What Does Quantizing a Field Really Mean in Quantum Field Theory?

[baka]

Hi, I am currently trying to read Chapter 2 Peskin & Schroeder (PS) QFT.

There are basics questions that bugging me so much.

1. The road from classical particle to QF.
So particle in classical mech is discrete right, and we take a continuous
aproach to make it into classical field. But what does it mean by "quantizing" the field? Do we make it into a discrete entity again, but with certain rules (commutation rule)? So why is it called a field and not particle? Does it means that we can say that particle is field and field is particle just like we see the duality of wave-particle?
So what does it mean by "single particle wave equation then?

2. Fields, Particle and Antiparticle
In the beginning of ch 2, peskin wrote that single particle relativistic wave equation give rise to +ve and -ve energy. That's why we have no right to assume that any relativistic process can be explained in terms of single particle only but rather particle and antiparticle.
But then near the end of ch 2, we get equation of pi in terms of creation and annihilation operator. Can I then interpret the field as consist of particle and antiparticle which is represented by creation and anihilation operator?

Thanks

 

[dextercioby]

Everything is just mathematics.Physics comes in and says:"hey,we've got a particle,an electron and another one a photon,you theorists do some mathematics and come up with a formula that fits my (experimental) results the best".

The most important concept in physics is the MODEL.The theoretical physicist has to create the best mathematical model possible whose predictions are confirmed by exp.The model of quantum field is simple:
One takes a classical field and applies the postulates of quantum mechanics to it.

Quantum statistical mechanics allows us to define uniparticle and multiparticle states once the second quantization is made (which means introducing creation+annihilation ops,Fock spaces,Heisenberg algebras,no.of occupation description of quantum states).

As for particle vs. antiparticle,do the classical description of the field and then quantize the Noether charge that comes from U(1) global invariance.

Daniel.

 

[BlackBaron]

I'm asking similar questions myself.
This is the answer I've come some far:

As dextercioby said,

The model of quantum field is simple:
One takes a classical field and applies the postulates of quantum mechanics to it.

It turns out that the energy spectrum of such a quantum field is discrete, i.e. it's composed by a integer number of "clusters" (or, as they are commonly called, "quantums of energy"). We interpret each of those quantums as particles (as dextercioby said, this is just a model).
This model has nothing to do with the wave-particle duality.
Creation and anhilation operators raise or lower, respectively, the number of quantums, so, as their name suggest, we interpret them as "creators" and "anhilators" of particles.
The fact that we can write the fields in terms of creation and anhilation operators shows that those fields merely describe where/when/how particles appear and dissapear. Surprisingly enough, that's enough to know their behavior.
Finally, it's a very interesting and ilustrating exercise to use QFT to calculate "the probability amplitude of a single particle being at the point x" and show that such amplitude behaves exactly as the single-particle wave-function of regular QM. For example, the single-particle wavefunctions coming from Klein-Gordon and Dirac fields obey, respectively, Klein-Gordon and Dirac's equations (which are, more or less, the relativistic extension of Schrödinger and Pauli's equations). I don't remember the exact details of the procedure, but I think it's done in a QFT book by Roman.

 

[CarlB]

Read the first chapter in the book by Zee, "Quantum Field Theory in a Nutshell", for a more explicit explanation of what the connection between QFT, QM and classical mechanics is.

If you're hard up for cash, and can't find it in your local library, you might try reading it in the bookstore. Just wash your hands first and don't open so wide that it ruins that "new book" look.

Carl