longer how many particles occupy a state, but whether a state is occupied or not. Another difference with the boson case is that, for fermions, the order of the states matters. If for instance the first two states ui and uj are exchanged, we get the opposite ket:
(A-11)
but it obviously does not change the physical meaning of the ket.
A-1-c. Fock space
The Fock states are the building blocks used to construct this whole chapter. We have until now considered separately the spaces S, A(N) associated with different values of the particle number N. We shall now regroup them into a single space, called the “Fock space”, using the direct sum4 formalism. For bosons:
(A-12)
and, for fermions:
(A-13)
(the sums go to infinity). In both cases, we have included on the right-hand side a first term associated with a total number of particles equal to zero. The corresponding space, S,A(0), is defined as a one-dimensional space, containing a single state called “vacuum” and denoted |0〉 or |vac〉. For bosons as well as fermions, an orthonormal basis for the Fock space can be built with the Fock states |n1, n2, …, ni, nl..〉, relaxing the constraint (A-4): the occupation numbers may then take on any (integer) values, including zeros for all, which corresponds to the vacuum ket |0〉. Linear combinations of all these basis vectors yield all the vectors of the Fock space, including linear superpositions of kets containing different particle numbers. It is not essential to attribute a physical interpretation to such superpositions since they can be considered as intermediate states of the calculation. Obviously, the Fock space contains many kets with well defined particle numbers: all those belonging to a single sub-space S(N) for bosons, or A(N) for fermions. Two kets having different particle numbers N are necessarily orthogonal; for example, all the kets having a non-zero total population are orthogonal to the vacuum state.
Comments:
(i) Contrary to the distinguishable particle case, the Fock space is not the tensor product of the spaces of states associated with particles numbered 1, 2,…, q, etc. First of all, for a fixed N, it only includes the totally symmetric (or antisymmetric) subspace of this tensor product; furthermore, the Fock space is the direct sum of such subspaces associated with each value of the particle number N.
The Fock space is, however, the tensor product of Fock spaces
(A-14)
This is because the Fock states, which are a basis for
(A-15)
It is often said that each individual state defines a “mode” of the system of identical particles. Decomposing the Fock state into a tensor product allows considering the modes as describing different and distinguishable variables. This will be useful on numerous occasions (see for example Complements BXV, DXV and EXV).
(ii) One should not confuse a Fock state with an arbitrary state of the Fock space. The occupation numbers of individual states are all well defined in a Fock state (also called “number state”), whereas an arbitrary state of the Fock space is a linear superposition of these eigenstates, with several non-zero coefficients.
A-2. Creation operators a†
Choosing a basis of individual states {|ui〉}, we now define the action in the Fock space of the creation operator5
A-2-a. Bosons
For bosons, we introduce the linear operator
As all the states of the Fock space may be obtained by a linear superposition of |n1, n2, .., ni, ..〉, the action of
Creation operators acting on the vacuum allow building occupied states. Recurrent application of (A-16), leads to:
Comment:
Why was the factor
A-2-b. Fermions
For fermions, we define the operator
where the newly created state |ui〉 appears first in the list of states in the ket on the right-hand side. If we start from a ket where the individual state |ui〉 is already occupied (ni = 1), the action of
