A point in the **phase space** defines all **configuration variables q ^{k}** (like the coordinates of all particles) of a physical system, together with the corresponding

The configuration space of the cellular lattice model consists of twelf functions (a^{i}_{μ}(n_{1},n_{2},n_{3})) on the lattice, where n_{1},n_{2},n_{3} are integer numbers which enumerate the cells of the lattice.

For each configuration variable a^{i}_{μ}(n_{1},n_{2},n_{3}), we have a corresponding **momentum variable** denoted by π^{i}_{μ}(n_{1},n_{2},n_{3}).

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

(
a^{i}_{μ}(n_{1},n_{2},n_{3}),
π^{i}_{μ}(n_{1},n_{2},n_{3})
).

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 = ^{1}⁄_{2m} p^{2} + 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.