Where Can I Find Hamiltonians of the Standard Model in QFT?

[metroplex021]

Hi all - as everyone knows, the fundamental laws of the Standard Model are almost always presented in Lagrangian form. Can anyone tell me of anywhere (such as a textbook) that I might find the Hamiltonians corresponding to these Lagrangians written out? (I'm confused on a couple of points and am having loads of trouble working it out myself.) Any tips received with gratitude!

 

[tom.stoer]

The Hamiltonian formulation is usually rather ugly due to several reasons:

Manifest Poincare covariance is lost b/c time t and the Hamiltonian H are singled out. Whereas in the Lagrangian the symmetry is explicitly visible, in the Hamiltonian (and all other generators of the Poincare algebra) the symmetry is hidden and has to be checked explicitly.

Something similar happens with the gauge symmetries. They are explicitly visible in te Lagrangian but "fixed" in the Hamiltonian. This can be achieved in various different ways, e.g. via eliminating unphysical degrees of freedom which means reducing 4 to 2 gauge degrees of freedom (the transversal polarization states). Doing this results in physical degrees of freedom but unfortunately generates rather complicated interaction terms which in some cases can not be written down explicitly.

In addition in order to construct the Hamiltonian you have to construct the (physical) Hilbert space with an inner product AND you have to regularize all expressions. You have to regularize in the Lagrangian (or path integral) approach as well, but there you can do it on the level of Greens functions or matrix elements, so in a sense one step later. Therefore the Lagrangian itself still looks nice whereas the complexity is present in the matrix elements. In the Hamiltonian approach things are more intertwined and already the Hamiltonian gets more complicated.

To cut a long story short: I do not know about a standard textbook treatment of QCD, but I can give you some references where you can check the details.

http://physik.uni-graz.at/itp/oberw/oberw08/Vortraege/reinhardt_oberwoelz08.pdf
http://cdsweb.cern.ch/record/292166/files/9511450.pdf
http://www.adsabs.harvard.edu/abs/1994AnPhy.233..317L

 

[metroplex021]

Thank you *very* much for that - awesome of you Part of what I was wondering was whether anyone has any use for eigenvalue equations in QFT. Now it seems that they do. Thanks mate.