}-x_{2} \dot{x}_{1}}{x_{1}^{2}+x_{2}^{2}}=\int_{\mathbf{x}_{a}}^{\mathbf{x}_{b}} \frac{\mathbf{x} \times d \mathbf{x}}{\mathbf{x}^{2}} $$

$$ \int d^{3} x e^{-a \mathbf{x}^{2}+2 b \mathbf{x} \cdot \mathbf{y}}=(2 \pi)^{3 / 2} a^{-3 / 2} e^{b^{2} \mathbf{y}^{2} / a}, \quad a>0 $$

$$ \frac{\delta \hat{W}_{L}[\mathbf{A}]}{\delta A_{i}^{a}(\mathbf{x})}=i \hat{P} \int_{L} d x_{i}^{\prime} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) T_{a}\left(\mathbf{x}^{\prime}\right) \hat{W}_{L}[\mathbf{A}] $$

$$ \frac{d \mathbf{v}}{d t}=\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v} \cdot \boldsymbol{\nabla}) \mathbf{v}=\frac{\partial \mathbf{v}}{\partial t}+\boldsymbol{\nabla}\left(\frac{1}{2} \mathbf{v}^{2}\right)-\mathbf{v} \times(\boldsymbol{\nabla} \times \mathbf{v}) $$

$$ \begin{align*} \mathbf{j} & =\sigma_{0}\left(\mathbf{E}+\frac{1}{c} \mathbf{v} \times \mathbf{B}\right) \\ & =\sigma_{0}\left(\mathbf{E}+\frac{1}{\rho e c} \mathbf{j} \times \mathbf{B}\right) . \end{align*} $$

$$ n=\frac{1}{2 \pi} \oint_{C} \frac{\mathbf{x} \times d \mathbf{x}}{\mathbf{x}^{2}} $$

$$ \begin{align*} N_{1} & =\int d^{3} x_{1}, d^{3} x_{2} \int_{0}^{L_{1}} d s \int_{0}^{L_{2}} d t \int d^{3} x_{1}^{\prime} d^{3} x_{2}^{\prime} G_{0}\left(\mathbf{x}_{1}-\mathbf{x}_{1}^{\prime} ; s\right) G_{0}\left(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1} ; L_{1}-s\right) \\ & \times G_{0}\left(\mathbf{x}_{2}-\mathbf{x}_{2}^{\prime} ; t\right) G_{0}\left(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2} ; L_{2}-t\right) \frac{l}{\left|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{2}^{\prime}\right|^{4}} \end{align*} $$

$$ \hat{W}_{L}[\mathbf{A}]=e^{i A_{i}\left(\overline{\mathbf{x}}^{1}\right) \Delta x_{i}^{1}} e^{i A_{i}\left(\overline{\mathbf{x}}^{2}\right) \Delta x_{i}^{2}} \cdots e^{i A_{i}\left(\overline{\mathbf{x}}^{n}\right) \Delta x_{i}^{n}} \cdots, $$

$$ M \frac{\partial \mathbf{v}}{\partial t}+\boldsymbol{\nabla}\left(\frac{M}{2} \mathbf{v}^{2}\right)=e \mathbf{E}+M \mathbf{v} \cdot\left(\boldsymbol{\nabla} \times \mathbf{v}+\frac{e}{M c} \mathbf{B}\right) . $$

$$ P_{L}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\sum_{m} \frac{1}{\sqrt{r_{b} r_{a}}}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)}, $$

$$ s^{\prime}=\frac{s}{L_{1}}, \quad t^{\prime}=\frac{t}{L_{2}}, \quad \mathbf{x}=\frac{\mathbf{x}_{1}-\mathbf{x}_{1}^{\prime}}{\sqrt{L_{1}}}, \quad \mathbf{y}=\frac{\mathbf{x}_{2}-\mathbf{x}_{2}^{\prime}}{\sqrt{L_{2}}} $$

$$ \frac{\delta \hat{W}_{L}[\mathbf{A}]}{\delta A_{i}^{a}(\mathbf{x})}=i \hat{P} \delta_{i}(\mathbf{x}, L) T_{a}(\mathbf{x}) \hat{W}_{L}[\mathbf{A}] $$

$$ \mathbf{X}=\boldsymbol{\nabla} \times \mathbf{v}+\frac{e}{M c} \mathbf{B} . $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m}=2 \frac{\sqrt{r_{b} r_{a}}}{L a} e^{-\left(r_{b}^{2}+r_{a}^{2}\right) / L a} I_{m}\left(2 \frac{r_{b} r_{a}}{L a}\right) $$

$$ \begin{align*} N_{2} & =\int d^{3} x_{1} d^{3} x_{2} \int d^{3} x_{1}^{\prime} d^{3} x_{1}^{\prime \prime} d^{3} x_{2}^{\prime} \\ & \times\left[\int_{0}^{L_{1}} d s \int_{0}^{S} d s^{\prime} G_{0}\left(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1} ; L_{1}-s\right) \nabla_{x_{1}^{\prime \prime}}^{j} G_{0}\left(\mathbf{x}_{1}-\mathbf{x}_{1}^{\prime \prime} ; s^{\prime}\right) \nabla_{x_{1}^{\prime}}^{i} G_{0}\left(\mathbf{x}_{1}^{\prime \prime}-\mathbf{x}_{1}^{\prime} ; s-s^{\prime}\right)\right] \\ & \times D_{i k}\left(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{2}\right) D_{j k}\left(\mathbf{x}_{1}^{\prime \prime}-\mathbf{x}_{2}^{\prime}\right)\left[\int_{0}^{L_{2}} d t G_{0}\left(\mathbf{x}_{2}-\mathbf{x}_{2}^{\prime} ; L_{2}-t\right) G_{0}\left(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2} ; t\right)\right] \end{align*} $$

$$ \delta \hat{W}_{L}=i d x_{i} d^{\prime} x_{j} \hat{P} F_{i j}^{a}(\mathbf{x}) T_{a}(\mathbf{x}) \hat{W}_{L} $$

$$ \frac{\partial}{\partial t} \mathbf{B}=-c \boldsymbol{\nabla} \times \mathbf{E}, $$

$$ \prod_{n=1}^{N} \int_{-\pi}^{\pi} \frac{d \varphi_{n}}{2 \pi} $$

$$ N_{2}=\frac{-\sqrt{2} V L_{2}^{-1 / 2} L_{1}^{-1} M^{-1 / 2}}{(4 n)^{6}} \int_{0}^{1} d t \int_{0}^{t} d t^{\prime} t^{\prime}(1-t) \sqrt{\frac{t-t^{\prime}}{1-t+t^{\prime}}} $$

$$ F_{i j}=\partial_{i} A_{j}-\partial_{j} A_{i}-i\left[A_{i}, A_{j}\right] $$

$$ \frac{\partial}{\partial t} \mathbf{X}=\boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{X}) . $$

$$ \prod_{n=1}^{N} \int_{-\infty}^{\infty} \frac{d \varphi_{n}}{2 \pi} $$

$$ \int_{0}^{1} d t \frac{t^{\kappa+\frac{1}{2}}}{\sqrt{1-t}}=B\left(\kappa+\frac{3}{2}, \frac{1}{2}\right) $$

$$ \begin{align*} \hat{W}_{\square} & =e^{i A_{i}\left(\mathbf{x}-d^{\prime} \mathbf{x} / 2\right) d x_{i}} e^{i A_{j}(\mathbf{x}+d \mathbf{x} / 2) d^{\prime} x_{j}} e^{-i A_{i}\left(\mathbf{x}+d^{\prime} \mathbf{x} / 2\right) d x_{i}} e^{-i A_{j}(\mathbf{x}-d \mathbf{x} / 2) d^{\prime} x_{j}} \\ & =e^{i\left[A_{i}(\mathbf{x}) d x_{i}-\partial_{j} A_{i}(\mathbf{x}) d x_{i} d^{\prime} x_{j}+\ldots\right]} e^{i\left[A_{j}(\mathbf{x}) d^{\prime} x_{j}+\partial_{i} A_{j}(\mathbf{x}) d x_{i} d^{\prime} x_{j}+\ldots\right]} \\ & \times e^{-i\left[A_{i}(\mathbf{x}) d x_{i}+\partial_{j} A_{i}(\mathbf{x}) d x_{i} d^{\prime} x_{j}+\ldots\right]} e^{-i\left[A_{j}(\mathbf{x}) d^{\prime} x_{j}-\partial_{i} A_{j}(\mathbf{x}) d x_{i} d^{\prime} x_{j}+\ldots\right]} \\ & =e^{i F_{i j}(\mathbf{x}) d x_{i} d^{\prime} x_{j}} \end{align*} $$

$$ \nabla \times \mathbf{j}=-\frac{\rho e^{2}}{M c} \mathbf{B}, $$

$$ \sum_{n} e^{i k\left(\varphi_{b}+2 \pi n-\varphi_{a}\right)}=\sum_{m=-\infty}^{\infty} \delta(k-m) e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \bar{W}_{L} \equiv \int \mathcal{D} \mathbf{A} e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}}} W_{L}[\mathbf{A}] $$

$$ \frac{\partial}{\partial t} \mathbf{v}+\nabla\left(\frac{M}{2} \mathbf{v}^{2}\right)=e \mathbf{E} . $$

$$ P_{L}^{n}\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right)=\frac{2}{L a} \int_{-\infty}^{\infty} d \mu e^{-\left(r_{b}^{2}+r_{a}^{2}\right) / L a} I_{|\mu|}\left(2 \frac{r_{b} r_{a}}{L a}\right) \frac{1}{2 \pi} e^{i \mu\left(\varphi_{b}-\varphi_{a}+2 \pi n\right)} $$

$$ \begin{align*} \delta \bar{W}_{L} & =\int \mathcal{D} \mathbf{A} \delta W_{L}[\mathbf{A}] e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}}} \\ & =i d x_{i} d^{\prime} x_{j} \int \mathcal{D} \mathbf{A} F_{i j}^{a}(\mathbf{x}) T_{a}(\mathbf{x}) W_{L}[\mathbf{A}] e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}}} \end{align*} $$

$$ \mathbf{j}=-\frac{\rho e^{2}}{M c} \mathbf{A} . $$

$$ P_{L}^{n}(\mathbf{x}, \mathbf{x})=\frac{2}{L a} \int_{-\infty}^{\infty} d \mu e^{-2 r^{2} / L a} I_{|\mu|}\left(2 \frac{r^{2}}{L a}\right) \frac{1}{2 \pi} e^{i 2 \pi \mu n} $$

$$ i \frac{4 \pi}{k} \epsilon_{i j k} \frac{\delta \mathcal{A}_{\mathrm{e}, \mathrm{CS}}}{\delta A_{k}^{a}(\mathbf{x})}=F_{i j}^{a}(\mathbf{x}) $$

$$ \nabla \times \mathbf{B}=\frac{4 \pi}{c} j=-\frac{\rho 4 \pi e^{2}}{M c^{2}} \mathbf{A} . $$

$$ P_{L}^{n}(\mathbf{x}, \mathbf{x})=\frac{2}{a} \frac{\omega}{\sinh \omega L} \int_{-\infty}^{\infty} d \mu e^{-2\left(r^{2} / a\right) \omega \operatorname{coth} \omega L} I_{|\mu|}\left(\frac{2}{a} \frac{r^{2} \omega}{\sinh \omega L}\right) \frac{1}{2 \pi} e^{i 2 \pi \mu n} $$

$$ \delta \bar{W}_{L}=-\frac{4 \pi i}{k} \int \mathcal{D} \mathbf{A} d S_{i} \delta_{i}(\mathbf{x}, L) T_{a}(\mathbf{x}) T_{a}(\mathbf{x}) W_{L}[\mathbf{A}] e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}}} $$

$$ \boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{A})+\lambda^{-2} \mathbf{A}=0 $$

$$ P_{L}^{n} \equiv \int d^{2} x P_{L}^{n}(\mathbf{x}, \mathbf{x})=\frac{1}{2 \sinh \omega L} \int_{-\infty}^{\infty} d \mu e^{-|\mu| \omega L} e^{2 \pi i \mu n} $$

$$ \int_{S} d S_{i} \delta_{i}(\mathbf{x}, L)=\left\{\begin{array}{l} 1 \\ 0 \end{array}\right\} \text { if the line } L\left\{\begin{array}{cc} \text { pierces } & \mathrm{S} \\ \text { misses } & \mathrm{S} \end{array}\right\} $$

$$ \lambda^{-2}=\frac{\rho 4 \pi e^{2}}{M c^{2}} $$

$$ \begin{align*} P_{L} & \equiv \int d^{2} x P_{L}(\mathbf{x}, \mathbf{x})=\sum_{n=-\infty}^{\infty} P_{L}^{n} \\ & =\frac{1}{2 \sinh \omega L}\left(\frac{2}{1-e^{-\omega L}}-1\right)=\frac{1}{[2 \sinh (\omega L / 2)]^{2}} \end{align*} $$

$$ \bar{W}_{L_{+}}-\bar{W}_{L_{-}} \equiv \Delta \bar{W}_{L}=-\frac{4 \pi i}{k} \int \mathcal{D} \mathbf{A} T_{a}(\mathbf{x}) T_{a}(\mathbf{x}) W_{L}[\mathbf{A}] e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}}} $$

$$ P_{L}^{n}=\frac{\omega L}{\sinh \omega L} \frac{1}{4 \pi^{2} n^{2}+\omega^{2} L^{2}} $$

$$ \left(T_{a}\right)_{\alpha \beta}\left(T_{a}\right)_{\gamma \delta}=\frac{1}{2} \delta_{\alpha \delta} \delta_{\beta \gamma}-\frac{1}{2 N} \delta_{\alpha \beta} \delta_{\gamma \delta} . $$

$$ \begin{align*} \frac{1}{2 \omega L} \operatorname{coth}(\omega L / 2) & =\sum_{n=-\infty}^{\infty} \frac{1}{4 \pi^{2} n^{2}+(\omega L)^{2}} \\ & =\frac{1}{L^{2}} \sum_{n=-\infty}^{\infty} \frac{1}{\omega_{n}^{2}+\omega^{2}} \end{align*} $$

$$ \left(1-\frac{\pi i}{N k}\right) \bar{W}_{L_{+}}-\left(1+\frac{\pi i}{N k}\right) \bar{W}_{L_{-}}=-\frac{2 \pi i}{k} \bar{W}_{L_{0}} . $$

$$ P_{L}^{n}=P_{L} \cdot \alpha_{n} $$

$$ \left(1-\frac{\pi i}{N k}\right) \bar{W}_{L_{T+}}-\left(1+\frac{\pi i}{N k}\right) \bar{W}_{L_{T-}}=-\frac{2 \pi i}{k} N \bar{W}_{L_{T 0}} . $$

$$ \begin{align*} \alpha_{n} & =\frac{1}{\omega_{n}^{2}+\omega^{2}}\left[\sum_{n=-\infty}^{\infty} \frac{1}{\omega_{n}^{2}+\omega^{2}}\right]^{-1} \\ & =\frac{1}{L^{2}} \frac{1}{\omega_{n}^{2}+\omega^{2}}\left[\frac{1}{2 \omega L} \operatorname{coth} \frac{\omega L}{2}\right]^{-1} \end{align*} $$

$$ \bar{W}_{L_{T+}}=a \bar{W}_{L_{T 0}}, \quad \bar{W}_{L_{T-}}=a^{-1} \bar{W}_{L_{T 0}} . $$

$$ B_{3}=\frac{g}{2 \pi} \epsilon_{3 j k} \partial_{j} \partial_{k} \varphi=g \delta^{(2)}\left(\mathbf{x}_{\perp}\right) $$

$$ a=1-\frac{\pi i}{N k}\left(N^{2}-1\right), \quad a^{-1}=1+\frac{\pi i}{N k}\left(N^{2}-1\right) . $$

$$ \mathcal{A}_{\mathrm{mag}}=\frac{e}{c} \int_{t_{a}}^{t_{b}} d t \dot{\mathrm{x}} \cdot \mathbf{A} $$

$$ \left[\left(1-\frac{\pi i}{N k}\right) a\right] H_{L_{+}}-\left[\left(1+\frac{\pi i}{N k}\right) a^{-1}\right] H_{L_{-}}=-\frac{2 \pi i}{k} H_{L_{0}} $$

$$ A_{i}=\frac{g}{2 \pi} \partial_{i} \varphi, \quad(i=1,2), $$

$$ \varphi(\mathbf{x}) \equiv \arctan \left(x_{2} / x_{1}\right) $$

$$ \int d^{2} x\left(\partial_{1} \partial_{2}-\partial_{2} \partial_{1}\right) \varphi=\oint d \varphi=2 \pi $$

$$ \Phi=\int d^{2} x B_{3} $$

$$ \Phi=g $$

$$ \mathcal{A}_{\mathrm{mag}}=-\hbar \mu_{0} \int_{t_{a}}^{t_{b}} d t \dot{\varphi} $$

$$ \mu_{0} \equiv-\frac{e g}{2 \pi \hbar c} $$

$$ n=\frac{1}{2 \pi} \int_{t_{a}}^{t_{b}} d t \dot{\varphi} $$

$$ \mathcal{A}_{\mathrm{mag}}=-\hbar \mu_{0} 2 \pi n $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\int_{-\infty}^{\infty} d \mu & \frac{1}{\sqrt{r_{b} r_{a}}}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\mu} \\ & \times \sum_{n=-\infty}^{\infty} \frac{1}{2 \pi} e^{i\left(\mu-\mu_{0}\right)\left(\varphi_{b}+2 \pi n-\varphi_{a}\right)} \end{align*} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\sum_{m=-\infty}^{\infty} \frac{1}{\sqrt{r_{b} r_{a}}}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m+\mu_{0}} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m+\mu_{0}}=\sqrt{r_{b} r_{a}} \frac{M}{\hbar} \frac{1}{\left(\tau_{b}-\tau_{a}\right)} \exp \left\{-\frac{M}{2 \hbar} \frac{r_{b}^{2}+r_{a}^{2}}{\tau_{b}-\tau_{a}}\right\} I_{\left|m+\mu_{0}\right|}\left(\frac{M}{\hbar} \frac{r_{b} r_{a}}{\tau_{b}-\tau_{a}}\right) $$

$$ \begin{gather*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m+\mu_{0}}=\sqrt{r_{b} r_{a}} \frac{M \omega}{2 \hbar \eta} \frac{\eta}{\sinh \eta} \exp \left[-\frac{M}{2 \hbar} \frac{\omega}{2} \operatorname{coth} \eta\left(r_{b}^{2}+r_{a}^{2}\right)\right] \\ \times I_{\left|m+\mu_{0}\right|}\left(\frac{M \omega r_{b} r_{a}}{2 \hbar \sinh \eta}\right) e^{\left(m+\mu_{0}\right) \eta} \end{gather*} $$

$$ \frac{e g}{2 \pi \hbar c}=\text { integer }, $$

$$ \Phi_{0} \equiv g_{0} \equiv \frac{2 \pi \hbar c}{e}=\frac{h c}{e} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{2 i M}{\hbar} \sum_{m=-\infty}^{\infty} I_{m}\left(\kappa r_{<}\right) K_{m}\left(\kappa r_{>}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{2 i M}{\hbar} \sum_{m=-\infty}^{\infty} I_{\left|m+\mu_{0}\right|}\left(\kappa r_{<}\right) K_{\left|m+\mu_{0}\right|}\left(\kappa r_{>}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \int_{-\infty}^{\infty} \frac{d E}{2 \pi \hbar} \operatorname{disc}\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\sum_{m=-\infty}^{\infty} \int_{0}^{\infty} d k k J_{\left|m+\mu_{0}\right|}\left(k r_{b}\right) J_{\left|m+\mu_{0}\right|}\left(k r_{a}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{b}\right)} $$

$$ \begin{align*} \int_{-\infty}^{\infty} \frac{d E}{2 \pi \hbar} \operatorname{disc}\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =\int \frac{d^{2} k}{(2 \pi)^{2}} e^{i \mathbf{k}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)}=\frac{1}{2 \pi} \int_{0}^{\infty} d k k J_{0}\left(k\left|\mathbf{x}_{b}-\mathbf{x}_{a}\right|\right) \\ & =\sum_{m=-\infty}^{\infty} \int_{0}^{\infty} d k k J_{m}\left(k r_{b}\right) J_{m}\left(k r_{a}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \end{align*} $$

$$ \begin{align*} \psi_{\mathbf{k}} & =\varphi_{\mathbf{k}}+\frac{1}{E-\hat{H}_{0}+i \eta} \hat{V} \psi_{\mathbf{k}} \\ & =\varphi_{\mathbf{k}}-\frac{i}{\hbar} \hat{R}(E) \hat{V} \varphi_{\mathbf{k}} \end{align*} $$

$$ \hat{H} \psi_{\mathbf{k}}=\left(\hat{H}_{0}+\hat{V}\right) \psi_{\mathbf{k}}=E \psi_{\mathbf{k}} $$

$$ \psi_{\mathbf{k}}(\mathbf{x})=\varphi_{\mathbf{k}}(\mathbf{x})-\frac{i}{\hbar} \int d^{D} x^{\prime}\left(\mathbf{x} \mid \mathbf{x}^{\prime}\right)_{E} V\left(\mathbf{x}^{\prime}\right) \varphi_{\mathbf{k}}\left(\mathbf{x}^{\prime}\right) $$

$$ \psi_{\mathbf{k}}(\mathbf{x}) \xrightarrow{|\mathbf{x}| \rightarrow \infty} e^{i \mathbf{k} x}+\frac{e^{i|\mathbf{k}||\mathbf{x}|}}{|\mathbf{x}|} f(\theta, \varphi)+\ldots $$

$$ \frac{d \sigma}{d \Omega}=|f(\theta, \varphi)|^{2} $$

$$ \psi_{\mathbf{k}}(\mathbf{x}) \xrightarrow{|\mathbf{x}| \rightarrow \infty} e^{i \mathbf{k} x}+\frac{e^{i|\mathbf{k}||\mathbf{x}|}}{\sqrt{|\mathbf{x}|}} f(\varphi)+\ldots $$

$$ \frac{d \sigma}{d \varphi}=|f(\varphi)|^{2} $$

$$ \psi(\mathbf{x})=\int d^{D} x^{\prime} \operatorname{disc}\left(\mathbf{x} \mid \mathbf{x}^{\prime}\right)_{E} \phi\left(\mathbf{x}^{\prime}\right) $$

$$ \psi(\mathbf{x})=\sum_{m=-\infty}^{\infty} a_{m} J_{\left|m+\mu_{0}\right|}(k r) e^{i m \varphi} $$

$$ \lim _{x \rightarrow \infty} \psi(\mathbf{x})=e^{-i k x} e^{-i \mu_{0} \varphi} $$

$$ e A_{i}=-\hbar c \mu_{0} \partial_{i} \varphi $$

$$ \hat{\mathbf{P}}=-i \hbar \nabla-\frac{e}{c} \mathbf{A}=-i \hbar\left(\nabla+i \mu_{0} \nabla \varphi\right) $$

$$ \mathbf{j}(\mathbf{x})=-i \frac{\hbar}{2 M} \psi^{\dagger} \stackrel{\leftrightarrow}{\nabla} \psi(\mathbf{x})-\frac{e}{M c} \mathbf{A}(\mathbf{x}) \psi^{\dagger} \psi(\mathbf{x}) . $$

$$ a_{m}=(-i)^{\left|m+\mu_{0}\right|} $$

$$ f(\varphi)=\frac{1}{\sqrt{2 \pi}} e^{-i \pi / 4} \sin \pi \mu_{0} \frac{e^{-i \varphi / 2}}{\cos (\varphi / 2)} $$

$$ \frac{d \sigma}{d \varphi}=\frac{1}{2 \pi} \sin ^{2} \pi \mu_{0} \frac{1}{\cos ^{2}(\varphi / 2)} $$

$$ \psi_{\mathbf{k}}=\psi^{(1)}+\psi^{(2)}+\psi^{(3)} $$

$$ \psi^{(1)}=\sum_{m=1}^{\infty}(-i)^{m+\mu_{0}} J_{m+\mu_{0}} e^{i m \varphi} $$

$$ \begin{align*} \psi^{(2)} & =\sum_{m=-\infty}^{-1}(-i)^{m+\mu_{0}} J_{\left|m+\mu_{0}\right|} e^{i m \varphi} \\ & =\sum_{m=1}^{\infty}(-i)^{m-\mu_{0}} J_{m-\mu_{0}} e^{-i m \varphi} \end{align*} $$

$$ \psi^{(3)}=(-i)^{\left|\mu_{0}\right|} J_{\left|\mu_{0}\right|} $$

$$ \psi^{(2)}\left(r, \varphi, \mu_{0}\right)=\psi^{(1)}\left(r,-\varphi,-\mu_{0}\right) $$

$$ \psi^{(1)}=\frac{1}{2}(-i)^{\mu_{0}} e^{-i \rho \cos \varphi} I(\rho) $$

$$ I(\rho) \equiv \int_{0}^{\rho} d \rho^{\prime} e^{i \rho^{\prime} \cos \varphi}\left(J_{1+\mu_{0}}-i J_{\mu_{0}} e^{i \varphi}\right) $$

$$ \partial_{\rho} \psi^{(1)}=-i \cos \varphi \psi^{(1)}+\frac{1}{2}(-i)^{\mu_{0}}\left(J_{1+\mu_{0}}-i J_{\mu_{0}} e^{i \varphi}\right), $$

$$ \begin{align*} \partial_{\rho} \psi^{(1)} & =\sum_{m=1}^{\infty}(-i)^{m+\mu_{0}} \partial_{\rho} J_{m+\mu_{0}} e^{i m \varphi} \\ & =\sum_{m=1}^{\infty}(-i)^{m+\mu_{0}} \frac{1}{2}\left(J_{m+\mu_{0}-1}-J_{m+\mu_{0}+1}\right) e^{i m \varphi} \\ & =-\frac{i}{2} \sum_{m=1}^{\infty}(-i)^{m+\mu_{0}} J_{m+\mu_{0}} e^{i m \varphi}\left(e^{i \varphi}+e^{-i \varphi}\right)+\frac{1}{2}(-i)^{\mu_{0}}\left(J_{1+\mu_{0}}-i J_{\mu_{0}} e^{i e}\right) \end{align*} $$

$$ \int_{0}^{\infty} d \rho e^{i \beta \rho} J_{\alpha}(k \rho)=\frac{1}{\left(k^{2}-\beta^{2}\right)^{1 / 2}} e^{i \alpha \arcsin (\beta / k)}, \quad 0<\beta-2 $$

$$ \begin{align*} I_{\infty} & \equiv \int_{0}^{\infty} d \rho^{\prime} e^{i \rho^{\prime} \cos \varphi}\left(J_{1+\mu_{0}}-i J_{\mu_{0}} e^{i \varphi}\right)=\frac{1}{|\sin \varphi|}\left[e^{i \mu_{0}(\pi / 2-|\varphi|)}-i e^{i \varphi} e^{i\left(1+\mu_{0}\right)(\pi / 2-|\varphi|)}\right] \\ & =\frac{i}{|\sin \varphi|} e^{i \mu_{0}(\pi / 2-|\varphi|)}\left(e^{-i|\varphi|}-e^{i \varphi}\right)= \begin{cases}0, & \varphi<0, \\ e^{-i \mu_{0} \varphi} 2 i^{\mu_{0}}, & \varphi>0,\end{cases} \end{align*} $$

$$ \psi_{\infty}^{(1)}= \begin{cases}0, & \varphi<0 \\ e^{-i k x} e^{-i \mu_{0} \varphi}, & \varphi>0\end{cases} $$

$$ \psi_{\infty}^{(2)}= \begin{cases}e^{-i k x} e^{-i \mu_{0} \varphi}, & \varphi<0 \\ 0, & \varphi>0\end{cases} $$

$$ \psi_{\mathrm{sc}}=\Delta \psi^{(1)}+\Delta \psi^{(2)}+\psi^{(3)} $$

$$ \Delta I(\rho) \equiv I(\rho)-I_{\infty}=\int_{\rho}^{\infty} d \rho^{\prime} e^{i \rho^{\prime} \cos \varphi}\left(J_{1+\mu_{0}}-i e^{i \varphi} J_{\mu_{0}}\right) $$

$$ J_{\alpha}(\rho) \sim \sqrt{2 / \pi \rho} \cos (\rho-\alpha / 2-\pi / 4) $$

$$ \Delta I(\rho)=\sqrt{\frac{2}{\pi}}[A(\rho)+B(\rho)] $$

$$ \begin{align*} A(\rho) & =\int_{\rho}^{\infty} \frac{d \rho^{\prime}}{\sqrt{\rho^{\prime}}} e^{i \rho^{\prime} \cos \varphi} \cos \left[\rho^{\prime}-\left(1+\mu_{0}\right) / 2-\pi / 4\right] \\ B(\rho) & =-i e^{i \varphi} \int_{\rho^{\prime}}^{\infty} \frac{d \rho^{\prime}}{\sqrt{\rho^{\prime}}} e^{i \rho^{\prime} \cos \varphi} \cos \left[\rho^{\prime}-\mu_{0} / 2-\pi / 4\right] \end{align*} $$

$$ A(\rho)=\left[\frac{(-i)^{1 / 2+\mu_{0}}}{\sqrt{1+\cos \theta}} \int_{\sqrt{\rho(1+\cos \varphi)}}^{\infty} d t e^{i t^{2}}+\frac{i^{3 / 2+\mu_{0}}}{\sqrt{1-\cos \theta}} \int_{\sqrt{\rho(1-\cos \varphi)}}^{\infty} d t e^{-i t^{2}}\right] $$

$$ \int_{x}^{\infty} d t e^{ \pm i t^{2}}= \pm \frac{i}{2} \frac{\exp \left( \pm i x^{2}\right)}{x}+\ldots $$

$$ \begin{align*} A(\rho)= & \frac{1}{2}\left[(-i)^{\frac{1}{2}+\mu_{0}} \frac{e^{i \rho}}{\sqrt{\rho(1+\cos \varphi)^{2}}}+i^{\frac{1}{2}+\mu_{0}} \frac{e^{-i \rho}}{\sqrt{\rho(1-\cos \varphi)^{2}}}\right] e^{i \rho \cos \varphi} \\ B(\rho)= & (-i) \frac{e^{i \varphi}}{2} \\ & \times\left[(-i)^{-\frac{1}{2}+\mu_{0}} \frac{e^{i \rho}}{\sqrt{\rho(1+\cos \varphi)^{2}}}+i^{-\frac{1}{2}+\mu_{0}} \frac{e^{-i \rho}}{\sqrt{\rho(1-\cos \varphi)^{2}}}\right] e^{i \rho \cos \varphi} \end{align*} $$

$$ \Delta \psi^{(1)}=\frac{\sqrt{-i}}{2 \sqrt{2 \pi \rho}}\left[(-1)^{\mu_{0}} e^{i \rho} \frac{1+e^{i \varphi}}{1+\cos \varphi}+i e^{-i \rho} \frac{1-e^{i \varphi}}{1-\cos \varphi}\right] $$

$$ \Delta \psi^{(1)}+\Delta \psi^{(2)}=\frac{\sqrt{-i}}{\sqrt{2 \pi \rho}}\left[e^{i \rho} \frac{\cos \left(\pi \mu_{0}-\varphi / 2\right)}{\cos (\varphi / 2)}+i e^{-i \rho}\right]+e^{-i\left(\rho \cos \varphi+\mu_{0} \varphi\right)} $$

$$ \psi(\mathbf{x}) \rightarrow e^{-i\left(\rho \cos \varphi+\mu_{0} \varphi\right)}+\psi_{\mathrm{sc}}(\mathbf{x}) $$

$$ \psi_{\mathrm{sc}}=\frac{1}{\sqrt{2 \pi i \rho}} e^{i \rho} \frac{\sin \pi \mu_{0}}{\cos (\varphi / 2)} e^{-i \varphi / 2} $$

$$ \psi(\mathbf{x})=\sqrt{\frac{i}{2}} e^{-i(\varphi / 2+\rho \cos \varphi)} \int_{0}^{\sqrt{\rho(1+\cos \varphi)}} d t e^{i t^{2}} $$

$$ \begin{align*} &\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}+\left(-\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{2 i M}{\hbar} \sum_{m} I_{|m+1|}\left(\kappa r_{<}\right) K_{|m+1|}\left(\kappa r_{>}\right) \\ & \times \frac{1}{2 \pi}\left[e^{i m\left(\varphi_{b}-\varphi_{a}\right)}+(-)^{m} e^{i m\left(\varphi_{b}-\varphi_{a}\right)}\right] \\ &=-\frac{4 i M}{\hbar} e^{-i\left(\varphi_{b}-\varphi_{a}\right)} \sum_{m=\text { odd }} I_{|m|}\left(\kappa r_{<}\right) K_{|m|}\left(\kappa r_{>}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \end{align*} $$

$$ \begin{align*} Z & =\int d^{2} x\left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) \\ & =\frac{1}{2} e^{\mu_{0} \eta} \sum_{m=-\infty}^{\infty} e^{m \eta} \int_{0}^{\infty} d \xi e^{-\xi \cosh \eta} I_{\left|m+\mu_{0}\right|}(\xi) \end{align*} $$

$$ \xi \equiv M \omega r^{2} / 2 \hbar \sinh \eta $$

$$ Z_{\mathrm{ex}}=\int d^{2} x\left(-\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{1}{2} e^{\mu_{0} \eta} \sum_{m=-\infty}^{\infty}(-)^{m} e^{m \eta} \int_{0}^{\infty} d \xi e^{-\xi \cosh \eta} I_{\left|m+\mu_{0}\right|}(\xi) $$

$$ Z_{1,1_{\mathrm{ex}}}=\frac{1}{2} \sum_{m=-\infty}^{\infty}( \pm)^{m} e^{\eta\left(m+\mu_{0}\right)} \frac{1}{\sinh \eta} e^{-\eta\left|m+\mu_{0}\right|} $$

$$ d^{2} x=l_{\mathrm{e}}^{2}(T) \frac{\sinh \eta}{\eta} d \xi $$

$$ V \equiv \int d^{2} x e^{-\epsilon \xi}=\frac{l_{\mathrm{e}}^{2}(T)}{\epsilon} \frac{\sinh \eta}{\eta} $$

$$ \cosh \eta^{\prime} \equiv \epsilon+\cosh \eta $$

$$ \begin{align*} e^{\eta^{\prime}} & =\cosh \eta^{\prime}+\sqrt{\cosh ^{2} \eta^{\prime}-1} \\ & =e^{\eta}\left(1+\frac{\epsilon}{\sinh \eta}-\frac{1}{2} e^{-\eta} \frac{\epsilon^{2}}{\sinh ^{3} \eta}+\ldots\right), \quad \eta>0 \end{align*} $$

$$ Z_{1,1_{\mathrm{ex}}}=\frac{1}{2} \sum_{m=-\infty}^{\infty}( \pm)^{m} e^{\eta\left(m+\mu_{0}\right)} \frac{1}{\sinh \eta^{\prime}} e^{-\eta^{\prime}\left|m+\mu_{0}\right|} $$

$$ Z_{1,1_{\mathrm{ex}}}=\frac{1}{2} e^{\eta \mu_{0}} \frac{1}{\sinh \eta^{\prime}}\left\{\frac{e^{-\eta^{\prime} \mu_{0}}}{1 \mp a}+\frac{e^{\eta^{\prime} \mu_{0}}}{1 \mp b}-e^{\eta^{\prime}\left|\mu_{0}\right|}\right\} $$

$$ Z=\frac{1}{2}\left(Z_{1}+Z_{1_{\mathrm{ex}}}\right)=\frac{1}{2} e^{\eta \mu_{0}} \frac{1}{\sinh \eta^{\prime}}\left\{\frac{e^{-\eta^{\prime} \mu_{0}}}{1-a^{2}}+\frac{e^{\eta^{\prime} \mu_{0}}}{1-b^{2}}-e^{\eta^{\prime}\left|\mu_{0}\right|}\right\} . $$

$$ \left(\mathbf{x}_{a} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=l_{\mathrm{e}}^{-2}(T) \frac{\eta}{\sinh \eta}, \quad\left(-\mathbf{x}_{a} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=l_{\mathrm{e}}^{-2}(T) \frac{\eta}{\sinh \eta} e^{-2 \cosh \eta \xi} $$

$$ \begin{align*} Z_{1,0} & =\frac{1}{2} \int_{0}^{\infty} d \xi e^{-\epsilon \xi}=\frac{1}{2 \epsilon} \\ Z_{1_{\mathrm{ex}}, 0} & =\frac{1}{2} \int_{0}^{\infty} d \xi e^{-(\epsilon+2 \cosh \eta) \xi}=\frac{1}{2(\epsilon+2 \cosh \eta)} \end{align*} $$

$$ \Delta Z=-\frac{1}{8 \sinh \eta}\left[\operatorname{coth} \eta+2\left(\mu_{0}-1\right)-2 e^{2\left(\mu_{0}+1\right) \eta} \frac{1}{\sinh 2 \eta}+4 e^{2 \eta \mu_{0}}\right] $$

$$ \frac{p V}{N k_{B} T}=1+\sum_{r=2}^{\infty} B_{r} n^{r-1} $$

$$ B_{2}=V\left(\frac{1}{2}-\frac{Z_{2}}{Z_{1}^{2}}\right) $$

$$ Z_{1}=V \frac{\eta}{l_{\mathrm{e}}^{2}(T) \sinh \eta} $$

$$ B_{2}=\frac{V}{Z_{1}}\left(Z_{1} / 2-2 Z_{\mathrm{rel}}\right)=\frac{l_{\mathrm{e}}^{2}(T) \sinh \eta}{2 \eta}\left(Z_{1}-4 Z_{\mathrm{rel}}\right) $$

$$ B_{2}=\frac{l_{\mathrm{e}}^{2}(T)}{4 \eta}\left[\operatorname{coth} \eta+2\left(\mu_{0}-1\right)-2 e^{2\left(\mu_{0}+1\right) \eta} \frac{1}{\sinh (2 \eta)}+4 e^{2 \eta \mu_{0}}\right] . $$

$$ B_{2}=\frac{l_{\mathrm{e}}^{2}(T)}{4}\left[1-2\left(1-\left|\mu_{0}\right|^{2}\right)^{2}\right], \quad \mu_{0} \in(-1,1) . $$

$$ A_{i i}=-1, \quad A_{i, i+1}=1 $$

$$ \begin{array}{ll} A_{i i}=1, A_{i i+1}=-t, A_{i k}=t-1, & \text { type r (right to left) }, \\ A_{i i}=-t, A_{i i+1}=1, A_{i k}=t-1, & \text { type l (left to right) } . \end{array} $$

$$ A_{i j}=\left(\begin{array}{rrr} 1 & -t & t-1 \\ t-1 & 1 & -t \\ -t & t-1 & 1 \end{array}\right) . $$

$$ A_{i j}=\left(\begin{array}{cccc} 1 & -t & 0 & t-1 \\ t-1 & -t & 1 & 0 \\ 0 & t-1 & 1 & -t \\ 1 & 0 & t-1 & -t \end{array}\right) . $$

$$ A(t)=t^{2}-t+1 . $$

$$ A(t)=t^{2}-3 t+1 . $$

$$ X(a)=(-a)^{-3 w} K(a) . $$

$$ K_{n}(a)=-\left(a^{2}+a^{-2}\right)^{n-1} . $$

$$ K(a)=a^{7}-a^{3}-a^{-5} . $$

$$ X(a)=-a^{16}+a^{12}+a^{4} . $$

$$ \frac{1}{t} J_{L_{+}}(t)-t J_{L_{-}}(t)=\left(\sqrt{t}-\frac{1}{\sqrt{t}}\right) J_{L_{0}}(t) $$

$$ J_{2}(t)=-(\sqrt{t}+1 / \sqrt{t}) \text {. } $$

$$ J_{n}(t)=[-(\sqrt{t}+1 / \sqrt{t})]^{n-1} $$

$$ A_{L_{+}}(t)-A_{L_{-}}(t)=(\sqrt{t}-1 / \sqrt{t}) A_{L_{0}}(t) $$

$$ J_{\text {trefoil }}(t) J_{L_{+}}(t)=t^{2} \cdot 1+t(\sqrt{t}-1 / \sqrt{t}) J_{L_{0}}(t) . $$

$$ \frac{1}{t} J_{L_{0}^{0}}(t)=t J_{2}(t)+(\sqrt{t}-1 / \sqrt{t}) J_{1}(t) . $$

$$ J_{L_{0}}(t)=-\sqrt{t}\left(1+t^{2}\right) $$

$$ J_{\text {trefoil }}=t+t^{3}-t^{4} $$

$$ \frac{1}{t} H_{L_{+}}(t, \alpha)-t H_{L_{-}}(t, \alpha)=\alpha H_{L_{0}}(t, \alpha) $$

$$ H_{2}(t, \alpha)=\left(t^{-1}-t\right) \alpha^{-1} $$

$$ \begin{align*} H_{\text {trefoil(rh) }}(t, \alpha) & =-t^{4}+2 t^{2}+t^{2} \alpha^{2} \\ H_{\text {trefoil(lh) }}(t, \alpha) & =-t^{-4}+2 t^{-2}+t^{-2} \alpha^{2} \\ H_{\text {Hopf(rh) }}(t, \alpha) & =\left(t-t^{3}\right) \alpha^{-1}+t \alpha \\ H_{\text {knot } 4_{1}}(t, \alpha) & =t^{-2}-1+t^{2}-\alpha^{2} \end{align*} $$

$$ A_{L}(t)=H_{L}\left(1, t^{1 / 2}-t^{-1 / 2}\right) $$

$$ \begin{align*} H_{\text {granny }}(t, \alpha) & =\left(2 t^{2}-t^{4}+t^{2} \alpha^{2}\right)^{2} \\ H_{\text {square }}(t, \alpha) & =\left(2 t^{2}-t^{4}+t^{2} \alpha^{2}\right)\left(2 t^{-2}-t^{-4}+t^{2} \alpha^{2}\right) \end{align*} $$

$$ f_{N}=C \mu^{N} N^{\alpha} $$

$$ C \approx 1.2325, \quad \mu \approx 0.9949, \quad \alpha \approx 0 . $$

$$ f_{N}(R)=e^{-A\left(N^{\beta} l / R\right)^{\gamma}}, $$

$$ G\left(C, C^{\prime}\right)=\frac{1}{4 \pi} \oint_{C} \oint_{C^{\prime}}\left[d \mathbf{x} \times d \mathbf{x}^{\prime}\right] \cdot \frac{\mathbf{x}-\mathbf{x}^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}} $$

$$ \oint_{C} d \mathbf{x} \oint_{C^{\prime}} \frac{d \mathbf{x}^{\prime} \times\left(\mathbf{x}-\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}}=-\int d^{3} x \boldsymbol{\delta}(\mathbf{x} ; C) \cdot\left[\int d^{3} x^{\prime} \boldsymbol{\delta}\left(\mathbf{x}^{\prime} ; C^{\prime}\right) \times \frac{\mathbf{R}^{\prime}}{R^{\prime 3}}\right] $$

$$ G\left(C, C^{\prime}\right)=-\frac{1}{4 \pi} \int d^{3} x \boldsymbol{\delta}(\mathbf{x} ; C) \cdot \nabla \Omega\left(\mathbf{x} ; C^{\prime}\right) $$

$$ \int d^{3} x \boldsymbol{\delta}(\mathbf{x} ; C) \cdot \nabla \Omega\left(\mathbf{x} ; S^{\prime}\right)=-\int d^{3} x \nabla \cdot \boldsymbol{\delta}(\mathbf{x} ; C) \Omega\left(\mathbf{x} ; S^{\prime}\right)=0 $$

$$ G\left(C, C^{\prime}\right)=-\int d^{3} x \boldsymbol{\delta}(\mathbf{x} ; C) \cdot \boldsymbol{\delta}\left(\mathbf{x} ; S^{\prime}\right) $$

$$ G\left(C, C^{\prime}\right)=-\oint_{C} d x_{i} \delta_{i}\left(\mathbf{x} ; S^{\prime}\right) $$

$$ G\left(C, C^{\prime}\right)=-\frac{1}{4 \pi} \oint_{C^{\prime}} d \Omega^{\prime}\left(\mathbf{x}^{\prime} ; C\right)=-\frac{1}{4 \pi} \oint_{C} d \Omega\left(\mathbf{x}, C^{\prime}\right) $$

$$ B_{i}=\partial_{i} \Omega $$

$$ G\left(C, C^{\prime}\right)=-\oint_{C} d x_{i} B_{i}=-\oint_{C^{\prime}} d x_{i}^{\prime} B_{i}^{\prime} $$

$$ \tau=L_{\mathrm{k}}-N_{w}, $$

$$ \sigma \equiv \frac{\tau}{N_{w}} $$

$$ G\left(C, C^{\prime}\right)=\frac{1}{4 \pi} \oint_{C} \oint_{C^{\prime}}\left[d \mathbf{x} \times d \mathbf{x}^{\prime}\right] \cdot \frac{\mathbf{x}-\mathbf{x}^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}} $$

$$ G\left(C, C^{\prime}\right)=\frac{1}{4 \pi} \oint_{\bar{C}} d \tau \oint_{\bar{C}} d \tau^{\prime}\left[\dot{\mathbf{x}}(\tau) \times\left(\dot{\mathbf{x}}\left(\tau^{\prime}\right)+\epsilon \dot{\mathbf{n}}\left(\tau^{\prime}\right)\right)\right] \cdot \frac{\mathbf{x}(\tau)-\mathbf{x}\left(\tau^{\prime}\right)-\epsilon \mathbf{n}\left(\tau^{\prime}\right)}{\left|\mathbf{x}(\tau)-\mathbf{x}\left(\tau^{\prime}\right)-\epsilon \mathbf{n}\left(\tau^{\prime}\right)\right|^{3}} $$

$$ L_{\mathrm{k}}=G(\bar{C}, \bar{C})+T_{\mathrm{w}} $$

$$ T_{\mathrm{w}} \equiv \frac{1}{2 \pi} \oint_{\bar{C}} d \tau \dot{\mathbf{x}}(\tau) \cdot[\mathbf{n}(\tau) \times \dot{\mathbf{n}}(\tau)] /|\dot{\mathbf{x}}(\tau)| $$

$$ W_{\mathrm{r}} \equiv G(\bar{C}, \bar{C})=\frac{1}{4 \pi} \oint_{\bar{C}} d \tau \oint_{\bar{C}} d \tau^{\prime}\left[\dot{\mathbf{x}}(\tau) \times \dot{\mathbf{x}}\left(\tau^{\prime}\right)\right] \cdot \frac{\mathbf{x}(\tau)-\mathbf{x}\left(\tau^{\prime}\right)}{\left|\mathbf{x}(\tau)-\mathbf{x}\left(\tau^{\prime}\right)\right|^{3}} $$

$$ L_{\mathrm{k}}=W_{\mathrm{r}}+T_{\mathrm{w}} $$

$$ W_{\mathrm{r}}=-\frac{1}{4 \pi} \oint_{\bar{C}} d \Omega(\mathbf{x}) $$

$$ \tau \in\left(\tau^{\prime}-\delta, \tau^{\prime}+\delta\right) $$

$$ \begin{align*} & \mathbf{x}(\tau) \approx \mathbf{x}\left(\tau^{\prime}\right)+\dot{\mathbf{x}}\left(\tau^{\prime}\right)\left(\tau-\tau^{\prime}\right) \\ & \dot{\mathbf{x}}(\tau) \approx \dot{\mathbf{x}}\left(\tau^{\prime}\right) \end{align*} $$

$$ \begin{align*} T_{\mathrm{w}} & =-\frac{1}{4 \pi} \oint_{C^{\prime}} d \tau^{\prime}\left[\mathbf{n}\left(\tau^{\prime}\right) \times \dot{\mathbf{x}}\left(\tau^{\prime}\right)\right] \cdot \dot{\mathbf{n}}\left(\tau^{\prime}\right) \epsilon^{2} \int_{\tau^{\prime}-\delta}^{\tau^{\prime}+\delta} d \tau \frac{1}{\sqrt{\left|\dot{\mathbf{x}}\left(\tau^{\prime}\right)\right|^{2}\left(\tau-\tau^{\prime}\right)^{2}+\epsilon^{2}}} \\ & =\frac{1}{2 \pi} \oint_{C^{\prime}} d \tau^{\prime}\left[\dot{\mathbf{x}}\left(\tau^{\prime}\right) \times \mathbf{n}\left(\tau^{\prime}\right)\right] \cdot \dot{\mathbf{n}}\left(\tau^{\prime}\right) /\left|\dot{\mathbf{x}}\left(\tau^{\prime}\right)\right| \end{align*} $$

$$ E=-\frac{I I^{\prime}}{c^{2}} \oint_{C} \oint_{C^{\prime}} d \mathbf{x} \cdot d \mathbf{x}^{\prime} \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{curr}}=-i \oint_{C} d \mathbf{x} \mathbf{A}(\mathbf{x})-i \oint_{C^{\prime}} d \mathbf{x} \mathbf{A}(\mathbf{x}) $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{CS}}=\frac{i}{2} \int d^{3} x \mathbf{A} \cdot(\boldsymbol{\nabla} \times \mathbf{A}) $$

$$ \mathbf{A}(\mathbf{x}) \rightarrow \mathbf{A}(\mathbf{x})+\boldsymbol{\nabla} \Lambda(\mathbf{x}) $$

$$ \nabla \cdot \oint d \mathbf{x}=0 $$

$$ \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{x})=\left(\oint_{C}+\oint_{C^{\prime}}\right) d \mathbf{x} $$

$$ A_{i}(\mathbf{x})=\left(\oint_{C}+\oint_{C^{\prime}}\right) G_{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) d x_{j}^{\prime} $$

$$ \epsilon_{i j k} \nabla_{j} G_{k l}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\delta_{i j}^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{T} $$

$$ \delta_{i j}^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{T}=\left(\delta_{i j}-\frac{\nabla_{i} \nabla_{j}}{\nabla^{2}}\right) \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \boldsymbol{\nabla} \cdot \mathbf{A}(\mathbf{x})=0 $$

$$ i \epsilon_{i j k} p_{j} G_{k l}(\mathbf{p})=\delta_{i l}-\frac{p_{j} p_{l}}{\mathbf{p}^{2}} $$

$$ G_{i j}(\mathbf{p})=i \epsilon_{i k j} p_{k} \frac{1}{\mathbf{p}^{2}} $$

$$ \mathcal{A}_{\mathrm{GF}}=\frac{1}{2 \alpha}(\boldsymbol{\nabla} \cdot \mathbf{A})^{2} $$

$$ \left(\epsilon_{i j k} \nabla_{j}+\frac{i}{\alpha} \nabla_{i} \nabla_{k}\right) G_{k l}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\delta_{i k} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right), $$

$$ \left(i \epsilon_{i j k} p_{j}-\frac{i}{\alpha} p_{i} p_{k}\right) G_{k l}(\mathbf{p})=\delta_{i k} $$

$$ G_{i k}(\mathbf{p})=\left(i \epsilon_{i j k} p_{j}+i \alpha \frac{p_{i} p_{k}}{\mathbf{p}^{2}}\right) \frac{1}{\mathbf{p}^{2}} $$

$$ G_{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\int \frac{d^{3} p}{(2 \pi)^{3}} e^{i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} \frac{i \epsilon_{i k j} p_{k}}{\mathbf{p}^{2}}=\frac{1}{4 \pi} \epsilon_{i k j} \nabla_{k} \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}=\frac{1}{4 \pi} \epsilon_{i j k} \frac{\left(x-x^{\prime}\right)_{k}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}} . $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{int}}=-i \oint_{C} \oint_{C^{\prime}} d x_{i} d x_{j}^{\prime} G_{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{int}}=-\frac{i}{2}\left(\oint_{C} \oint_{C}+\oint_{C^{\prime}} \oint_{C^{\prime}}\right) d x_{i} d x_{j}^{\prime} G_{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}}=\mathcal{A}_{\mathrm{e}, \mathrm{CS} 12}+\mathcal{A}_{\mathrm{e}, \mathrm{curr}} $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{CS} 12}=i \int d^{3} x \mathbf{A}_{1} \cdot\left(\boldsymbol{\nabla} \times \mathbf{A}_{2}\right) $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{curr}}=-i \oint_{C_{1}} d \mathbf{x} \mathbf{A}_{1}(\mathbf{x})-i \oint_{C_{2}} d \mathbf{x} \mathbf{A}_{2}(\mathbf{x}) $$

$$ D_{a b}^{\mu \nu}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \equiv\left\langle A_{a}^{\mu}(\mathbf{x}) A_{b}^{\nu}\left(\mathbf{x}^{\prime}\right)\right\rangle, \quad a, b=1,2 $$

$$ \begin{align*} D_{11}^{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) & =0, \quad D_{22}^{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=0, \\ D_{12}^{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) & =D_{21}^{i j}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\int \frac{d^{3} p}{(2 \pi)^{3}} e^{i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} \frac{i \epsilon_{i k j} k^{k}}{\mathbf{p}^{2}}=\frac{1}{4 \pi} \epsilon_{i k j} \nabla_{k} \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} \\ & =\frac{1}{4 \pi} \epsilon_{i j \kappa} \frac{\left(x-x^{\prime}\right)^{k}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{3}} . \end{align*} $$

$$ \mathcal{A}_{\mathrm{GF}}=\frac{1}{2 \alpha}\left[\left(\boldsymbol{\nabla} \cdot \mathbf{A}_{1}\right)^{2}+\left(\boldsymbol{\nabla} \cdot \mathbf{A}_{2}\right)^{2}\right], $$

$$ Z=\int_{C_{1}} \mathcal{D} \mathbf{x}_{1} \int_{C_{2}} \mathcal{D} \mathbf{x}_{2} \int \mathcal{D} \mathbf{A}_{1} \mathcal{D} \mathbf{A}_{2} e^{-\mathcal{A}_{\mathrm{e}}-\mathcal{A}_{\mathrm{GF}}} $$

$$ Z=\mathrm{const} \times \int_{C_{1}} \mathcal{D} \mathbf{x}_{1} \int_{C_{2}} \mathcal{D} \mathbf{x}_{2} e^{i G\left(C_{1}, C_{2}\right)}, $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{curr}, \lambda}=-i \oint_{C_{1}} d \mathbf{x} \mathbf{A}_{1}(\mathbf{x})-i \lambda \oint_{C_{2}} d \mathbf{x} \mathbf{A}_{2}(\mathbf{x}) $$

$$ \begin{align*} Z(\lambda) & =\int_{C_{1}} \mathcal{D} \mathbf{x}_{1} \int_{C_{2}} \mathcal{D} \mathbf{x}_{2} \int \mathcal{D} \mathbf{A}_{1} \mathcal{D} \mathbf{A}_{2} e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS} 12}-\mathcal{A}_{\mathrm{e}, \mathrm{curr}, \lambda}-\mathcal{A}_{\mathrm{GF}}} \\ & =\text { const } \times \int_{C_{1}} \mathcal{D} \mathbf{x}_{1} \int_{C_{2}} \mathcal{D} \mathbf{x}_{2} e^{i m \lambda} \end{align*} $$

$$ \left\langle m^{2}\right\rangle=\frac{\int d^{3} x_{1} d^{3} x_{2} \int_{-\infty}^{+\infty} d m m^{2} P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; m\right)}{\int d^{3} x_{1} d^{3} x_{2} \int_{-\infty}^{+\infty} d m P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; m\right)} $$

$$ Z \equiv \int d^{3} x_{1} d^{3} x_{2} \int_{-\infty}^{+\infty} d m P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; m\right) $$

$$ P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; m\right)=P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}-\mathbf{x}_{2} ; m\right) $$

$$ \int d^{3} x_{1} d^{3} x_{2} P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; m\right)=V \int d^{3} x P_{L_{1}, L_{2}}(\mathbf{x} ; m) $$

$$ P_{z}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right)=\lim _{n_{1}, n_{2} \rightarrow 0} \int \mathcal{D}(\text { fields }) \psi_{1}^{\alpha_{i}}\left(\mathbf{x}_{1}\right) \psi_{1}^{* \alpha_{1}}\left(\mathbf{x}_{1}^{\prime}\right) \psi_{2}^{\alpha_{2}}\left(\mathbf{x}_{2}\right) \psi_{2}^{* \alpha_{2}}\left(\mathbf{x}_{2}^{\prime}\right) e^{-\mathcal{A}} $$

$$ \mathcal{A}=\mathcal{A}_{\mathrm{CS} 12}+\mathcal{A}_{\mathrm{e}, \mathrm{curr}}+\mathcal{A}_{\mathrm{pol}}+\mathcal{A}_{\mathrm{GF}} $$

$$ \mathcal{A}_{\mathrm{pol}}=\sum_{i=1}^{2} \int d^{3} \mathbf{x}\left[\left|\overline{\mathbf{D}}^{i} \psi_{i}\right|^{2}+m_{i}^{2}\left|\Psi_{i}\right|^{2}\right] $$

$$ \mathbf{D}^{i}=\boldsymbol{\nabla}+i \gamma_{i} \mathbf{A}^{i} $$

$$ \gamma_{1}=1, \quad \gamma_{2}=\lambda $$

$$ m_{i}^{2}=2 M z_{i} $$

$$ \Psi_{i}=\left(\psi_{i}^{1}, \ldots, \psi_{i}^{n_{i}}\right) $$

$$ \left|\mathbf{D}^{i} \bar{\Psi}_{i}\right|^{2}=\sum_{\alpha_{i}=1}^{n_{i}}\left|\mathbf{D}^{i} \psi_{i}^{\alpha_{i}}\right|^{2}, \quad\left|\Psi_{i}\right|^{2}=\sum_{\alpha_{i}=1}^{n_{i}}\left|\psi_{i}^{\alpha_{i}}\right|^{2} $$

$$ \mathcal{D}(\text { fields })=\int \mathcal{D} A_{1}^{i} \mathcal{D} A_{2}^{j} \mathcal{D} \Psi_{1} \mathcal{D} \Psi_{1}^{*} \mathcal{D} \Psi_{2} \mathcal{D} \Psi_{2}^{*} $$

$$ P_{L_{1}, L_{2}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; m\right)=\lim _{\substack{\mathbf{x}_{1}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \int_{-\infty}^{\infty} d k e^{-i m \lambda} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right) $$

$$ Z=\int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{\mathbf{x}_{\prime}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}} \int_{c-i \infty}^{c+\infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \int_{-\infty}^{\infty} d m \int_{-\infty}^{+\infty} d \lambda e^{-i m \lambda} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right) $$

$$ Z=\int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{\mathbf{x}_{1}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1} M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; 0\right) $$

$$ \mathcal{A}=\mathcal{A}_{0}+\lambda \mathcal{A}_{1}+\lambda^{2} \mathcal{A}_{2} $$

$$ \mathcal{A}_{0} \equiv \mathcal{A}_{\mathrm{CS} 12}+\mathcal{A}_{\mathrm{GF}}+\int d^{3} x\left[\left|\mathbf{D}_{1} \Psi_{1}\right|^{2}+\left|\nabla \Psi_{2}\right|^{2}+\sum_{i=1}^{2}\left|\Psi_{i}\right|^{2}\right], $$

$$ \mathcal{A}_{1} \equiv \int d^{3} x \mathbf{j}_{2}(\mathbf{x}) \cdot \mathbf{A}_{2}(\mathbf{x}) $$

$$ \mathbf{j}_{2}(\mathbf{x})=i \Psi_{2}^{*}(\mathbf{x}) \nabla \Psi_{2}(\mathbf{x}) $$

$$ \mathcal{A}_{2} \equiv \frac{1}{4} \int d^{3} \mathbf{x} \mathbf{A}_{2}^{2}\left|\Psi_{2}(\mathbf{x})\right|^{2} $$

$$ P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; 0\right)=\int \mathcal{D}(\text { fields }) e^{-\mathcal{A}_{0}} \psi_{1}^{\alpha_{1}}\left(\mathbf{x}_{1}\right) \psi_{1}^{* \alpha_{1}}\left(\mathbf{x}_{1}^{\prime}\right) \psi_{2}^{\alpha_{2}}\left(\mathbf{x}_{2}\right) \psi_{2}^{\alpha_{2}}\left(\mathbf{x}^{\prime}\right) $$

$$ \boldsymbol{\nabla} \times \mathbf{A}_{1}=0 $$

$$ P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; 0\right)=G_{0}\left(\mathbf{x}_{1}-\mathbf{x}_{1}^{\prime} ; z_{1}\right) G_{0}\left(\mathbf{x}_{2}-\mathbf{x}_{2}^{\prime} ; z_{2}\right), $$

$$ G_{0}\left(\mathbf{x}_{i}-\mathbf{x}_{i}^{\prime} ; z_{i}\right)=\left\langle\psi_{i}^{\alpha_{i}}\left(\mathbf{x}_{i}\right) \psi_{i}^{* \alpha_{i}}\left(\mathbf{x}_{i}^{\prime}\right)\right\rangle . $$

$$ \left\langle\tilde{\psi}_{i}^{\alpha_{i}}\left(\mathbf{k}_{i}\right) \tilde{\psi}_{i}^{* \alpha_{i}}\left(\mathbf{k}_{i}^{\prime}\right)\right\rangle=\delta^{(3)}\left(\mathbf{k}_{i}-\mathbf{k}_{i}^{\prime}\right) \frac{1}{\mathbf{k}_{i}^{2}+m_{i}^{2}}, $$

$$ G_{0}\left(\mathbf{x}_{i}-\mathbf{x}_{i}^{\prime} ; z_{i}\right)=\int \frac{d^{3} k}{(2 \pi)^{3}} e^{i \mathbf{k} \cdot \mathbf{x}} \frac{1}{\mathbf{k}_{i}^{2}+m_{i}^{2}}, $$

$$ \begin{align*} G_{0}\left(\mathbf{x}_{i}-\mathbf{x}_{i}^{\prime} ; L_{i}\right) & =\int_{c-i \infty}^{c+i \infty} \frac{M d z_{i}}{2 \pi i} e^{z_{i} L_{i}} G_{0}\left(\mathbf{x}_{i}-\mathbf{x}_{i}^{\prime} ; z_{i}\right) \\ & =\frac{1}{2}\left(\frac{M}{4 \pi L_{i}}\right)^{3 / 2} e^{-M\left(\mathbf{x}_{i}-\mathbf{x}_{i}^{\prime}\right) / 2 L_{i}} \end{align*} $$

$$ Z=2 \pi \int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{\mathbf{x}_{1}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}} G_{0}\left(\mathbf{x}_{1}-\mathbf{x}_{1}^{\prime} ; L_{1}\right) G_{0}\left(\mathbf{x}_{2}-\mathbf{x}_{2}^{\prime} ; L_{2}\right) $$

$$ Z=\frac{2 \pi M^{3} V^{2}}{(8 \pi)^{3}}\left(L_{1} L_{2}\right)^{-3 / 2} $$

$$ \int_{c-i \infty}^{c+i \infty} \frac{d z}{2 \pi} e^{z L} \lim _{\mathbf{x}^{\prime} \rightarrow \mathbf{x}} G_{0}\left(\mathbf{x}-\mathbf{x}^{\prime} ; z\right)=\int_{c-i \infty}^{c+i \infty} \frac{d z}{2 \pi i} e^{z L} G_{0}(\mathbf{0}, z) $$

$$ \left.G_{0}(\mathbf{0} ; z)=\left.\langle | \psi(\mathbf{x})\right|^{2}\right\rangle . $$

$$ \left.\left.\langle | \psi(\mathbf{x})\right|^{2}\right\rangle=\int \frac{d^{3} k}{k^{2}+m^{2}} \rightarrow \infty $$

$$ N \equiv \int d^{3} x_{1} d^{3} x_{2} \int_{-\infty}^{\infty} d m m^{2} P_{L_{1}, L_{2}}\left(\mathbf{x}_{1}, \mathbf{x}_{2} ; m\right) $$

$$ \begin{align*} N & =\int d^{3} x_{1} d^{3} r_{2} \int_{-\infty}^{\infty} d m m^{2} \lim _{\substack{\mathbf{x}_{1}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}} \int_{c-i \infty}^{c_{\tau} i \infty} \frac{M d z_{i}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} \\ & \times e^{z_{1} L_{1}+z_{2} L_{2}} \int_{-\infty}^{\infty} d \lambda e^{-i m \lambda} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right) \end{align*} $$

$$ \int_{-\infty}^{\infty} d m m^{2} e^{-i m \lambda} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right)=-\int_{-\infty}^{\infty} d m\left(\frac{\partial^{2}}{\partial \lambda^{2}} e^{-i m \lambda}\right) P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right) $$

$$ \begin{align*} N & =\int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{\mathbf{x}_{1}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}}(-1) \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \\ & \times \int_{-\infty}^{\infty} d \lambda \delta(\lambda)\left[\frac{\partial^{2}}{\partial \lambda^{2}} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; \lambda\right)\right] \end{align*} $$

$$ N=\int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{\mathbf{x}_{1}^{\prime} \rightarrow \mathbf{x}_{1} \\ \mathbf{x}_{2}^{\prime} \rightarrow \mathbf{x}_{2}}}(-1) \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}}\left[\frac{\partial^{2}}{\partial \lambda^{2}} P_{\vec{z}}\left(\overrightarrow{\mathbf{x}}_{1}, \overrightarrow{\mathbf{x}}_{2} ; 0\right)\right] $$

$$ \begin{align*} N= & \int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{n_{1} \rightarrow 0 \\ n_{2} \rightarrow 0}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \\ & \times \int \mathcal{D}(\text { fields }) \exp \left(-\mathcal{A}_{0}\right)\left|\psi_{1}^{\alpha_{1}}\left(\mathbf{x}_{1}\right)\right|^{2}\left|\psi_{2}^{\alpha_{2}}\left(\mathbf{x}_{2}\right)\right|^{2} \\ & \times\left[\left(\int d^{3} x \mathbf{A}_{2} \cdot \Psi_{2}^{*} \nabla \Psi_{2}\right)^{2}+\frac{1}{2} \int d^{3} x \mathbf{A}_{2}^{2}\left|\Psi_{2}\right|^{2}\right] \end{align*} $$

$$ \mathcal{A}_{0}^{0} \equiv \mathcal{A}_{\mathrm{CS}}+\int d^{3} x\left[\left|\mathbf{D}^{1} \Psi_{1}\right|^{2}+\left|\nabla \Psi_{2}\right|^{2}+\sum_{i=1}^{2} 2\left|\Psi_{i}\right|^{2}\right] $$

$$ \mathcal{A}_{1}^{0} \equiv \int d^{3} x \mathbf{j}_{1}(\mathbf{x}) \cdot \mathbf{A}_{1}(\mathbf{x}) $$

$$ \mathbf{j}_{1}(\mathbf{x}) \equiv i \Psi_{1}^{*}(\mathbf{x}) \nabla \Psi_{1}(\mathbf{x}) $$

$$ \mathcal{A}_{0}^{2} \equiv \frac{1}{4} \int d^{3} \mathbf{x} \mathbf{A}_{1}^{2}\left|\Psi_{1}(\mathbf{x})\right|^{2} $$

$$ e^{\mathcal{A}_{0}}=e^{\mathcal{A}_{0}^{0}+\mathcal{A}_{0}^{1}+\mathcal{A}_{0}^{2}}=e^{\mathcal{A}_{0}}\left[1-\mathcal{A}_{0}^{1}+\frac{\left(\mathcal{A}_{0}^{1}\right)^{2}}{2}-\mathcal{A}_{0}^{2}+\ldots\right] $$

$$ \begin{align*} N & =\kappa^{2} \int d^{3} x_{1} d^{3} x_{2} \lim _{\substack{n_{1} \rightarrow 0 \\ n_{2} \rightarrow 0}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \\ & \times \int \mathcal{D}(\text { fields }) \exp \left(-\mathcal{A}_{0}^{0}\right)\left|\psi_{1}^{\alpha_{1}}\left(\mathbf{x}_{1}\right)\right|^{2}\left|\psi_{2}^{\alpha_{2}}\left(\mathbf{x}_{2}\right)\right|^{2} \\ & \times\left[\left(\int d^{3} x \mathbf{A}_{1} \cdot \Psi_{1}^{*} \boldsymbol{\nabla} \Psi_{1}\right)^{2}+\frac{1}{2} \int d^{3} x \mathbf{A}_{1}^{2}\left|\Psi_{1}\right|^{2}\right] \\ & \times\left[\left(\int d^{3} x \mathbf{A}_{2} \cdot \Psi_{2}^{*} \boldsymbol{\nabla} \Psi_{2}\right)^{2}+\frac{1}{2} \int d^{3} x \mathbf{A}_{2}^{2}\left|\Psi_{2}\right|^{2}\right] \end{align*} $$

$$ \begin{align*} N_{1} & =\frac{\kappa^{2}}{4} \lim _{\substack{n_{1} \rightarrow 0 \\ n_{2} \rightarrow 0}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \int d^{3} x_{1} d^{3} x_{2} \int d^{3} x_{1}^{\prime} d^{3} x_{2}^{\prime} \\ & \left.\times\left.\langle | \psi_{1}^{\alpha_{1}}\left(\mathbf{x}_{1}\right)\right|^{2}\left|\psi_{2}^{\alpha_{2}}\left(\mathbf{x}_{2}\right)\right|^{2}\left(\left|\Psi_{1}\right|^{2} \mathbf{A}_{1}^{2}\right)_{\mathbf{x}_{1}^{\prime}}\left(\left|\Psi_{2}\right|^{2} \mathbf{A}_{2}^{2}\right)_{\mathbf{x}_{2}^{\prime}}\right\rangle \end{align*} $$

$$ \begin{align*} N_{1} & =\frac{V}{4 \pi} \frac{M^{4}}{(8 \pi)^{6}}\left(L_{1} L_{2}\right)^{-\frac{1}{2}} \int_{0}^{1} d s[(1-s) s]^{-\frac{3}{2}} \int d^{3} x e^{-M \mathbf{x}^{2} / 2 s(1-s)} \\ & \times \int_{0}^{1} d t[(1-t) t]^{-\frac{3}{2}} \int d^{3} y e^{-M \mathbf{y}^{2} / 2 t(1-t)} \int d^{3} x_{1}^{\prime \prime} \frac{1}{\left|\mathbf{x}_{1}^{\prime \prime}\right|^{4}} \end{align*} $$

$$ \int d^{3} x_{1}^{\prime \prime} \frac{1}{\left|\mathbf{x}_{1}^{\prime \prime}\right|^{4}} \sim 4 \pi^{2} \int_{\xi}^{\infty} \frac{d l}{l^{2}} $$

$$ N_{1}=V \pi^{1 / 2} \frac{M}{(4 \pi)^{3}}\left(L_{1} L_{2}\right)^{-1 / 2} \xi^{-1} $$

$$ \begin{align*} N_{2} & =\kappa^{2} \lim _{\substack{n_{1} \rightarrow 0 \\ n_{2} \rightarrow 0}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \int d^{3} x_{1} d^{3} x_{2} \int d^{3} x_{1}^{\prime} d^{3} x_{1}^{\prime \prime} d^{3} x_{2}^{\prime} \\ & \left.\times\left.\langle | \psi_{1}^{\alpha_{1}}\left(\mathbf{x}_{1}\right)\right|^{2}\left|\psi_{2}^{\alpha_{2}}\left(\mathbf{x}_{2}\right)\right|^{2}\left(\mathbf{A}_{1} \cdot \Psi_{1}^{*} \boldsymbol{\nabla} \Psi_{1}\right)_{\mathbf{x}_{1}^{\prime}}\left(\mathbf{A}_{1} \cdot \Psi_{1}^{*} \boldsymbol{\nabla} \Psi_{1}\right)_{\mathbf{x}_{1}^{\prime \prime}}\left(\mathbf{A}_{2}^{2}\left|\Psi_{2}\right|^{2}\right)_{\mathbf{x}_{2}^{\prime}}\right\rangle \end{align*} $$

$$ N_{2}=-4 \sqrt{2} V L_{2}^{-1 / 2} L_{1}^{-1} \frac{M^{3}}{\pi^{6}} \int_{0}^{1} d t \int_{0}^{t} d t^{\prime} C\left(t, t^{\prime}\right) $$

$$ \begin{align*} C\left(t, t^{\prime}\right) & =\left[(1-t) t^{\prime}\left(t-t^{\prime}\right)\right]^{-3 / 2} \int d^{3} x d^{3} y d^{3} z e^{-M(\mathbf{y}-\mathbf{x})^{2} / 2(1-t)} \\ & \times\left(\boldsymbol{\nabla}_{\mathbf{y}}^{j} e^{-M \mathbf{y}^{2} / 2 t^{\prime}}\right)\left(\boldsymbol{\nabla}_{\mathbf{x}}^{i} e^{-M \mathbf{x}^{2} / 2\left(t-t^{\prime}\right)}\right) \frac{\left[\delta_{i j} \mathbf{z} \cdot(\mathbf{z}+\mathbf{x})-(z+x)_{i} z_{j}\right]}{|\mathbf{z}|^{3}|\mathbf{z}+\mathbf{x}|^{3}} \end{align*} $$

$$ N_{2}=-\frac{V L_{2}^{-1 / 2} L_{1}^{-1}}{(2 \pi)^{6}} M^{3 / 2} 4 K $$

$$ K \equiv \frac{1}{6} B\left(\frac{3}{2}, \frac{1}{2}\right)+\frac{1}{2} B\left(\frac{5}{2}, \frac{1}{2}\right)-B\left(\frac{7}{2}, \frac{1}{2}\right)+\frac{1}{3} B\left(\frac{9}{2}, \frac{1}{2}\right)=\frac{19 \pi}{384} \approx 0.154, $$

$$ N_{3}=\left.N_{2}\right|_{L_{1} \leftrightarrow L_{2}} $$

$$ \begin{gather*} N_{4}=-4 \kappa^{2} \frac{1}{2} \lim _{\substack{n_{1} \rightarrow 0 \\ n_{2} \rightarrow 0}} \int_{c-i \infty}^{c+i \infty} \frac{M d z_{1}}{2 \pi i} \frac{M d z_{2}}{2 \pi i} e^{z_{1} L_{1}+z_{2} L_{2}} \int d^{3} x_{1} d^{3} x_{2} \int d^{3} x_{1}^{\prime} d^{3} x_{2}^{\prime} d^{3} x_{1}^{\prime \prime} d^{3} x_{2}^{\prime \prime} \\ \times\left.\langle | \psi_{1}^{\alpha_{1}}\left(\mathbf{x}_{1}\right)\right|^{2}\left|\psi_{2}\left(\mathbf{x}_{2}^{\alpha_{2}}\right)\right|^{2}\left(\mathbf{A}_{1} \cdot \Psi_{1}^{*} \nabla \Psi_{1}\right)_{\mathbf{x}_{1}^{\prime}}\left(\mathbf{A}_{1} \cdot \Psi_{1}^{*} \nabla \Psi_{1}\right)_{\mathbf{x}_{1}^{\prime \prime}} \\ \left.\times\left(\mathbf{A}_{2} \cdot \Psi_{2}^{*} \nabla \Psi_{2}\right)_{\mathbf{x}_{2}^{\prime}}\left(\mathbf{A}_{2} \cdot \Psi_{2}^{*} \nabla \Psi_{2}\right)_{\mathbf{x}_{2}^{\prime \prime}}\right\rangle \end{gather*} $$

$$ N_{4}=-\frac{1}{16} \frac{M^{5} V}{(2 \pi)^{11}}\left(L_{1} L_{2}\right)^{-1 / 2} \int_{0}^{1} d s \int_{0}^{s} d s^{\prime} \int_{0}^{1} d t \int_{0}^{t} d t^{\prime} C\left(s, s^{\prime}, t, t^{\prime}\right) $$

$$ \begin{align*} & C\left(s, s^{\prime} ; t, t^{\prime}\right)=\left[(1-s) s^{\prime}\left(s-s^{\prime}\right)\right]^{-3 / 2}\left[(1-t) t^{\prime}\left(t-t^{\prime}\right)\right]^{-3 / 2} \\ & \quad \times \int \frac{d^{3} p}{(2 \pi)^{3}}\left[\epsilon_{i k \alpha} \frac{p^{\alpha}}{\mathbf{p}^{2}} \epsilon_{j l \beta} \frac{p^{\beta}}{\mathbf{p}^{2}}+\epsilon_{i l \alpha} \frac{p^{\alpha}}{\mathbf{p}^{2}} \epsilon_{j k \beta} \frac{p^{\beta}}{\mathbf{p}^{2}}\right] \\ & \quad \times\left[\int d^{3} x^{\prime} d^{3} y^{\prime} e^{-i \sqrt{L_{1}} \mathbf{p}\left(\mathbf{x}^{\prime}-\mathbf{y}^{\prime}\right)} e^{-M \mathbf{x}^{\prime 2} / 2(1-s)}\left(\boldsymbol{\nabla}_{\mathbf{y}^{\prime}}^{j} e^{-M \mathbf{y}^{\prime 2} / 2 t^{\prime}}\right)\left(\boldsymbol{\nabla}_{\mathbf{x}^{\prime}}^{i} e^{-M(\mathbf{x}-\mathbf{y})^{2} / 2\left(s-s^{\prime}\right)}\right)\right] \\ & \quad \times\left[\int d^{3} u^{\prime} d^{3} v^{\prime} e^{-i \sqrt{L_{2}} \mathbf{p}\left(\mathbf{u}^{\prime}-\mathbf{v}^{\prime}\right)} e^{-M \mathbf{v}^{\prime 2} / 2(1-t)}\left(\boldsymbol{\nabla}_{\mathbf{u}^{\prime}}^{l} e^{-M \mathbf{u}^{\prime 2} / 2 t^{\prime}}\right)\left(\boldsymbol{\nabla}_{\mathbf{v}^{\prime}}^{k} e^{-M\left(\mathbf{u}^{\prime}-\mathbf{v}^{\prime}\right)^{2} / 2\left(t-t^{\prime}\right)}\right)\right] \end{align*} $$

$$ N_{4} \propto L_{1}^{-1} $$

$$ N_{4} \propto L_{2}^{-1} $$

$$ N_{4} \propto L_{1}^{-3 / 2} $$

$$ \begin{align*} N_{4} & \approx-\frac{128 V}{\pi^{5}} \frac{M}{\pi^{3 / 2}}\left(L_{1} L_{2}\right)^{-1 / 2} \int_{0}^{1} d s \int_{0}^{1} d t(1-s)(1-t)(s t)^{1 / 2} \\ & \times\left[L_{1} t(1-s)+L_{2}(1-t) s\right]^{-1 / 2} \end{align*} $$

$$ \left\langle m^{2}\right\rangle=\frac{N_{1}+N_{2}+N_{3}+N_{4}}{Z}, $$

$$ l=\frac{M}{V} $$

$$ M=\sum_{i=1}^{N_{p}} m_{a} \frac{L_{k}}{a} $$

$$ L_{2} \approx \frac{a V l}{m_{a}} . $$

$$ \left\langle m^{2}\right\rangle \approx \frac{N_{1}+N_{2}}{Z}, $$

$$ \left\langle m^{2}\right\rangle=\frac{a l}{m_{a}}\left[\frac{\xi^{-1} L_{i}}{2 \pi^{1 / 2} M^{2}}-\frac{2 K L_{1}^{1 / 2}}{\pi^{4} M^{3 / 2}}\right], $$

$$ G\left(C, C^{\prime}\right)=\frac{1}{2 \pi} \int d \tau \dot{\mathbf{x}}(\tau) \nabla \varphi(\mathbf{x}(\tau))=\frac{1}{2 \pi} \int d \tau \dot{\varphi}(\mathbf{x}(\tau)) $$

$$ \mathcal{A}_{\mathrm{e}, \text { int }}=i 2 \hbar \theta G\left(C, C^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{CS}}=\frac{1}{4 \theta \hbar i} \int d^{3} x \mathbf{A} \cdot(\boldsymbol{\nabla} \times \mathbf{A}) $$

$$ \theta=\pi \mu_{0} $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{CS}}=\frac{1}{4 \pi i} \frac{e^{2}}{c^{2} \hbar \mu_{0}} \int d^{3} x \mathbf{A} \times(\boldsymbol{\nabla} \times \mathbf{A}) $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{curr}}=-i \frac{e}{c} \oint_{C} d \mathbf{x} \mathbf{A}(\mathbf{x})-i \frac{e}{c} \oint_{C^{\prime}} d \mathbf{x} \mathbf{A}(\mathbf{x}) $$

$$ \mathbf{j}(\mathbf{x}) \equiv e c \sum_{\alpha} \oint_{C_{\alpha}} d \mathbf{x}_{\alpha} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}_{\alpha}\right) $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{curr}}=-i \frac{1}{c^{2}} \int d^{3} x \mathbf{j}(\mathbf{x}) \mathbf{A}(\mathbf{x}) $$

$$ \mathbf{B} \equiv \boldsymbol{\nabla} \times \mathbf{A} $$

$$ \mathbf{B}(\mathbf{x})=\mu_{0} \frac{2 \pi \hbar c}{e^{2}} \mathbf{j}(\mathbf{x}) $$

$$ \mathbf{B}(\mathbf{x})=\mu_{0} \Phi_{0} \frac{1}{e} \mathbf{j}(\mathbf{x}) $$

$$ \begin{align*} & A_{3}=i \phi=i A_{0}, \quad A_{1}=A_{x}, \quad A_{2}=A_{y}, \\ & B_{3}=i B_{z}, \quad B_{1}=-i E_{y}, \quad B_{2}=i E_{x} . \end{align*} $$

$$ \begin{align*} & j_{3}=i j_{0}=i c \rho\left(\mathbf{x}_{\perp}\right), \quad \rho(\mathbf{x}) \equiv e \sum_{\alpha} \delta^{(2)}\left(\mathbf{x}_{\perp}-\mathbf{x}_{\perp \alpha}\right) \\ & j_{1}=i j_{x}\left(\mathbf{x}_{\perp}\right)=e \sum_{\alpha} \dot{x}_{\alpha} \delta^{(2)}\left(\mathbf{x}_{\perp}-\mathbf{x}_{\perp \alpha}\right) \\ & j_{2}=i j_{y}\left(\mathbf{x}_{\perp}\right)=e \sum_{\alpha} \dot{y}_{\alpha} \delta^{(2)}\left(\mathbf{x}_{\perp}-\mathbf{x}_{\perp \alpha}\right) \end{align*} $$

$$ \mathcal{A}_{\mathrm{int}}=\int d t d^{2} x\left[\rho \phi-\frac{1}{c}\left(j_{x} A_{x}+j_{x} A_{y}\right)\right] $$

$$ B_{z}=\mu_{0} \Phi_{0} \rho, \quad E_{x}=\mu_{0} \Phi_{0} \frac{1}{c} j_{y}, \quad E_{y}=\mu_{0} \Phi_{0} \frac{1}{c} j_{x} $$

$$ \mathcal{A}_{\mathrm{anyon}}=\mathcal{A}_{\mathrm{CS}}+\mathcal{A}_{\mathrm{boson}} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{boson}}=\int d^{2} x \int_{t_{a}}^{t_{b}} d t\left\{\psi^{*}(\mathbf{x}, t)\right. & {\left[i \hbar\left(\partial_{t}+i \frac{e}{\hbar} \phi(\mathbf{x}, t)\right)+\mu\right] \psi(\mathbf{x}, t) } \\ & \left.-\frac{\hbar^{2}}{2 M}\left|\left[\boldsymbol{\nabla}-i \frac{e}{\hbar c} \mathbf{A}(\mathbf{x}, t)\right] \psi(\mathbf{x}, t)\right|^{2}\right\} \end{align*} $$

$$ p_{0} \rightarrow p_{0}-\frac{e}{c} \phi, \quad \mathbf{p} \rightarrow \mathbf{p}-\frac{e}{c} \mathbf{A} . $$

$$ B_{z}(\mathbf{x}, t)=\mu_{0} \Phi_{0} \psi^{\dagger}(\mathbf{x}, t) \psi(\mathbf{x}, t) $$

$$ \partial_{x} A_{y}(\mathbf{x}, t)-\partial_{y} A_{x}(\mathbf{x}, t)=\mu_{0} \Phi_{0} \psi^{\dagger}(\mathbf{x}, t) \psi(\mathbf{x}, t) $$

$$ \left(\partial_{x} \partial_{y}-\partial_{y} \partial_{x}\right) \Lambda(\mathbf{x}, t)=0 $$

$$ \left(A_{x}, A_{y}\right)=\left(\partial_{x} \alpha, \partial_{y} \alpha\right) $$

$$ \left(\partial_{x} \partial_{y}-\partial_{y} \partial_{x}\right) \alpha(\mathbf{x}, t)=\mu_{0} \frac{2 \pi \hbar c}{e} \psi^{\dagger}(\mathbf{x}, t) \psi(\mathbf{x}, t) $$

$$ \varphi\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \equiv \arctan \left[\left(y-y^{\prime}\right) /\left(x-x^{\prime}\right)\right] . $$

$$ \left(\partial_{x} \partial_{y}-\partial_{y} \partial_{x}\right) \varphi\left(\mathbf{x}-\mathbf{x}^{\prime}\right)=2 \pi \delta^{(2)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \varphi\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-\varphi\left(\mathbf{x}^{\prime}-\mathbf{x}\right)=\pi $$

$$ \alpha(\mathbf{x}, t)=\mu_{0} \frac{\hbar c}{e} \int d^{2} x \varphi\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \psi^{\dagger}(\mathbf{x}, t) \psi(\mathbf{x}, t) $$

$$ \Psi(\mathbf{x}, t)=e^{-i(e / \hbar c) \int^{\mathbf{x}} d \mathbf{x}^{\prime} \mathbf{A}\left(\mathbf{x}^{\prime}, t\right)} \psi(\mathbf{x}, t) $$

$$ D_{i} \psi(\mathbf{x}, t)=\left(\partial_{i}-i \frac{e}{\hbar c} A_{i}\right) \psi(\mathbf{x}, t) $$

$$ \begin{align*} {\left[\hat{\psi}(\mathbf{x}, t), \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)\right] } & =\delta^{(2)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \\ {\left[\hat{\psi}^{\dagger}(\mathbf{x}, t), \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)\right] } & =0 \\ {\left[\hat{\psi}(\mathbf{x}, t), \hat{\psi}\left(\mathbf{x}^{\prime}, t\right)\right] } & =0 \end{align*} $$

$$ \begin{align*} \hat{\psi}(\mathbf{x}, t) \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)-e^{i \pi \mu_{0}} \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right) \hat{\psi}(\mathbf{x}, t) & =\delta^{(2)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \\ \hat{\psi}^{\dagger}(\mathbf{x}, t) \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right)-e^{i \pi \mu_{0}} \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right) \hat{\psi}^{\dagger}(\mathbf{x}, t) & =0 \\ \hat{\psi}(\mathbf{x}, t) \hat{\psi}\left(\mathbf{x}^{\prime}, t\right)-e^{i \pi \mu_{0}} \hat{\psi}\left(\mathbf{x}^{\prime}, t\right) \hat{\psi}(\mathbf{x}, t), & =0 \end{align*} $$

$$ \tilde{\psi}(\mathbf{x}, t)=e^{-i \frac{e}{\hbar c} \alpha(\mathbf{x}, t)} \psi(\mathbf{x}, t), \quad \tilde{\psi}^{\dagger}(\mathbf{x}, t)=\psi(\mathbf{x}, t) e^{i \frac{e}{\hbar c} \alpha(\mathbf{x}, t)} . $$

$$ \hat{\psi}^{\dagger}(\mathbf{x}, t) e^{i \frac{e}{\hbar c} \hat{\alpha}(\mathbf{x}, t)} \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right) e^{i \frac{e}{\hbar c} \hat{\alpha}\left(\mathbf{x}^{\prime}, t\right)}=e^{i \pi \mu_{0}} \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right) e^{i \frac{e}{\hbar c} \hat{\alpha}\left(\mathbf{x}^{\prime}, t\right)} \hat{\psi}^{\dagger}(\mathbf{x}, t) e^{i \frac{e}{\hbar c} \hat{\alpha}(\mathbf{x}, t)} $$

$$ e^{i \int d^{2} x^{\prime} f\left(\mathbf{x}^{\prime}, t\right) \hat{\psi}^{\dagger}(\mathbf{x}, t) \hat{\psi}(\mathbf{x}, t)} \hat{\psi}^{\dagger}(\mathbf{x}, t) e^{-i \int d^{2} x^{\prime} f\left(\mathbf{x}^{\prime}, t\right) \hat{\psi}^{\dagger}(\mathbf{x}, t) \hat{\psi}(\mathbf{x}, t)}=e^{i f(\mathbf{x}, t)} \hat{\psi}^{\dagger}(\mathbf{x}, t) $$

$$ e^{i \hat{A}} \hat{B} e^{-i \hat{A}}=1+i[\hat{A}, \hat{B}]+\frac{i^{2}}{2!}[\hat{A},[\hat{A}, \hat{B}]]+\ldots $$

$$ \begin{align*} \hat{\psi}^{\dagger}(\mathbf{x}, t) \hat{\psi}^{\dagger} & \left(\mathbf{x}^{\prime}, t\right) e^{i \mu_{0} \varphi\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} e^{i \frac{e}{\hbar c}\left[\hat{\alpha}(\mathbf{x}, t)+\hat{\alpha}\left(\mathbf{x}^{\prime}, t\right)\right]} \\ & =e^{i \pi \mu_{0}} \hat{\psi}^{\dagger}\left(\mathbf{x}^{\prime}, t\right) \hat{\psi}^{\dagger}(\mathbf{x}, t) e^{i \mu_{0} \varphi\left(\mathbf{x}^{\prime}-\mathbf{x}\right)} e^{i \frac{e}{\hbar c}\left[\hat{\alpha}\left(\mathbf{x}^{\prime}, t\right)+\hat{\alpha}(\mathbf{x}, t)\right]} \end{align*} $$

$$ E_{x}=-\frac{1}{\rho e c} j_{y} B_{z} $$

$$ R_{x y} \equiv \frac{B_{z}}{\rho e c} $$

$$ \nu=\frac{1}{5}, \frac{2}{7}, \frac{1}{3}, \frac{2}{5}, \frac{2}{3}, \ldots $$

$$ \rho L_{x} L_{y}=2 \times \int \frac{d p_{x} d p_{y} L_{x} L_{y}}{(2 \pi \hbar)^{2}} $$

$$ p_{F}=\sqrt{2 \pi \rho} \hbar $$

$$ \rho=M \omega \frac{1}{2 \pi \hbar} \sum_{n=0}^{n_{F}} $$

$$ \rho_{\max }=\frac{M \omega}{2 \pi \hbar} $$

$$ \rho=\frac{M \omega}{2 \pi \hbar} \nu . $$

$$ \frac{B_{z}}{\rho}=\frac{h c}{e} \frac{1}{\nu} $$

$$ \Phi \equiv \frac{B_{z}}{\rho} $$

$$ \frac{\Phi}{\Phi_{0}}=\frac{1}{\nu} $$

$$ R_{x y}=\frac{h}{e^{2}} \frac{1}{\nu} $$

$$ B_{z}^{\text {eff }}=B_{z}-B_{z}^{\text {stat }}, \quad B_{z}^{\text {stat }}=2 m \Phi_{0} \rho $$

$$ \omega^{\mathrm{eff}}=e B_{z}^{\mathrm{eff}} / M c . $$

$$ B_{z}^{\mathrm{eff}}= \pm \rho \Phi_{0} / \nu^{\mathrm{eff}}, \quad \nu^{\mathrm{eff}}=1,2,3, \ldots $$

$$ \pm \frac{1}{\nu^{\mathrm{eff}}}=\frac{1}{\nu}-2 m $$

$$ \nu=\frac{\nu^{\mathrm{eff}}}{2 m \nu^{\mathrm{eff}} \pm 1} $$

$$ \mathcal{A}=\frac{1}{8 \pi} \int d t d^{3} x\left[\mathbf{E}^{2}-(\boldsymbol{\nabla} \times \mathbf{A})^{2}\right] $$

$$ \mathbf{E}=-\frac{1}{c} \frac{\partial \mathbf{A}}{\partial t}-\nabla A_{0} . $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{1}{8 \pi c} \int d^{4} x\left[\mathbf{E}^{2}+(\boldsymbol{\nabla} \times \mathbf{A})^{2}\right] $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{L}{8 \pi c} \int d^{3} x(\boldsymbol{\nabla} \times \mathbf{A})^{2} $$

$$ \frac{L}{4 \pi c} \boldsymbol{\nabla} \times(\boldsymbol{\nabla} \times \mathbf{A})+i \frac{e^{2}}{2 \pi c^{2} \hbar \mu_{0}} \boldsymbol{\nabla} \times \mathbf{A}=i \frac{1}{c^{2}} \mathbf{j} . $$

$$ \frac{L}{4 \pi c}\left(\boldsymbol{\nabla} \times \mathbf{B}+i \lambda^{-1} \mathbf{B}\right)=i \frac{1}{c^{2}} \mathbf{j}, $$

$$ \lambda \equiv \frac{c \hbar \mu_{0}}{2 e^{2}} L=\frac{\mu_{0}}{2 \alpha} L $$

$$ \frac{L}{4 \pi c}\left(-\boldsymbol{\nabla}^{2}+\lambda^{-2}\right) \mathbf{B}=i \frac{1}{c^{2}} \boldsymbol{\nabla} \times \mathbf{j}+\frac{1}{c^{2}} \lambda^{-1} \mathbf{j} . $$

$$ \boldsymbol{\nabla} \times \mathbf{j} \propto \mathbf{B} $$

$$ \left(\boldsymbol{\nabla} \times \mathbf{j}_{\perp}\right)_{z} \propto B_{z} $$

$$ B_{z}=\mu_{0} \Phi_{0} \rho $$

$$ \frac{L}{4 \pi c}\left(-\boldsymbol{\nabla}^{2}+\lambda^{-2}\right) \Delta B_{z}=\frac{1}{c^{2}}\left(\boldsymbol{\nabla} \times \mathbf{j}_{\perp}\right)_{z} . $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{CS}}=\frac{k}{4 \pi i} \int d^{3} x \epsilon_{i j k} \operatorname{tr}_{N}\left(A_{i} \nabla_{j} A_{k}+\frac{2}{3} A_{i} A_{j} A_{k}\right), $$

$$ A_{i} \leftrightarrow U A_{i} U^{-1}+i\left(\partial_{i} U\right) U^{-1} $$

$$ \mathcal{A}_{\mathrm{e}, \mathrm{CS}} \rightarrow \mathcal{A}_{\mathrm{e}, \mathrm{CS}}+2 \pi i n k \hbar, \quad n=\text { integer } . $$

$$ B_{i} \equiv \epsilon_{i j k} F_{j k} $$

$$ F_{i j} \equiv \partial_{i} A_{j}-\partial_{j} A_{i}-i\left[A_{i}, A_{j}\right] $$

$$ W_{L}[\mathbf{A}] \equiv \operatorname{tr}_{N} \hat{W}[\mathbf{A}] \equiv \operatorname{tr}_{N} \hat{P} e^{i \oint_{L} d \mathbf{x} \mathbf{A}} $$

$$ \left\langle W_{L}[\mathbf{A}]\right\rangle \equiv \frac{\int \mathcal{D} A_{i} e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}} / \hbar} W_{L}[\mathbf{A}]}{\int \mathcal{D} A_{i} e^{-\mathcal{A}_{\mathrm{e}, \mathrm{CS}} / \hbar}} $$

$$ \left\langle W_{0}[\mathbf{A}]\right\rangle=\frac{q^{N / 2}-q^{-N / 2}}{q^{1 / 2}-q^{-1 / 2}} $$

$$ q \equiv e^{-2 \pi i /(N+k)} $$

$$ q^{N / 2}\left\langle W_{L_{+}}[\mathbf{A}]\right\rangle-q^{-N / 2}\left\langle W_{L_{-}}[\mathbf{A}]\right\rangle=\left(q^{1 / 2}-q^{-1 / 2}\right)\left\langle W_{L_{0}}[\mathbf{A}]\right\rangle . $$

$$ \frac{\left\langle W_{L}[\mathbf{A}]\right\rangle}{\left\langle W_{0}[\mathbf{A}]\right\rangle}=H_{L}\left(t,-\left(t^{1 / N}-t^{-1 / N}\right)\right), \quad t=e^{\pi i / k} $$

$$ c=e^{-i 2 \pi\left(N^{2}-1\right) / 2 N k} $$