Schwinger-Dyson equations
To calculate S-Matrix elements we have to calculate correlation functions $\braket{\Omega|T{\phi(x_1)…\phi(x_n)}|\Omega}$, which contain the vacuum state of the interacting theory $\ket{\Omega}$ and the interacting fields $\phi$.
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, since the euler lagrange equations for $$\mathcal{L}=-\frac{1}{2}\phi(\Box + m^2)\phi + \mathcal{L}_\text{int}[\phi]$$ are $$(\Box+m^2)\phi - \mathcal{L}^\prime_\text{int}[\phi]=0$$
- So when pulling the differential operator into the time ordering we get some $\delta$-functions also called contact interactions.
- In a classical theory $[\phi,\pi]=0$, which means the contact interactions are a quantum feature.
- These allow for loops and thus make the difference to classical field theories.
- Schwinger Dyson equations contain in principle non-perturbative information
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 $[\phi(\vec{x},t),\pi(\vec{y},t)]=i\delta^3(\vec{x}-\vec{y})$. 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 to higher correlators works by writing the time ordering in terms of multiple theta functions and acting on all of them.