143
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
dent from the plane-wave expansion functions in the expansion of φ ˆ , (5.129),
which in turn originate from the fact that φ ˆ obeys the wave equation (5.121).
The latter follows from the discrete nature of the energy spectrum and the
associated operators ˆ
a, ˆ
a
† which refer to individual quanta i.e. particles.
Comment (3)
Next, we may ask: what is the meaning of the ground state |0> for a quantum
field? It is undoubtedly the state with n(k) = 0 for all k, i.e. the state with
no quanta in it – and hence no particles in it, on our new interpretation. It is
therefore the vacuum! As we shall see later, this understanding of the vacuum
as the ground state of a field system is fundamental to much of modern particle
physics – for example, to quark confinement and to the generation of mass for
the weak vector bosons. Note that although we discarded the overall (infinite)
constant in H ˆ , differences in zero-point energies can be detected; for example,
in the Casimir effect (Casimir 1948, Kitchener and Prosser 1957, Sparnaay
1958, Lamoreaux 1997, 1998). These and other aspects of the quantum field
theory vacuum are discussed in Aitchison (1985).
Comment (4)
Consider the two-particle state (5.138): |k 1 , k 2 > ∝ a ˆ
† (k 1 )ˆ a
† (k 2 )|0>. Since the
a ˆ
† operators commute, (5.130), this state is symmetric under the interchange
k 1 ↔ k 2 . This is an inevitable feature of the formalism as so far developed –
there is no possible way of distinguishing one quantum of energy from another,
and we expect the two-quantum state to be indifferent to the order in which
the quanta are put in it. However, this has an important implication for
the particle interpretation: since the state is symmetric under interchange
of the particle labels k 1 and k 2 , it must describe identical bosons. How the
formalism is modified in order to describe the antisymmetric states required
for two fermionic quanta will be discussed in section 7.2.
Comment (5)
Finally, the reader may well wonder how to connect the quantum field theory
formalism to ordinary ‘wavefunction’ quantum mechanics. The ability to see
this connection will be important in subsequent chapters and it is indeed quite
simple. Suppose we form a state containing one quantum of the φ ˆ field, with
momentum k
′ :
|k
′
> = N a ˆ
† (k
′ )|0>
(5.143)
where N is a normalization constant. Now consider the amplitude <0|φ ˆ (x, t)|k
′
>.
We expand this out as
∫
dk
<0|φ ˆ (x, t)|k
′
> = <0|
√ [ˆ a(k)e
ikx−iωt + ˆ
a
† (k)e
−ikx+iωt ]N a ˆ
† (k
′ )|0>.
2π 2ω
(5.144)
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
dent from the plane-wave expansion functions in the expansion of φ ˆ , (5.129),
which in turn originate from the fact that φ ˆ obeys the wave equation (5.121).
The latter follows from the discrete nature of the energy spectrum and the
associated operators ˆ
a, ˆ
a
† which refer to individual quanta i.e. particles.
Comment (3)
Next, we may ask: what is the meaning of the ground state |0> for a quantum
field? It is undoubtedly the state with n(k) = 0 for all k, i.e. the state with
no quanta in it – and hence no particles in it, on our new interpretation. It is
therefore the vacuum! As we shall see later, this understanding of the vacuum
as the ground state of a field system is fundamental to much of modern particle
physics – for example, to quark confinement and to the generation of mass for
the weak vector bosons. Note that although we discarded the overall (infinite)
constant in H ˆ , differences in zero-point energies can be detected; for example,
in the Casimir effect (Casimir 1948, Kitchener and Prosser 1957, Sparnaay
1958, Lamoreaux 1997, 1998). These and other aspects of the quantum field
theory vacuum are discussed in Aitchison (1985).
Comment (4)
Consider the two-particle state (5.138): |k 1 , k 2 > ∝ a ˆ
† (k 1 )ˆ a
† (k 2 )|0>. Since the
a ˆ
† operators commute, (5.130), this state is symmetric under the interchange
k 1 ↔ k 2 . This is an inevitable feature of the formalism as so far developed –
there is no possible way of distinguishing one quantum of energy from another,
and we expect the two-quantum state to be indifferent to the order in which
the quanta are put in it. However, this has an important implication for
the particle interpretation: since the state is symmetric under interchange
of the particle labels k 1 and k 2 , it must describe identical bosons. How the
formalism is modified in order to describe the antisymmetric states required
for two fermionic quanta will be discussed in section 7.2.
Comment (5)
Finally, the reader may well wonder how to connect the quantum field theory
formalism to ordinary ‘wavefunction’ quantum mechanics. The ability to see
this connection will be important in subsequent chapters and it is indeed quite
simple. Suppose we form a state containing one quantum of the φ ˆ field, with
momentum k
′ :
|k
′
> = N a ˆ
† (k
′ )|0>
(5.143)
where N is a normalization constant. Now consider the amplitude <0|φ ˆ (x, t)|k
′
>.
We expand this out as
∫
dk
<0|φ ˆ (x, t)|k
′
> = <0|
√ [ˆ a(k)e
ikx−iωt + ˆ
a
† (k)e
−ikx+iωt ]N a ˆ
† (k
′ )|0>.
2π 2ω
(5.144)