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The harmonic oscillator as a quantum field
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The harmonic oscillator has two different interpretations:

(i) It can be viewed as a single particle in 1+1 dimensions (which is 
how it is introduced in QM), where it describes a particle in 
1-dimensional space R^1 with states in L^2(R^1), bound in an external 
field and oscillating in time.

(ii) It can be viewed as a free field in 1+0-dimensions, where it 
describes an arbitrary number of noninteracting particles in 
0-dimensional R^0={0} with states in C^1=L^2(R^0).

Mathematically, it is precisely the same - physically, the 
interpretation is radically different! Here we look at the field 
interpretation. In a field interpretation, Single-particle QM is 
1+0-dimensional field theory.

For the harmonic oscillator, a^* creates one particle at a fixed 
(unmentioned) time. Since there is no space, there is no momentum to 
distinguish single particle states. 
   |0>=|> is the vacuum (ground state), 
   |1>=a^*|0> is the 1-particle state, 
   |2>=a^*|1> the 2-particle state, etc..
The frequency omega of the harmonic oscillator is the particle mass 
(setting c=1 and hbar=1): m=omega, and the N-particle state has the 
mass E_N=N*omega of N particles. The Hamiltonian is H=omega a^*a.
 
In higher dimension, each particle comes together with its quantum 
numbers (for a scalar field just the momentum p); thus there is one 
creation operator a^*(p) for each allowed quantum number p, and there 
are many 1-particle states |p>=a^*(p)|>, and even more 2-particle 
states |p',p>=a^*(p')|p>. 
 
Thus the notation is a bit different. To make the analogy perfect, 
one should assign the 1-particle state in dimension 0 a zero momentum 
and write a^*(0) for a^*, |> for the vacuum, |0> for the 1-particle 
state, |0,0> for the 2-particle state, etc., and the Hamiltonian as
   H = sum_{p in R^{0}} m a^*(p)a(p), 
which indeed equals omega a^*a, since R^{0} contains only one element.