# Field Operators in Klein-Gordon theory

1. Aug 12, 2008

### philipp_w

Currently I am working through a script concerning QFT. To introduce the concept of canonical filed quantisation one starts with the (complex valued) Klein-Gordon field. I think the conept of quantising fields is clear to me but I am not sure if one can claim that the equations of motion for the field operators have to fulfil same equation as the KG field does.

Mostly this is done in textbooks by pure calculating (see Peskin/Schroeder sec. 2.4). Is this somewhat of luck or this is not everytime the case that fieldoperators do not fulfil the equations of motions of their corresponding (classical) fields?

Or is it always allowed to demand the same equations of motions as the fields would fulfil?

(look here

http://theorie.physik.uni-giessen.de/~hees/publ/lect.pdf

formula 3.72)

maybe someone understood what I was trying to say and has some help for me. thx

2. Aug 12, 2008

### philipp_w

Am I right in the assumption that this is a sort of some natural requirement to those field operators fullfiling the equations of motions? Because since we started with our intuitive wave-partivle thinking from the Schrödinger theory to predict a relativistic wave equation, we got stucked in some trouble concerning the negative energy states and the interpretation to one particle theory was no longer possible.

Thats why we moved on to multi-particle description with the help of canonical fieldquantisaiton, but should we give up our aim of relativistic one-particle equation? because, what is the interpretation of this field operator? Doesnt create it a particle at position x? so the physics should stay the same, in the sense that this creation-opertor has to fulfil the KG equation from its interpreation as a creation-operator.

I think the effect is just that we are now able to understand better those obscure negative energy states, since we are now "living" in the fock-space with multi-particles and are not longer bounded to the interpretation of one-particle wave function.

But the physics remains the same, we have found a relativistic one-particle (operator-valued)waveequation? Or what else is the interpretation of field operators fulfilling the KG equations?

(the Gupta-Bleuer method came to my mind, where the quantisized vector potential isnt equal to the lorentzgauged field, means the divergence of the fieldop. is not the zero valued operator in fock space, in fact they are not really operators)

--> concerning all that I am now a little bit confused

3. Aug 12, 2008

### Avodyne

A classical field obeys the Hamilton-Jacobi equation of motion

$${\partial\varphi\over\partial t}=\{\varphi,H\},$$

where $\{\cdot,\cdot\}$ is the Poisson bracket. A quantum field obeys the Heisenberg equation of motion

$${\partial\varphi\over\partial t}={1\over i\hbar}[\varphi,H],$$

where $[\cdot,\cdot]$ is the commutator.

We quantize a theory by postulating that the classical coordiates and momenta $q, p$ are replaced by quantum operators $Q, P$ that obey

$$[Q,P]={1\over i\hbar}\{q,p\}.$$

For a hamiltonian that does not have any operator-ordering ambiguities, the quantum equation will be the same as the classical equation.

4. Aug 13, 2008

### philipp_w

thats my problem

5. Aug 13, 2008

### Avodyne

6. Aug 13, 2008