Editing Interaction picture (section)

==Schwinger–Tomonaga equation==
The term interaction representation was invented by Schwinger.<ref name=Schwinger>{{Citation  | last1 = Schwinger  | first1 = J. | title = Selected papers on Quantum Electrodynamics   | publisher = Dover  | year = 1958 | isbn =0-486-60444-6 | page = 151}}</ref><ref name=Schwinger0>{{Citation  | last1 = Schwinger  | first1 = J. | title =  Quantum electrodynamics. I. A covariant formulation.|year = 1948 |journal=Physical Review | volume= 74 |issue = 10 | pages= 1439–1461 | doi = 10.1103/PhysRev.74.1439 | bibcode = 1948PhRv...74.1439S | url= https://doi.org/10.1103/PhysRev.74.1439| url-access = subscription }}</ref>
In this new mixed representation the state vector is no longer constant in general,  but it is constant if there is no coupling between fields.  The change of representation leads directly to the Tomonaga–Schwinger equation:<ref name=Schwinger1>{{Citation  | last1 = Schwinger  | first1 = J. | title = Selected papers on Quantum Electrodynamics   | publisher = Dover  | year = 1958 | isbn =0-486-60444-6 | page = 151,163,170,276 }}</ref><ref name=Schwinger0/>
:<math>ihc \frac {\partial  \Psi[\sigma]}{\partial \sigma(x)} = \hat{H}(x)\Psi(\sigma) </math>
:<math> \hat{H}(x) = - \frac{1}{c} j_{\mu}(x) A^{\mu}(x) </math>
Where the Hamiltonian in this case is the QED interaction Hamiltonian, but it can also be a generic interaction, and <math>\sigma</math> is a spacelike surface that is passing through the point <math>x</math>. The derivative formally represents a variation over that surface given <math>x</math> fixed. It is difficult to give a precise mathematical formal interpretation of this equation.<ref name=Wakita1976>{{Citation | last1= Wakita | first1 =  Hitoshi | journal = Communications in Mathematical Physics |pages = 61–68 | title = Integration of the Tomonaga-Schwinger Equation|volume = 50|year = 1976| issue = 1 | doi = 10.1007/BF01608555 | bibcode = 1976CMaPh..50...61W | s2cid = 122590381 | url = http://projecteuclid.org/euclid.cmp/1103900149 }}</ref>

This approach is called the 'differential' and 'field' approach by Schwinger, as opposed to the 'integral' and 'particle' approach of the Feynman diagrams.<ref name=SchwingerNP> {{Citation | title=Schwinger Nobel prize lecture| url=https://www.nobelprize.org/uploads/2018/06/schwingerlecture.pdf|pages=140|quote="Schwinger informally calls differential as local approach, and calls integral as a type of global approach. The term global here is used with respect to the integration domain"}}</ref><ref name=Schwinger3>{{Citation  | last1 = Schwinger  | first1 = J. | title = Selected papers on Quantum Electrodynamics  | publisher = Dover  | year = 1958 | isbn =0-486-60444-6 | page = preface xiii |quote="Schwinger informally calls local approach referring to fields also in the context of local actions. Particle are emergent properties from  an integral approach applied to the field, or averaged approach. He is at the same time making an analogy to the classical distinction between particles and fields, and to show how this is realized for quantum fields}}</ref>

The core idea is that if the interaction has a small coupling constant (i.e. in the case of electromagnetism of the order of the fine structure constant) successive perturbative terms will be powers of the coupling constant and therefore smaller.<ref name=Schwinger2>{{Citation  | last1 = Schwinger  | first1 = J. | title = Selected papers on Quantum Electrodynamics  | publisher = Dover  | year = 1958 | isbn =0-486-60444-6 | page = 152 }}</ref>