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(1,ψ(t))2 =cos2 , (2,ψ(t))2 =sin2 , (67)
| | ~ | | ~
respectively. This results in an oscillation of the transition probabilities.
Let us shortly mention one particular realization of a two-state system which, among many
others, has been discussed in the Feynman lectures [39]. Consider an ammonia (NH ) molecule.
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If one fixes the plane spanned by the three hydrogen atoms, one observes two possible spatial
configurations ,1) and ,2), corresponding to position of the nitrogen atom in the lower or the
upper hemisphere, respectively (cf. Fig. 2). The nondiagonal elements of the Hamiltonian H =
12
H = A correspond to a nonvanishing transition probability from one such configuration into
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−
the other. If the ammonia has been originally in state ,1), it will constantly swing back and forth
between the two states, with a probability given by equations (67).
B From single to multiple quanta — “second” field quanti-
zation
The quantum formalism introduced in the main text is about single quantized objects. What if
one wants to consider many such objects? Do we have to add assumptions in order to treat such
multi-particle, multi-quanta systems appropriately?
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The answer is yes. Experiment and theoretical reasoning (the representation theory of the
Lorentzgroup[93]andthespin-statisticstheorem[54,66,20,52])indicatethatthereare(atleast)
two basic types of states (quanta, particles): bosonic and fermionic states. Bosonic states have
what is called “integer spin;” i.e., s =0,~,2~,3~,..., whereas fermionic states have “half-integer
b
1~ ,3~ ,5~
spin;” s = .... Most important, they are characterized by the way identical copies of
f 2 2 2
them can be “brought together.” Consider two boxes, one for identical bosons, say photons, the
other one for identical fermions, say electrons. For the first, bosonic, box, the probability that
anotheridenticalbosonisaddedincreases with the number of identical bosonswhicharealreadyin
thebox. Thereisatendencyofbosonsto“condensate”intothesamestate. Thesecond,fermionic
box, behaves quite differently. If it is already occupied by one fermion, another identical fermion
cannot enter. This is expressed in the Pauli exclusion principle: A system of fermions can never
occupy a configuration of individual states in which two individual states are identical.
How can the bose condensation and the Pauli exclusion principle be implemented? There are
several forms of implementation (e.g., fermionic behavior via Slater-determinants), but the most
compact and widely practiced form uses operator algebra. In the following we shall present this
formalism in the context of quantum field theory [50, 64, 54, 66, 20, 52, 43].
AclassicalfieldcanberepresentedbyitsFouriertransform(“ ”standsforcomplexconjugation)
∗
A(x,t) = A(+)(x,t)+A( )(x,t) (68)
−
A(+)(x,t) = [A( )(x,t)] (69)
− ∗
A(+)(x,t) = a ki,siu ki,si(x)e −iωkit , (70)
k Xi,si
where ν = ω /2π stands for the frequency in the field mode labeled by momentum k and s is
ki i i
some observable such as spin or polarization. u stands for the polarization vector (spinor) at
ki,si
k ,s , and, most important with regardsto the quantized case, complex-valued Fourier coefficients
i i
a C.
ki,si
∈
From now on, the k ,s -mode will be abbreviated by the symbol i; i.e., 1 k ,s , 2 k ,s ,
i i 1 1 2 2
≡ ≡
3 k ,s , ..., i k ,s , ....
3 3 i i
≡ ≡
In (second16) quantization, the classical Fourier coefficients a become re-interpreted as opera-
i
tors, which obey the following algebraic rules (scalars would not do the trick). For bosonic fields
(e.g., for the electromagnetic field), the commutator relations are (“ ” stands for self-adjointness):
†
a i,a†j = a ia†j −a†ja =δ , (71)
i ij
h i
[a i,a j] = a†i,a†j =0 . (72)
h i
For fermionic fields (e.g., for the electron field), the anti-commutator relations are:
{a i,a†j} = a ia†j +a†ja =δ , (73)
i ij
{a i,a = {a†i,a†j}=0 . (74)
j
}
The anti-commutator relations, in particular {a†j,a†j} = 2(a†j)2 = 0, are just a formal expression
of the Pauli exclusion principle stating that, unlike bosons, two or more identical fermions cannot
co-exist.
The operators a†i and a are called creation and annihilation operators, respectively. This
i
terminology suggests itself if one introduces Fock states and the occupation number formalism. a†i
and a are applied to Fock states to following effect.
i
TheFockspaceassociatedwithaquantizedfieldwillbethedirectproductofallHilbertspaces
H ; i.e.,
i
H , (75)
i
i I
Y∈
where I is an index set characterizing all different field modes labeled by i. Each boson (photon)
field mode is equivalent to a harmonic oscillator [43, 65]; each fermion (electron, proton, neutron)
field mode is equivalent to the Larmor precession of an electron spin.
16ofcourse,thereisonly“theoneandonly”quantization, theterm“second” oftenreferstooperatortechniques
formultiquantasystems;i.e.,quantum fieldtheory
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