  In order to solve quantum field theory in a non-perturbative way, Lagrangian
lattice simulations have been very successful. Here we discuss a recently
proposed alternative Hamiltonian lattice formulation - the Monte Carlo
Hamiltonian. In order to show its working in the case of the scalar
$\Phi^{4}_{1+1}$ model, we have computed thermodynamic functions like free
energy, average energy, entropy and specific heat. We find good agreement
between the results from the Monte Carlo Hamiltonian and standard Lagrangian
lattice computations. However, the Monte Carlo Hamiltonian results show less
fluctuations under variation of temperature. We address properties of the MC
Hamiltonian, like a finite temperature window, and scaling properties. Also we
discuss possible future applications - like quantum chaos in many-body systems,
the non-perturbative computation of wave functions of elementary particles, as
well as scattering amplitudes in high energy physics.
