Uni— title: Schwinger-Dyson equations id: 2023-01-07T165748 category: tags:
- rework bibFile: content/notes/bib_lit.json
Definition: Schwinger-Dyson equations
Let $\braket{\phi_x\phi_1\dotsc} = \braket{\Omega|T{\phi(x)\phi(x_1)\dotsc}|\Omega}$. Then the Schwinger-Dyson equations are $$(\Box + m^2)\braket{\phi_x\phi_1\dotsc\phi_n} = \braket{\mathcal{L}_\text{int}^\prime[\phi_x]\phi_1\dotsc\phi_n} - i\sum_j\delta^4(x - x_j)\braket{\phi_1\dotsc\phi_{j-i}\phi_{j+1}\dotsc\phi_n},$$ where $\mathcal{L}\text{int}^\prime = \frac{d}{d\phi}\mathcal{L}\text{int}$ is the derivative of the interaction lagrangian
So when pulling the differential operator into the time ordering we get some $\delta$-functions also called contact interactions. These allow for loops and thus make the difference to classical field theories.
Derivation of the Schwinger-Dyson equations
Calculate the time derivateives of $\braket{\phi_x\phi_1}$ by expanding the time ordering in terms of $\Theta$-functions and using $\partial\Theta = \delta$: $$\partial_t\braket{\phi_x\phi_1} = \braket{\partial_t \phi_x \phi_1} + \delta(t-t_1)\braket{\Omega|[\phi(x),\phi(x_1)]|\Omega}$$ With $\delta(t-t_1)\braket{\Omega|[\phi(x),\phi(x_1)]|\Omega} = 0$ since the commutator vanishes at equal times The second time derivitave becomes $$\partial^2_t\braket{\phi_x\phi_1} = \braket{\partial^2_t \phi_x \phi_1} + \delta(t-t_1)\braket{\Omega|[\partial_t\phi(x),\phi(x_1)]|\Omega}$$ where we can insert the expression for the canonic comutator. Since the spacial derivative does not depend on $t$ we can pull it inside the time ordering leading to $$(\Box + m^2)\braket{\phi_x\phi_1} = \braket{(\Box + m^2)\phi_x\phi_1}-i\delta^4(x-x_1).$$ Inserting the Euler Lagrange equations for the scalar field $(\Box + m^2)\phi(x) = \mathcal{L}^\prime_\text{int}[\phi]$ leads to the dyson schwinger equations for the two point corralator. The generalization