Books that emphasize Heisenberg picture

[IvanPavlov]

Hi fellas,

one friends that is Mathematician asked me to recommend some textbook that emphasizes Heisenberg picture and where this picture is rigorously explained. If anyone knows some good book for this I would be grateful :)

Regards,
Ivan

 

[bhobba]

Hi Ivan

Trouble is that picture was well and truly superseded when Dirac came up with his transformation theory which generally goes under the name of QM today. It's virtually impossible to rigorously develop QM by the Heisenberg picture alone these days - you can only explain it within the full QM machinery.

I recommend - Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Here QM is fairly rigorously developed from just 2 axioms and the Heisenberg picture correctly placed in that development.

Thanks
Bill

 

[vanhees71]

Any picture is equivalent (in usual QM; there are subtle troubles in QFT, known as Haag's theorem). It's just your choice of how to share the time dependence between the state operator (i.e., the statistical operator) of the system and the (self-adjoint) operators representing observables. This is only defined up to a time-dependent unitary transformation and is known as the choice of the picture. The Heisenberg picture is the one that is most closely related to the way classical mechanics is formulated in terms of the Hamilton formalism using Poisson brackets and lumps all time dependence to the observable operators. The Schrödinger picture is the one where the entire time dependence is put to the statistical operator and the observable operators are time independent. The most general picture is due to Dirac, where you choose one part of the Hamiltonian, $\hat{H}_0$ to propagate the observable operators and one part $\hat{H}_1$ that propagate the statistical operator. In any case you have $\hat{H}_0+\hat{H}_1=\hat{H}$, and the outcome for observable quantities (probability distributions for finding a certain possible value for an observable, average values for observables, transition probabilities like S-matrix elements, etc.) is independent of the choice of the picture of time evolution.

A good explanation of the Heisenberg picture for relativistic QFT is given in

Weinberg, Quantum Theory of Fields, Vol. 1

Usually QFT textbooks use the Heisenberg picture to start with and then derive perturbative QFT (Feynman diagrams) using the interaction picture (usually ignoring Haag's theorem of course ;-)).