discussion

The phase space

A point in the phase space defines all configuration variables qk (like the coordinates of all particles) of a physical system, together with the corresponding momentum variables pk.

The phase space of the lattice model

The configuration space of the cellular lattice model consists of twelf functions (aiμ(n1,n2,n3)) on the lattice, where n1,n2,n3 are integer numbers which enumerate the cells of the lattice.

For each configuration variable aiμ(n1,n2,n3), we have a corresponding momentum variable denoted by πiμ(n1,n2,n3).

Therefore, an element of the phase space is defined by the following pair of functions:

( aiμ(n1,n2,n3), πiμ(n1,n2,n3) ).

Symplectic structure and Poisson brackets on the phase space

The fundamental structure which is characteristic for the phase space is the so-called symplectic structure. It is defined by some symplectic form. It allows to define an important operation named "Poisson bracket", which transforms two functions f(p,q), g(p,q) into a third one {f,g}(p,q), defined by

{f,g}(p,q) = (∂p f) (∂q g) - (∂q f) (∂p g)

Using this Poisson bracket, we can easily describe the evolution equation for arbitrary functions f(p,q) as

∂t f = {H,f}.

Here H(p,q) is simply the energy, expressed in the phase space variables p,q. For the special case of f=p and f=q we obtain the Hamilton equations:

∂t q = {H,q} = ∂p H(p,q)

∂t p = {H,p} = - ∂q H(p,q)

For example, in the simplest case, for a particle moving in a potential V(q), we have

H = 12m p2 + V(q),

which gives as the Hamilton equations

p = m ∂t q,

∂t p = - ∂q V(q).

The symplectic structure defined here is important for the definition of a complex structure, which is required for the explanation why the gauge groups of the SM are unitary gauge groups.