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planetology
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Why is it that when normal ordering the terms in the Hamiltonian for bosons, the commutation rules are ignored, but when normal ordering fermion operators the anti-commutation rules are used to justify a change in sign?
Originally posted by Tom
Here is an easy-to-read document that will be of some assistance:
http://xxx.lanl.gov/pdf/physics/0212061
I will get back to this later with a more definitive post.
Normal ordering is a mathematical operation that rearranges the creation and annihilation operators of bosons and fermions in a specific order in order to obtain the correct expectation value of a quantum mechanical operator.
The main difference between normal ordering for bosons and fermions is the commutation and anti-commutation rules that govern their creation and annihilation operators. In bosonic systems, the operators commute, while in fermionic systems, they anti-commute. This results in different normal ordering procedures for the two types of particles.
Normal ordering is important because it allows us to correctly calculate the expectation values of operators in quantum mechanical systems. It also helps us to properly account for the symmetries and anti-symmetries of bosonic and fermionic systems.
Normal ordering is performed by rearranging the creation and annihilation operators according to their commutation or anti-commutation rules. For bosonic systems, the operators are rearranged in a specific order, while for fermionic systems, the operators are rearranged in pairs with a minus sign in between.
Yes, normal ordering is used in many areas of physics, including quantum field theory, statistical mechanics, and condensed matter physics. It is also used in the development of quantum algorithms and in the study of quantum information and computation.