Second Quantization: What Is It?

[hagopbul]

hello all

i want to know what is the second quantization?

 

[haushofer]

Second quantization is roughly the procedure to go from a classical (non-quantum!) relativistic field theory to a quantum field theory.

A classical field can loosely be seen as a continuous map from space time to some configuration space. In relativistic theories the equations of motion are governed by relativistic expressions.The question then is how to promote this theory to a quantum theory.

One argumentation is to consider the meaning of space and time in ordinary quantum mechanics. Position is represented by an operator, while time is a label which labels the evolution of a system. Relativistically, this seems strange: time and space are treated on equal footing in relativistic theories.

One way is to state that position is also a label, just like time. The operator is then the field itself. This is nice, because with this operator we can then act on the vacuum to "create" and "annihilate" particles, which we know phenomenologically a good relativistic quantum theory should be able to describe! I personally never saw the treatment of promoting time to be an operator, but this makes things much more complex i suppose; I'm not sure if it would give an equivalent theory.

The way to do this is called "second quantization". One expands the field in a Fourier expansion, and the coefficients are then the creation/annihilation operators.