Second Quantization: Motivation & Field Operators

[Euphemia]

Hi,all
What is the motivation of using Second Quantization ?
What kind of situation does people want to use field operators ?
Euphemia

 

[atyy]

In non-relativistic quantum mechanics, second quantization is mathematically equivalent to the Schroedinger equation. Take a lattice of atoms and write the Schroedinger equation. Second quantization essentially just taking the Fourier transform, and the "particles" are Fourier modes. Since it is natural for Fourier modes to be created and destroyed, these are useful for describing "particles" which are created and destroyed. In the relativistic case, one postulates the second quantized version directly without an underlying lattice.

 

[tom.stoer]

The framework of non-rel. QM based on the Schroedinger equation is not suitabe for the description of creation and annihilation of particles, e.g. electron - positron annihilation (and production of two photons). In order to achieve that one needs 'second quantization'

 

[Demystifier]

In non-relativistic QM, second quantization is a mathematical trick useful to write many-particle wave functions in a symmetric (Bose-Einstein statistics) or antisymmetric (Fermi-Dirac statistic) form automatically.

 

[dextercioby]

There's no <second> quantization, there's only <quantization> which means passing from classical Hamiltonian observables to quantum counterparts where they exist and are uniquely defined (Groeneveld-van Hove problem).

 

[qsa]

http://en.m.wikipedia.org/wiki/Canonical_quantization

 

[coolSmith]

Calculating Occupation Numbers. It's a math trick at best.

 

[Fra]

There's no <second> quantization, there's only <quantization> which means passing from classical Hamiltonian observables to quantum counterparts where they exist and are uniquely defined (Groeneveld-van Hove problem).

Meaning you can either see it as a one time thing (and then face the question, why stop there?), or (the way I prefer it) as defining an induction step and then worry about howto interpret that tower of turtles.

http://math.ucr.edu/home/baez/nth_quantization.html contains some interesting ponderings.

/Fredrik

 

[A. Neumaier]

What is the motivation of using Second Quantization ?
What kind of situation does people want to use field operators ?

It is very basic for an understanding of quantum statistical mechanics, where almost everything gets much simpler in the second quantization formulation using fields.

Some people don't like the terminology, but it is very standard.
(But one should not imagine that it is literally a quantization of something already quantized.)