QFT: Justifying Position-Momentum Commutation Relation

[jack47]

I'm taking a QFT course and there is something that is bugging me. When we write out the equations in terms of generalised coordinates (q and p like in Lagrangian mechanics), it is always then taken as obvious that q and p satisfy the position momentum commutation relation. They arn't position and momentum so is there any prior justification for using the given commutator? (apart from the fact that when we do use it we get a nice working theory)

 

[dextercioby]

One quantizes the classical field theory in which the graded fundamental Poisson brackets at equal times will become graded Lie brackets at equal times.

So they don't really come from nowhere.

Daniel.

 

[R0man]

The justification for the position-momentum commutation relation in quantum field theory (QFT) can be traced back to the fundamental principles of quantum mechanics. In classical mechanics, the position and momentum of a particle are treated as independent variables, and their values can be measured simultaneously with arbitrary precision. However, in quantum mechanics, the Heisenberg uncertainty principle states that the more precisely we measure the position of a particle, the less precisely we can know its momentum, and vice versa. This means that there is a fundamental uncertainty in the simultaneous measurement of position and momentum in quantum systems.

In QFT, the position and momentum operators are represented by the canonical conjugate variables q and p, respectively. These operators do not commute, which means that their order in a given expression matters. This is reflected in the commutation relation [q,p]=iħ, where ħ is the reduced Planck constant. This relation is a direct consequence of the uncertainty principle and is a fundamental property of quantum systems.

Furthermore, in QFT, the position and momentum operators play a crucial role in the quantization of fields, which is a key step in constructing a quantum field theory. The position operator is associated with the field amplitude, while the momentum operator is associated with the field momentum. The commutation relation between these operators is crucial in ensuring that the resulting quantum field theory is consistent and does not violate the fundamental principles of quantum mechanics.

In summary, the justification for the position-momentum commutation relation in QFT lies in the fundamental principles of quantum mechanics and its crucial role in the quantization of fields. While it may seem arbitrary at first, the commutation relation is a necessary consequence of the uncertainty principle and plays a fundamental role in the development of QFT as a working theory.