Why we can perform normal ordering?

[Jufa]

TL;DR Summary

In quantum field theory, when quantising the Klein Gordon field reorder some operators without caring of their commutators for the hamiltonian

As explained in the summary, it seems that the commutators of some operators (creation and anihilation) can be ignored when quantising the hamiltonian of the Klein Gordon Field. I wonder why we are allowed to do such a thing.
Is that possible because we are solely within a semiquantum (semiclassical) level? But if this was the case, how can one know which commutators can be ignored and which cannot? Or may be all of them can be ignored?
I need some help on this.

Thanks in advance.

 

[Demystifier]

The difference between "default" ordered and normal ordered free Hamiltonian is a c-number constant that does not depend on fields or any dynamic variables. This constant does not influence any physical quantity, which is the reason why we can ignore it.

 

[vanhees71]

All you need to describe quantized free fields is to realize the Poincare-Lie algebra with help of local fields, i.e., to define the Hamiltonian (energy), momentum, angular-momentum and boost operators.

You can use canonical quantization, i.e., use the Lagrangian for the KG equation, calculate the canonial field momenta and assume the equal-time commutation relations (for bosons, since trying it with anticommutation relations for fermions leads to a contradiction with microcausality of the local observables and boundedness of the Hamiltonian from below).

Then you have an operator ordering problem, and you have to define the operator ordering when multiplying field operators to get finite total energy, momentum, and angular momentum. So it is not a priori clear, how to write the densities of the said additive conserved quantities (a la Noether from Poincare symmetry) known from the corresponding classical field theory in terms of local field operators.

To have an interpretible relativistic dynamics of the quantum field theory all you need is that the corresponding total self-adjoint operators of these conserved quantities fulfill the Lie algebra of the Poincare group. So all that counts are commutation relations between these operators and thus additive contribution proportional to the unit operator are irrelevant.

Now writing down these operators in a naive way using their classical counterparts as educated guess leads to divergences, but these can be removed by subtracting an expression proportional to the unit operator. This subtraction ("renormalization") can be formalized in terms of a definition of how the operators should be ordered when calculating the products of field operators to define these conserved quantities, and that's "normal ordering", i.e., putting all creation operators to the left and all annihilation operators to the right. The result are well-defined operators of total energy, momentum, and angular momentum (as well as the generators of Lorentz boosts) obeying the commutation relations as they should in order to build a unitary representation of the proper orthochronous Poincare group, as was our goal from the very beginning.

That's of course pretty "hand waving" and it's simply a sloppy shortcut of physicists dealing with local operators and their products. There are more rigorous treatments like "causal perturbation theory" a la Epstein and Glaser. A very nice book using this approach is

G. Scharf, Finite Quantum Electrodynamics, Springer-Verlag (1989).

 

[Jufa]

The difference between "default" ordered and normal ordered free Hamiltonian is a c-number constant that does not depend on fields or any dynamic variables. This constant does not influence any physical quantity, which is the reason why we can ignore it.

Get it, thanks!

 

[Jufa]

All you need to describe quantized free fields is to realize the Poincare-Lie algebra with help of local fields, i.e., to define the Hamiltonian (energy), momentum, angular-momentum and boost operators.

You can use canonical quantization, i.e., use the Lagrangian for the KG equation, calculate the canonial field momenta and assume the equal-time commutation relations (for bosons, since trying it with anticommutation relations for fermions leads to a contradiction with microcausality of the local observables and boundedness of the Hamiltonian from below).

Then you have an operator ordering problem, and you have to define the operator ordering when multiplying field operators to get finite total energy, momentum, and angular momentum. So it is not a priori clear, how to write the densities of the said additive conserved quantities (a la Noether from Poincare symmetry) known from the corresponding classical field theory in terms of local field operators.

To have an interpretible relativistic dynamics of the quantum field theory all you need is that the corresponding total self-adjoint operators of these conserved quantities fulfill the Lie algebra of the Poincare group. So all that counts are commutation relations between these operators and thus additive contribution proportional to the unit operator are irrelevant.

Now writing down these operators in a naive way using their classical counterparts as educated guess leads to divergences, but these can be removed by subtracting an expression proportional to the unit operator. This subtraction ("renormalization") can be formalized in terms of a definition of how the operators should be ordered when calculating the products of field operators to define these conserved quantities, and that's "normal ordering", i.e., putting all creation operators to the left and all annihilation operators to the right. The result are well-defined operators of total energy, momentum, and angular momentum (as well as the generators of Lorentz boosts) obeying the commutation relations as they should in order to build a unitary representation of the proper orthochronous Poincare group, as was our goal from the very beginning.

That's of course pretty "hand waving" and it's simply a sloppy shortcut of physicists dealing with local operators and their products. There are more rigorous treatments like "causal perturbation theory" a la Epstein and Glaser. A very nice book using this approach is

G. Scharf, Finite Quantum Electrodynamics, Springer-Verlag (1989).

Many thanks for your explanation!