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Quantum Field Theory - Why and When?
The quantum mechanics that you have studied so far has dealt with systems
with a fixed number of ”particles”. In many cases these ”particles” are usual
particles like electrons or nuclei, but they can also be single spins on an atom, or
the polarization states of light. What is common for these systems is that one
can consider the ”particles” one by one - one electron, one single spin etc., and
that more complicated systems can be built by combining a fixed number of
these particles (from now one I will skip the quotation marks, since the point
has been made).1 For instance, combining electrons and nuclei, one builds
atoms, and by combining many spins you get a magnet. For these systems to
be interesting and realistic, one must usually2 introduce interactions between
the particles, and the interacting system can have properties very different
from the constitutent parts. From negatively charged pointlike electrons, and
positively charged nuclei with radii of a few Fermi (1fm = 10−15 m), one can
form neutral atoms with radii of an ˚
Angstrom (1˚
A= 10−10 m) or more. In
spite of all complications of atomic physics, it is nevertheless true that the
atom can be viewed as composed of its original, unbound constituents, and we
can imagine a process where a stripped nucleus, successively captures electrons
in order to form an atom. Quantum mechanically, an atom with N electrons
can be described by a wave function of the form3
~ ~r1 . . . ~rN ) =
Ψatom (R,
X
~ n1 (~r1 ) . . . ψnn (~rn )
CN,n1 ,...nn ψN (R)ψ
(1)
N,n1 ,...nn
~ is the position of the nucleus, and ~ri the
where CN,n1 ,...nn are constants R
th
position of the i electron. Mathematically, this amounts to saying that the
Hilbert space of the composite system - the atom - is the (direct) product
space of the Hilbert spaces of all the constituents.
A problem occurs when we attempt to describe processes where the number
of particles changes as a result of the interactions. The example of atom
formation from nuclei and electrons, alluded to above, is a good illustration.
When an electron is captured by a positive ion, it emits energy in the form
of electromagnetic radiation or, equivalently, by the emission of one or several
1
In the following we shall often refer to the concept ”degree of freedom, or d.o.f.”. Note
however that a single particle can be associated with several d.o.f. depending on its spin
and the dimension of space in which it moves. A proper definition of d.o.f. will come later.
2
But not always, light consisting of non-interacting photons, and hot crystals described
by a gas of non interacting phonons are important examples.
3
Well, this is a bit oversimplified since I forgot about the spin, but that is not the main
point here.
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photons. Quantum mechanically, such a process, involving an ion with unit
positive charge, is schematically described by4
|ion, +i ⊗ |electron, −i → |atom, 0i ⊗ |γ1 , γ2 , . . . γn i
(2)
where the arrow indicates time-evolution, and the last ket denotes a state of n
photons. This means that the Hilbert space before and after the interaction is
different, and we cannot expand the state vector of the final state in terms of
products of states describing the electrons and the nucleus only. Although new
particles are created, the basic laws of physics still holds: energy, momentum,
angular momentum and charge is conserved in the process. It looks, however,
as if a completely new degree of freedom, that of the photon, has been created
out of nothing! In no reasonable way can we think of the photon as ”consisting
of” electrons and nuclei. We can also think of the inverse process i.e. that of
photoionization. Here a photon is absorbed by an atom which then dissociates
into an ion and a free electron. In this case it looks like a degree of freedom
has disappeared!
In elementary particle physics this phenomenon - the creation and destruction of particles - is rule rather than exception. Figure 1 shows a picture of
a high energy collision, where pions are coming in from the left and hitting
hydrogen nuclei, i.e. protons, and creating showers of new particles. Just
as the energy needed to create the photons in the case of electron capture
was provided by the kinetic and potential energy of the electron-ion system,
the energy to create all the new particles is provided by the kinetic energy of
the incoming pions. Since in this case the created particles are massive, it is
important to use the relativistic expression E 2 = m2 c4 + p2 c2 for the energy.
To describe processes where particles are created and destroyed, the space
of quantum mechanical states, or the Hilbert space, must contain states with
an arbitrary number of particles. Such a Hilbert space is called a Fock space,
and mathematically it is a (direct) sum of all the different Hilbert spaces with
fixed number of particles,
F=
∞
M
Hn ,
(3)
n=0
and in general there is one such Fock space for each species of particles. A
creation process corresponds to a transition from a vector |ni ∈ Hn to a vector
4
The outer product |ai ⊗ |bi between two state vectors |ai and |bi is the state vector
equivalent of multiplying the two wave functions ψa and ψb . This is not to be confused
with the inner product ha|bi. An outer product expresses that a system consists of two
independent parts.
2
Figure 1:
Bubble chamber picture of elementary particle reaction. You can find this
picture, and many others, together with some explanatory text at the multimedia webpage
of CERN, http://multimedia-gallery.web.cern.ch/multimedia-gallery/
|n + 1i ∈ Hn+1 . Mathematically this is described by a creation operator a† ,
|n + 1i ∼ a†α |ni
(4)
where the label α denotes the quantum numbers of the created particle and is
usually taken to be some conserved quantities such as momentum or angular
momentum
It is no coincidence that the same symbol, a† , is used for the creation
operator and the raising operator familiar from the harmonic oscillator. In
fact the creation operators satisfy the same commutation relation,
[aα , a†α0 ] = δα,α0 ,
(5)
and all you know about the harmonic oscillator can be directly applied to
creation and annihilation processes by interpreting the n-state |ni for the harmonic oscillatoras a state containing n particles rather than being a state of
energy n¯
hω.
In a translationally invariant system, momentum ~k is a good quantum
number, and it would seem natural to consider operators a~†k creating a particle with momentum ~k. On the other hand, particle creation takes place in
some limited region of space, typically in the intersection region of two colliding particle beams, so from this point of view it seems natural to consider
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creation of particles at a specific point ~r. The uncertainty principle tells us
that a particle confined to a very small region has very high average momentum, and thus very high energy. Nevertheless, just as the idealized position
eigenstates |~ri are useful in describing quantum mechanical particles, it is useful to consider the process of creating a particle at a point, ~r, in space. In this
case it is conventional not to write a~†r , but to denote the creation and annihilation operators by ϕˆ(+) (~r) and ϕˆ(−) (~r), respectively.5 As you might expect,
from analogy with ordinary quantum mechanics, these are related to a~†k and
a~k by a Fourier transformation, but all this will be explained in detail later.
The quantum field defined by
ϕ(~
ˆ r) = ϕˆ(+) (~r) + ϕˆ(−) (~r)
(6)
is thus an operator that creates or destroys a particle at a certain point. As
just mentioned, the quantum field ϕ(~
ˆ r) is the (Fourier transform of the) sum of
a† and a and remembering that for the harmonic oscillator we have xˆ ∼ a† + a,
we see that we can think of ϕ(~
ˆ r) as analogous to position, and below we shall
see that there is also a corresponding velocity, or momentum, field operator
π
ˆ (~r). In this analogy, you should think of position and momenta in the sense
of satisfying canonical commutation relations of the form [ˆ
x, pˆ] = i¯h; it has
nothing to do with real space, ~r or real momentum. In quantum field theory,
the coordinate ~r is just a label on the quantum variable ϕ,
ˆ just as the particle
index i, can label the positions ~ri of the particles.
At this point you should realize that the analogue between the particle
number states and the states of a harmonic oscillator is valid only if the particles are bosons - only then is it possible to have more than one particle in the
same quantum state. To describe fermions we must use creation operators, b†α
(it is conventional to use b† instead of a† in the case of fermions) that satisfies
b† b† = 0 implying b† |1i = 0 which leaves us with the states |0i and |1i only.
This relation is in fact a special case of the anti-commutation relation,
{bα , bα0 } = {b†α , b†α0 } = 0 ,
(7)
and there is also a counterpart to (5),
{bα , b†α0 } = δα,α0 .
(8)
As we shall se later, fermions can only be produced in pairs, since the fermion
number, i.e. the the total number of fermions minus the total number of antifermions, turns out to be a conserved quantum number. The details of how
5
Here I use the usual convention to put hats on operators, except for the creation and
annihilation operators a, a† etc. which are alway understood to be operators.
4
to make use of these anti-commuting operators to describe fermions will be
explained later in the course.
Remembering that the Maxwell theory of electrodynamics is Lorentz invariant, and the examples given earlier, you might get the impression that
quantum field theory is of use only in relativistic theories. This is not true!
In many condensed matter systems, the theoretical description is in terms of
elementary excitations or quasiparticles. The perhaps simplest example, and
the one treated in some detail in the first chapter of the textbook, is that of
phonons in a crystal. A phonon is a quantum of sound, just as the photon
is a quantum of light. The difference is that while photons are quanta of the
electromagnetic field which can exist and propagate through empty space,6
phonons are quanta of the crystal displacement field7 which obviously can exist only inside the crystal, just as a bubble can exist only inside a liquid. In
the realm of condensed matter physics, there is a whole zoo of quasiparticles.
Examples are magnons - the quanta of classical spin waves, and rotons, the
quanta of the vorticity field in a superfluid.
So the criterion for when quantum field theory, rather than usual quantum
mechanics, must be used is whether or not creation and annihilation processes
are important. This again depends on the energy cost, E0 , for creating a
new particle. In elementary particle physics this cost is just the rest energy
E0 = mc2 , while in a condensed matter system it is given by the energy
gap, ∆, which is the energy of the lowest lying excitation above the ground
state. In elementary particle physics, the lightest, strongly interacting, boson
is the pi meson with a mass corresponding to E0 = 139 Mev. If we want to
describe a collision of protons at energies of several hundred MeV, we must
use quantum field theory in order to take into account the possibility of producing pions. At T = 2.17 helium becomes superfluid, and the roton gap is
∆rot = 8.65kT ≈ 1meV so only few rotons are thermally excited. In a neutron scattering experiments, however, the required energy is provided by the
neutron beam and the rotons can be created and observed.
A very important special case of quantum field theories are those containing
6
In modern elementary particle physics, the concept of empty space, or vacuum, is rather
intricate. The vacuum can in fact be argued to look much more like a ”boiling soup” of
”virtual” particles, or even strings or branes, than just an empty void. One can in fact make
many striking analogues between elementary particle physics and condensed matter physics,
where the photon and other ”elementary” particles appear as excitations in some underlying
substrate, much like the phonons are excitations of a crystal lattice. In this course we shall
not dwell on such hypothetical parallells.
7
The crystal displacement field η(~r, t) denotes the displacement of an infinitesimal part
of the crystal at position ~r from its equilibrium position.
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massless or gapless particles, such as photons or phonons.8 In these theories
particles can be created regardless of how small the available energy is, and
in general such theories are difficult to handle. The most famous example
is quantum electrodynamics, or QED, the theory of electrons and photons.
What makes this theory tractable is that the coupling constant, which here is
the electric charge, as measured by the dimensionless fine structure constant
α = e2 /¯hc ≈ 1/137, is very small, so that the probability of creating photons
is small even when the energy needed is present. Another example is quantum
chromodynamics, or QCD, the theory of quarks and gluons. Here the coupling
is strong, and multi-gluon states are indeed formed with a very high probability. In fact, this tendency is so strong that the empty vacuum is unstable and
the real physical vacuum is rather to be thought of as a soup of condensed
gluons. These important problems, related to copious production of gapless
particles, are usually referred to as infrared problems and will not be addressed
further in this course.
In all the examples considered so far, the created particles were bosons,
and could be thought of as a quantized vibration of a classical field. How this
works in detail is an important part of this course, but you should already
at this point appreciate the fundamental difference between a classical field
such as the crystal displacement field η(~r, t) on one hand, and the Heisenberg
quantum field operator ϕ(~r, t)9 on the other. The precise relation between
the bosonic quantum field in the Heisenberg picture, and the corresponding
classical field is that the latter is a quantum mechanical expectation value of
the former,
η(~r, t) = chϕ(~r, t)|ϕ(~
ˆ r, t)|ϕ(~r, t)i
(9)
where c is a constant,10 and the state vector |ϕ(~r, t)i is a coherent superposition of states with different number of particles.11 Here we see why the
quantum field had to be a sum of creation and annihilation operators. It is this
8
Although a particle physicist would normally talk about ”mass-gap” and a condensed
matter physicist about ”energy-gap” one often just uses the term ”gap”.
9
Remember that in the Heisenberg picture operators are time-dependent. The correspondence between classical and quantum fields are most easily understood using Heisenberg
quantum fields.
10
To get the same normalization as used in chapt. on in the textbook by Stone one should
√
take c = 1/ ρ0 , where ρ0 is the mean density of the crystal.
11
It has to bee! Since the quantum field operator only connects states with different
number of particles, the expectation value is zero in any state with fixed number of particles.
This is completely analogous with the harmonic oscillator, and a classical electromagnetic
field is indeed nothing but the expectaion value of the quantum field operator Fˆµν in a
coherent, or quasiclassical, state of photons.
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combination which is hermitian, just as the position operator xˆ for a harmonic
oscillator, while the raising and lowering operators are not. The quantum field
counterpart to the operator pˆ ∼ i(a† − a) is π
ˆ (~r), and the expectation value of
this operator is the classical velocity field.
Fermions cannot be described in this way - they are not to be thought of
as quantized vibrations of some classical field. We can indeed form fermionic
quantum fields, usually denoted by ψˆ† (~r), by taking a Fourier transform of
the operators b~†k and b~k that creates and destroys fermions of momentum ~k.
The expectation values of these operators can however not be interpreted as a
normal classical field.12 One can, however, form composite bosonic operators
by combining an even number of fermion field operators. The simplest example
ˆ r) which is related to the classical density
is the density operator ρˆ(~r) = ψˆ† (~r)ψ(~
of an arbitrary state |Ψi by,
ρ(~r) = hΨ|ˆ
ρ(~r)|Ψi
(10)
Note that although the expression for ρˆ(~r) in terms of the quantum field opˆ r) looks identical to the expression for the density of a one particle
erator ψ(~
quantum mechanical wave function ψ(~r), the interpretation is very different:
ψˆ† and ψˆ are operators acting on state vectors |Ψi that describe a many-body
system, while ψ and ψ ? are functions that describe the state of a one-particle
system. Having stressed the difference, one should also add that the formal
similarity is not a coincidence, as will become clear when you become familiar
with the formalism of quantum field theory.13
Just as there are gapless bosons, there can be gapless fermions. In elementary particle physics there was two long-term candidates for such a particle,
namely the electron and muon neutrinos. Today there is experimental evidence for a tiny (fractions of electronvolts) neutrino mass, so there might be
no elementary gapless fermions. In condensed matter physics, however, such
particles are important. The quasi particles in an ordinary metal, the so called
quasielectrons and quasiholes, are gapless. The presence or absence of a gap
is very important for the properties of the metal. At low enough temperature
12
There is a ”classical” description of fermions which utilizes so called Grassmann numbers
that is the subject of chapter 14 in the textbook.
13
In fact, in early days the field operators ϕˆ or ψˆ were often referred to as second quantized
objects, the idea being that the wave function ψ was obtained by a first quantization, which
was then quantized a second time to obtain the field operators. This way of thinking is
just confusing, but the unfortunate terminology has stuck, and we still use the term second
quantization to refer to a quantum field theoretical formulation of many particle quantum
mechanics, and first quantzation to the direct use of many body Hamiltonians and wave
functions.
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a very weak attractive force between the quasiparticles becomes important
and may result in the opening of a gap, which is physically manifested in the
metal becoming a superconductor. Quantum field theory is the natural language for describing this phenomenon of ”gap opening”, and understanding
why it implies superconductivity.
In these notes I stressed that quantum field theory is the natural language
to describe quantum mechanical processes where particles are created and annihilated. Since such processes occur both in the relativistic theories describing
elementary particles, and in the non-relativistic theories describing condensed
matter systems,14 quantum field theory provides a unified way of describing a
large number of seemingly unrelated phenomena. This unity of language has
proven to be fruitful in that methods and intuition from high energy physics
has been of great help to understand many condensed matter systems, and
vice versa. Nevertheless one should be aware that there is a body of very
important results that are specific to relativistic quantum field theory, and
goes beyond just having a convenient language for creation and annihilation
processes. For instance, while thinking of the quasiparticles and quasiholes in
a fermi gas or fermi liquid as particles and antiparticles is in a sense a matter
of convinience - one can always consider the full N -body electron wave function - in a relativistic theory of electrons, the existence of the antiparticles,
the positrons, is necessary, and they provide a new degree of freedom that
cannot be understood in any natural way using electrons only.15 In addition
to the necessity of having antiparticles there are several other deep results
that are special to relativistic quantum field theory. The most famous are the
spin-statistics theorem that states that fermions have half-integer spin while
bosons carry integer spin, and the CPT theorem that states that any consistent relativistic quantum field theory has to be invariant under the combined
symmetry transformation of parity, time-reversal and charge conjugation.
Summary: To describe processes where (quasi)particles are created and destroyed, the usual Hilbert space has to be extended to a Fock space which
contains states with different number of particles. The operators, a†α and aα
has matrix elements between states with different number of particles and correspond to creation and annihilation of a particle with quantum numbers α.
We can also consider the idealized process of creating or destroying a parti14
There are in fact cases where condensed matter systems are described by an approximatively Lorentz invariant theory, where the speed of light is replaced by some other velocity
such as a the speed of sound.
15
There is a intuitive description based on the so called ”Dirac sea” that provides a very
useful analogy, but is plagued with difficulties if taken too literally.
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cle at a point ~r - this is described by the quantum field operator ϕ(~r). For
bosons, the (quasi)particles can be viewed as quantized vibrations of a corresponding classical field. Conversely, starting from the quantum field theory,
the classical field is nothing but the expectation value of the quantum field in
a quasiclassical, or coherent, state.
Advanced courses: This is the first in a package of three courses in quantum field theory given in the masters program in theoretical physics. It is
followed by two more advanced courses, one about the QFT of elementary particle physics, and one aimed at condensed matter physics and modern atomic
physics. In the former you will learn about the standard model of particle
physics, which is a non-abelian gauge theory, and in the latter you will learn
about the more specialized QFT techniques that are used to describe macroscopic quantum phenomena such as superconductivity and Bose-Einstein condensation.
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