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date: 2021-05-10
excerpts: 2
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# Path integrals and the Dyson formula

One of the deepest results in quantum field theory, to me, is the Dyson formula {cite}`nlab2021dysonformula`. It describes the solution to the differential equation

$$ i\frac{\partial}{\partial t} \Psi(t) = A(t)\Psi(t) $$

in terms of the exponential of the path integral of the operator $A(t)$,

$$\begin{array}{rl} \Psi(t) & = U(t,0) \, \Psi(0) \\ & \\
U(t,0) & = \displaystyle \mathcal{T}\exp \left\{ i\int_0^t A(s) ds \right\} \end{array} $$

where $\mathcal{T}$ is the time-ordering operator. Here, $U(t,0)$ is known as the _time-evolution operator_. 

The proof of this formula is given by Picard integration and iterated integrals {cite}`brown2013iterated`.

On the other hand, we have the Feynman path integral

$$ \begin{array}{rl}
\psi(y,t) & = K(y,t; x,0)  \, \psi(x,0) \\ & \\ 
K(y,t; x,0) & = \displaystyle \int \exp\left\{iS(q,\dot{q})\right\} Dq \\ & \\ &  = \langle y \vert \, U(t,0)  \, \vert x\rangle\end{array}
$$

where $\psi(y,t)$ is the value of the state vector $\Psi(t)$ at position $y,$ $\int Dq$ is an integral over all paths $q$ from $x$ to $y$, and $S(q,\dot{q})$ is the action of the path $q.$ Here, $K(y,t;x,0)$ is known as the _kernel_ or _propagator_.

The time-ordered exponential in the Dyson series can be related to the Feynman path integral by viewing the Hamiltonian operators in a von Neumann infinite tensor product Hilbert space {cite}`gill2017feynman`.

The Dyson formula frames the dynamics of the system in terms of the time-evolution operator, while the path integral frames the dynamics in terms of the kernel.

The action gives a "particle" interpretation to the quantum or stochastic dynamical system. These "particles" are real only to the extent that this action is well-defined. 

It would be interesting to study the Dyson formula through the algebraic lens of regularity structures.

There should also be a generalization of the Dyson formula or the path integral to coends in category theory.

[Freeman Dyson - Linking the ideas of Feynman, Schwinger and Tomanaga]<https://www.youtube.com/watch?v=i3RcN5UGwgI>

## References

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