Creation and Annnihilator Operators: generality and meaning

Question

I am studying my fisrst course in quantum mechanichs where we treated the example of the Harmonic Oscillator through the Weyl Heisenberg Spectrum Generating Algebra Method. In that context we defined creation adn annihilation operators as $$a^+=\sqrt{\frac{m\omega}{2h}}(x-i\frac{p}{2m})\ \ \ \ \ \ \ a=\sqrt{\frac{m\omega}{2h}}(x+i\frac{p}{2m})$$

Moreover we searched minimum uncertainty states (coherent states) for this system as the eigenstates of $a$. Searching around I found that this is not a case, since one can charachterize states minimizing uncertainty for a couple of operators $X,P$ in the eigenstates of an operator of the form $(X+i\delta P)$ with $\delta\in\mathbb{R}$. An operator showing a structure analogous to the annihilation one.

Finally, I glared by chance at Wick quantization, where one, instead of quantizing operators $X,P$ quantizes polynomial expressions of the form $(X+i\delta P)^i(X-i\delta P)^j$.

Then I wonder:

• Is the definition of creation annihilation operators 'canonically' extendable to any couple of operators?
• Are the two instances where I encountered somehow related?
• What is their deep meaning in the theory?

Answer 1

A creation operator creates a particle. An annihilation operator destroys it. Thus it it not any two expressions of operators that can do the job. One has to define the field on which these operators work, in your case the solutions of the harmonic oscillator, not any two operators.

In quantum field theory as used in high energy physics , one defines the field as the solution of the corresponding quantum equation with no potential, the free particle, plane wave solution.

For example for the electron: the field covers the whole of space, and the function at each point in spacetime is the plane wave in the Dirac equation. The creation operator creates an electron at (x,y,z,t) and the annihilation annihilates it , then one can to the (x+dx.y+dy,z+dz,t+dt) point and repeat the process thus propagating an electron. For real particles one cannot use the plane wave because it has infinite extent, one uses wave packets, but that is another story.

The use comes with writing interactions with Feynman diagrams, which allow to calculate perturbatively the crossection and decays of many body systems. There are specific one-to-one rules for the integrations that have to take place.

I do not know about the Wick quantization , but it must have a one to one mathematical correspondence with the usual field quantization.

You could have a look at this link .