From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 235

that determines the time evolution of the system for any kind of Hamiltonian
functionH(p, q) by means of the Hamilton motion equations

x ̇={x, H}, p ̇={p, H},

where{·,·}is the Poisson bracket defined by the symplectic formω 0.
Peierls remarks that the phase space can be identified with the space of trajec-
tories of the system inMinduced by any given LagrangianL 0. In this sense, the
Cauchy data are not fixed at a given initial time but by the trajectories themselves,
which illustrates why it is the suitable framework for a covariant formulation. The
standard canonical approach corresponds to the choice of the singular Lagrangian
L 0 =0.
Next, Peierls introduceda Poisson structure in the space of trajectories in the
following way. Given time-dependent functionsAinTM×Rwe can consider a new
dynamical system with LagrangianLA=L 0 +λA. The trajectories of the new
dynamicsxλ(s) differ from those of that governed byL 0 .Ifwecomparethe
deviation of the trajectories with the same asymptotic values att=−∞in the
limitλ→0 we can associate to any other functionBdefined inTM×Ranew
function in the space of trajectories given by

DAB(t) = lim

λ→ 0

1

λ

[∫t

−∞

ds B(xλ(s),s)−

∫t

−∞

ds B(x 0 (s),s)

]

.

In a similar way, one can associate another function by comparing the deviation
of the trajectories with the same asymptotic values att=+∞

DAB(t) = limλ→ 0

1

λ

[∫∞

t

ds B(xλ(s),s)−

∫∞

t

ds B(x 0 (s),s)

]

. (3.64)

The Peierls bracket{·,·}L 0 is defined by

{A, B}L 0 =DAB− DAB.

It has been proven by Peierls in 1952 that, when the LagrangianL 0 is regular
it defines a Hamiltonian system, and the above bracket satisfies the following
properties:

{A, B+C}L 0 ={A, B}L 0 +{A, C}L 0 Distributive,

{A, BC}L 0 ={A, B}L 0 C+B{A, C}L 0 Leibnitz rule,

{A, B}L 0 =−{B,A}L 0 Antisymmetry,

{A,{B,C}L 0 }L 0 ={{C, A}L 0 ,B}L 0 +{{A, B}L 0 ,C}L 0 Jacobi identity,

and thus, defines a Poisson structure in the space of classical trajectories.