# Ising Hamiltonian for relativistic particles

An Ising system is described by the simple Hamiltonian: $$H = \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j \,\,\,\,\,\,\,\,\,\,(1)$$ Here the $x_i$ are spins (+1 or -1 in units of $\hbar$), the $c_{1i}$ are local magnetic fields and the $c_{2ij}$ are (nearest neighbor) interaction constants. So one usually only keeps constants $c_{2ij}$ if $x_i$ and $x_j$ are located close enough in space; after a certain spatial range the $c_{2ij}$ are supposed to become negligible. Note that (1) can be interpreted as two low-order terms of a general Taylor expansion of a general Hamiltonian H($x_1$,$x_2$,…,$x_n$): $$H = c_0 + \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j + \sum\limits_{i,j,k} c_{3ijk} x_i x_j x_k + … \,\,\,\,\,\,(2)$$
My questions are the following:
Q1) Does (1) also hold if the particles are moving ? Looking at Feynman’s book on stat mech it seems that (1) is then also valid: the form (1) arises according to Feynman due to magnetic dipole interaction or due to a “Coulomb potential combined with Pauli’s exclusion principle”; nothing in the derivation demands that the particles are static.
Q2) Suppose all particles are on a 2D lattice, except two particles (1 and 2) which are emitted from a source in the middle of the lattice and move in opposite direction through the lattice at relativistic speeds. Can we apply (1) here or do we need a corrected version ? (One way to correct (1) if some particles are moving might be to include higher order terms as in (2).)

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Welcome to physics.stackexchange Louis! This site uses LaTeX typesetting code. I tidied up your post but I am unfamiliar with the actual Hamiltonian you are using. Please ensure that I put the correct indices where they should go. – John M May 8 '13 at 22:45
@LouisV, When particles have a fixed location, then it makes sense to talk about them interacting with their "neighbours". If you have a bunch of moving particles, like a box of gas, then how will you specify which pairs of particles interact? Are you going to give a distance dependent pairwise interaction for every pair? – Siva May 9 '13 at 0:48
@Siva. Thanks Siva for the answer. Yes, as you say, I would give a distance dependent interaction to every pair. As far as I know this is exactly what one does for 'lattice gases' which are formally described by exactly the same Ising Hamiltonian, with xi = occupation of site i, and c2ij = interaction potential between a pair. But my question is: can the same form of the Hamiltonian be used if say 2 particles are moving at relativistic speed ? – LouisV May 9 '13 at 15:37
@John M. Thanks John for putting the indices ! The formula looks better now. – LouisV May 10 '13 at 19:18