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Operators in the Canonical Quantization of a Field |
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A previous study demonstrated that for a field of a harmonic oscillator there exists an operator A and its adjoint A* such that when A*A is applied to an eigenfunction of the Hamiltonian operator H^ it returns back that eigenfunction multiplied by a nonnegative integer. Furthermore when\ A* is applied to an eigenfunction of H^ it increases the eigenvalue for that eigenfunction by one unit and A is so applied it decreases the eigenvalue by one unit. Therefore A* is called a creator operator, A an annihilator operator and A*A is a number operator.
In the literature on the quantification of fields and second quantification these matters are treated as strictly as a mattter of physics. The creator, annihilator and number operators act on the number of particles in a physical field. The nature of the field particles being augmented, diminished or counted is left unspecified. They could be the charged particles creating the field. But physical intuition indicates that they would be perturbations in the field, such as photons in an electromagnetic field. When a photon is created by a charged particle it is thought to go off on its own at the speed of light. But no matter how far it goes that photon remains a perturbation in the universe pervading field of the charged particle that created it. It remains so until it is absorbed and annihilated by some other charged particle. The previous study which concerned harmonic oscillators found the creation, annihilation and counting operators affect the number of energy quanta and there are no particles as such unless lobes of the probability density distribution could be considered as particles.
However all of the physical interpretation of the quantification of fields is misleading. The exitence of creator, annihilator and counting operators is a strictly mathematical matter, as is shown below. All the talk of particles is an unjustified interpretation of the mathematics.
Here a field is a function over space and an operator is a function that take a field function as an input and returns a field function as an output.
Let A be an operator and A* as it adjoint (conjugate) operator. Suppose the operators A and A* are such that their commutator [A, A*] is equal to the identity operator. This is called canonical quantification.
This means that
Let L be an eigenfunction of A*A; i.e.,
Now consider AL. From the above
However, from canonical quantification AA* = A*A + I so
So AL is also an eigenfunction of A*A but with a eigenvalue of (λ−1).
Likewise this means that A^{n}L is an eigenfunction of A*A with an eigenvalue of (λ−n).
Note that A*A is Hermitian since (A*A)*=A*(A*)*=A*A. The eigenvalues of the eigenfunctions of A*A must therefore be nonnegative so there must be some integer m such that (λ−m) is equal to zero. Thus the eigenvalues of all of the eigenfunctions of A*A must be nonnegative integers.
Now consider A*L. From the previous.
Thus A*L is an eigenfunction of A*A but with an eigenvalue of (λ+1).
Thus the cononical quantification of A*A results in A* being a creation operator and A being an annihilator operator. Since all of the eigenvalues of A*A are nonnegative integers A*A can be called the number operator, a sort-of counting operator.
The existence of creator, annihilator and counting operators is strictly a mathematical matter of the product of an operator and its conjugate satisfying the canonical quantification condition; i.e., the commutator of the operator and its conjugate being equal to the identity operator.
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