Path integrals and the Dyson formula

One of the deepest results in quantum field theory, to me, is the Dyson formula [nLa]. 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 \(A(t)\),

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

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 [Bro13].

On the other hand, we have the Feynman path integral

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

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 [Gil17].

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

Bro13

Francis Brown. Iterated integrals in quantum field theory. 6th Summer School on Geometric and Topological Methods for Quantum Field Theory, pages 188–240, 2013.

Gil17

Tepper L Gill. The feynman-dyson view. In Journal of Physics: Conference Series, volume 845, 012023. IOP Publishing, 2017.

nLa

nLab. Dyson formula. https://ncatlab.org/nlab/show/Dyson+formula. Accessed: 2021.

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