Poisson brackets

In Chapter 3 we mentioned Hamilton and his view that time could be discussed in terms of algebra. Hamilton’s equations allow us to do just that and vindicate his assertion. Suppose f (p, q, t) is some time-dependent function over phase space. Then as we move along a dynamical trajectory, we find

where {f, g}PB is the Poisson bracket of any two functions f, g over phase space, define by

37 A dynamical trajectory is one that satisfies Hamilton’s equations of motion.

An immediate conclusion is that if we choose f to be the Hamiltonian itself, dH dH

then we deduce - = -: if a Hamiltonian is explicitly independent of time,

dt c d t d H

that is,- = 0, then the Hamiltonian is constant over any dynamical trajectory.

d t

The use of Poisson brackets to discuss evolution in time has an enormous advantage. Just after QM was discovered at the end of the first quarter of the twentieth century, Dirac showed that canonical quantization, the process of finding quantum mechanical version of a classical mechanical theory, could be understood as a replacement of Poisson brackets in the classical theory by commutators of operators in the quantum theory [Dirac, 1925], that is, pq-qpi = ih pi, qj}PB, and so on, where pi is the quantum operator corresponding to the classical variable pi, and so forth.

The algebraic approach using Poisson brackets was also the technology that Dirac used in his development of constraint mechanics [Dirac, 1964]: this has been applied to all fundamental theories in particle physics, general relativity, and quantum cosmology.

An important refinement of these ideas was the formulation of the so-called equation of small disturbances [DeWitt, 1965] and unequal-time Poisson brackets [Peierls, 1952]. This set the scene for unequal-time commutation relations in relativistic quantum field theory, currently the most successful technology available to study the physics of time and space.

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