Quantum Field Theory I

1.2. MANY-BODY QUANTUM MECHANICS 17
1.2 Many-Body Quantum Mechanics
The main charactersof QFT are quantum fields or perhaps creation and annihilation
operators (since the quantum fields are some specific linear combinations
of the creation and annihilation operators). In most of the available textbooks
on QFT, the creation and annihilation operators are introduced in the process
of the so-calledcanonical quantization 27 . This, however, is not the most natural
way. In opinion of the present author, it may be even bewildering, as it may
distort the student’s picture of relative importance of basic ingredients of QFT
(e.g. by overemphasizing the role of the second quantization). The aim of this
second introduction is to present a more natural definition of the creation and
annihilation operators, and to demonstrate their main virtues.
1.2.1 Fock space, creation and annihilation operators
Fock space
1-particle system
the states constitute a Hilbert space H 1 with an orthonormal basis |i〉, i ∈ N
2-particle system 28
the states constitute the Hilbert space H 2 or HB 2 or H2 F , with the basis |i,j〉
non-identical particles H 2 = H 1 ⊗H 1 |i,j〉 = |i〉⊗|j〉
identical bosons HB 2 ⊂ H1 ⊗H 1 |i,j〉 = √ 1 2
(|i〉⊗|j〉+|j〉⊗|i〉)
identical fermions HF 2 ⊂ H1 ⊗H 1 |i,j〉 = √ 1 2
(|i〉⊗|j〉−|j〉⊗|i〉)
n-particle system (identical particles)
the Hilbert space is either HB n or Hn F ⊂ H1 ⊗...⊗H
} {{ 1 , with the basis
}
n
|i,j,...,k〉 = 1 √
n!
∑
permutations
where p is the parity of the permutation, the upper
lower
(±1) p |i〉⊗|j〉⊗...⊗|k〉
} {{ }
n
sign applies to
bosons
fermions
0-particle system
1-dimensional Hilbert space H 0 with the basis vector |0〉 (no particles, vacuum)
Fock space
direct sum of the bosonic or fermionic n-particle spaces
∞⊕
∞⊕
H B = HB n H F =
n=0
27 There are exceptions. In the Weinberg’s book the creation and annihilation operators are
introduced exactly in the spirit we are going to adopt in this section. The same philosophy is
to be found in some books on many-particle quantum mechanics. On the other hand, some
QFT textbooks avoid the creation and annihilation operators completely, sticking exclusively
to the path integral formalism.
28 This is the keystone of the whole structure. Once it is really understood, the rest follows
smoothly. To achieve a solid grasp of the point, the reader may wish to consult the couple of
remarks following the definition of the Fock space.
n=0
H n F