1. Introduction
Both Heisenberg (HP) and Schrödinger pictures (SP) are used in quantum theory. Schrödinger solved Schrö- dinger eigenvalue equation for a hydrogen atom, and obtained the atomic energy levels. Heisenberg discussed the uncertainty principle based on the fundamental commutation relations. Both pictures are equivalent in dealing with a one-electron system. In dealing with many electrons or many photons a theory must be developed in the HP, incorporating the indistinguishability and Pauli’s exclusion principle. A quantum theory must give a classical result in some limit. We will see that this limit is represented by
. The HP, and not the SP, give the correct results for a many-particle system. The quantum field equation is nonlinear if a pair interaction exists.
2. One Electron Systems
We consider an electron in a potential energy field
, where 
is a position vector. In the Cartesian representation
(1)
The canonical momentum 
is
(2)
The Hamiltonian 
of the system is
(3)
In the HP the coordinates and momenta 
are regarded as Hermitean operators satisfying the fundamental commutation relations (quantum conditions):
(4)
where 
is Kronecker’s delta
(5)
and
, 
Planck constant.
The equations of motion for 
and 
are
(6)
The two equations can be included in a single equation:
(7)
where 
represent any physical observable made out of the components of the position 
and momentum
. The angular momentum 
can be included also. Dirac has shown that in the small 
limit:
(8)
where
(9)
is the classical Poisson brackets.
In the SP we use the equivalence relations:
(10)
and write down the Schrödinger wave equation as
(11)
The function 
is called the wave function. Normally, it is normalized such that
(12)
where 
is a normalization volume. The quantum average of an observable 
is defined by
(13)
If we use Dirac’s ket and bra notations, then we can see the theoretical structures more compactly [1] . The quantum state 
is represented by the ket vector 
or the bra vector
. The Schrödinger equation of motion is
(14)
whose Hermitean conjugate is
(15)
If we use the position representation and write
(16)
then we obtain Equation (11) from Equation (14).
We introduce the density operator 
defined by
(17)
Using Equations (14) and (15), we obtain
(18)
This equation, called the quantum Liouville equation, has a reversed sign compared with the equation of motion for
, see Equation (7).
We can express the quantum average 
of an observable 
as
(19)
where 
denotes a one-particle trace. Operators under a trace commute.
We assume that the Hamiltonian 
in Equation (11) is a constant of motion. Then, Equation (14) can be reduced to the energy 
eigenvalue equation:
(20)
after using a separation of variable method for solving Equation (11). Equation (20) is known as the Schrödinger energy-eigenvalue equation. The hydrogen atom energy-levels can be obtained from Equation (20) with
, 
permittivity.
Except for simple systems such as free electrons and simple harmonic oscillators, the Heisenberg equation of motion (7) [or the quantum Liouville Equation (18)] are harder to solve. This is so because the numbers of unknowns in the 
matrix are more numerous than in the 
vector.
3. Difficulties with the SP
The following items have difficulties in the SP. They cannot be addressed properly.
(a) The Classical Mechanical Limit 
Dirac showed that the fundamental commutation relations (8) can also be applied to a many particle system only if the Cartesian coordinates and momenta are used. The equation of motion (7) in the HP can be reduced to the classical equation of motion:
(21)
in the classical limit
(22)
The Schrödinger equation of motion (11) does not have such a simple limit.
(b) Indistinguishability All electrons are identical (indistinguishable) to each other. This is known as the indistinguishability. This property can be stated as follows:
Consider a system of 
electrons interacting with each other characterized by the Hamiltonian:
(23)
where 
is the kinetic energy and 
is the pair interaction energy. Here the upper indices 
and 
denote the electrons. The indistinguishability requires that
(24)
where 
are the permutation operators. For a three-particle system the permutation operators are
(25)
The order of the permutation group for an 
-particle system is
. The total momentum, the total angular momentum, and the total mass satisfy the same Equation (24). We may express this by
(26)
(c) Boson Creation and Annihilation Photons are bosons with full spin. They can be created and annihilated spontaneously. These processes can only be described by using creation and annihilation operators 
both of which move, following the Heisenberg equations of motion. One can no more limit the number of bosons in the system.
(d) The Second Quantization for Bosons Bosons can be treated using second-quantized operators 
satisfying the Bose commutation rules:
(27)
where 
indicates particle states. Both operators 
and 
move, following the Heisenberg equations of motion, e.g.
(28)
where 
is a many-boson Hamiltonian. The Hamiltonian for free photons is given by
(29)
where 
is angular frequency and 
denotes the polarization indices.
(e) The Second Quantization for Fermions Many fermions can be treated by using the complex dynamical operators 
satisfying the Fermi anticommutation rules:
(30)
Both operators 
and 
move, following the Heisenberg equations of motion.
If the system contains many electrons, then we must consider Pauli’s exclusion principle that no more than one fermion can occupy the same particle state. This is a restriction which cannot be described without considering permutation symmetry.
(f) Holes Dirac showed [1] that there is symmetry between the occupied and unoccupied states for fermions, based on second quantization calculations. Holes are as much physical particles as electrons, and are fermions.
All six properties (a) - (f) can be discussed in the HP, but not in the SP. The last five (b) - (f) concern many-particle systems.
4. Discussion
We first discuss two relevant topics.
(a) Wave packets Dirac assumed [1] that an experimentally observed particle correspond to a wave packet composed of the quantum waves, and showed that any wave packet moves obeying the classical mechanical laws of motion.
(b) The classical statistical limit Free fermions (bosons) in equilibrium obey the Fermi (Bose) distribution law:
(31)
where 
is the kinetic energy, 
the Boltzmann constant, 
the absolute temperature and 
the chemical potential; the upper (lower) signes correspond to the Fermi (Bose) distribution functions. In the classical statistical limit, which is realized in either low density 
limit or high temperature 
limit, both distribution functions 
approach the classical Boltzmann distribution function:
(32)
(33)
For illustration we consider a free electron model for a metallic body-centred cubic (bcc) crystal such as sodium. Electrons subject to the exclusion principle are fermions which obey the Fermi distribution law. The heat capacity 
at the low temperatures 
shows a 
-linear behavior. Phonons which are quanta of the lattice vibrations are bosons and they obey the Planck distribution law:
(34)
since the chemical potential 
for phonons. The heat capacity arising from the phonons at low temperatures shows Debye 
-law [2] . The electron wave packets have a linear size of the order of the lattice constant of the bcc crystal. The average phonon size is much greater, and is distributed with the Planck’s law.
The effects of quantum statistics which involve 
permutations, see Equation (24), are much stronger than the effects of quantum entangling that grows linearly with
.
Let us introduce boson field operators 
and 
which satisfy the Bose commutation rules:
(35)
where
(36)
and 
is Dirac’s delta-function.
A quantum many-boson Hamiltonian 
corresponding to the Hamiltonian in Equation (23) is
(37)
where 
are the field operators satisfying the equal-time commutation rules (35). The field equation is obtained from the Heisenberg Equation (28) as follows:
(38)
where 
is the velocity vector. We note that the field equation is nonlinear in the presence of a pair potential. This equation can be used to derive, and obtain the time dependent version of the Ginzburg-Landau (GL) equation [3] . This topic will be treated separately.
For a many-fermion system, fermion field operators 
and
, satisfying the Fermi anticommutaion rules are introduced. The field equation is given by
(39)
This equation is also nonlinear in the presence of a pair potential
.