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Is it possible to approximately calculate the dynamics of a "phi-fourth" interacting Klein-Gordon field by using a
finite dimensional Hilbert state space where the possible values of momentum are limited to a discrete set
##-p_{max},-\frac{N-1}{N}p_{max},-\frac{N-2}{N}p_{max},..., -\frac{2}{N}p_{max}, -\frac{1}{N}p_{max}, 0 ,\frac{1}{N}p_{max}, \frac{2}{N}p_{max}, ..., \frac{N-2}{N}p_{max}, \frac{N-1}{N}p_{max} , p_{max}##
and each momentum state can be excited at most M times from the ground (vacuum) state? The energy eigenstates could then be solved from a finite-dimensional matrix eigenvalue problem and I could calculate the probabilities of particles with different momenta being spontaneously created from the vacuum of the noninteracting Klein-Gordon field (which of course isn't the real ground state of the system anymore once the phi-fourth interaction is added).
finite dimensional Hilbert state space where the possible values of momentum are limited to a discrete set
##-p_{max},-\frac{N-1}{N}p_{max},-\frac{N-2}{N}p_{max},..., -\frac{2}{N}p_{max}, -\frac{1}{N}p_{max}, 0 ,\frac{1}{N}p_{max}, \frac{2}{N}p_{max}, ..., \frac{N-2}{N}p_{max}, \frac{N-1}{N}p_{max} , p_{max}##
and each momentum state can be excited at most M times from the ground (vacuum) state? The energy eigenstates could then be solved from a finite-dimensional matrix eigenvalue problem and I could calculate the probabilities of particles with different momenta being spontaneously created from the vacuum of the noninteracting Klein-Gordon field (which of course isn't the real ground state of the system anymore once the phi-fourth interaction is added).