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Summary
In the present paper we have constructed the renormalized LF Hamiltonian for the $\lambda \varphi^4$ model
in (2+1)-dimensional space-time.
We have found the explicit expression for the counterterm, necessary for the renormalization,
using the PV regularization. To do this we compare the diagrams of the covariant perturbation
theory in Lorentz coordinates with the analogous diagrams of the perturbation
theory generated by the LF Hamiltonian which has also the cutoff in the momentum
$ p_-$ ($|p_-|\geq \delta >0$). We show that both perturbation theories can be described by
the same set of diagrams, with the values of the compared diagrams coinciding in the limit
$\delta \to 0 $. Then we renormalize the LF Hamiltonian by the counterterm found in
the calculation of the divergent part of the corresponding diagram in the covariant perturbation
theory in Lorentz coordinates.
Furthermore we have taken into account the possibility of the spontaneous symmetry
breaking in this model and obtained the LF Hamiltonians corresponding to two different
vacua. We arrive at these LF Hamiltonians by considering the limit transition from the
theories quantized on the spacelike planes approaching the LF. It is
possible to describe the vacuum on these planes using the Gaussian approximation.
The Hamiltonians obtained with this approximation still require UV renormalization.
And the above-mentioned comparison of perturbative theories, generated by these
LF Hamiltonians, and the covariant perturbation theory in Lorentz coordinates allows
to renormalize both of these Hamiltonians in the PV regularization.
