In theoretical physics, quantum field theory
(QFT) is a theoretical framework that combines
classical field theory, special relativity,
and quantum mechanics and is used to construct
physical models of subatomic particles (in
particle physics) and quasiparticles (in condensed
matter physics).
QFT treats particles as excited states (also
called quanta) of their underlying fields,
which are—in a sense—more fundamental
than the basic particles. Interactions between
particles are described by interaction terms
in the Lagrangian involving their corresponding
fields. Each interaction can be visually represented
by Feynman diagrams, which are formal computational
tools, in the process of relativistic perturbation
theory.
== History ==
As a successful theoretical framework today,
quantum field theory emerged from the work
of generations of theoretical physicists spanning
much of the 20th century. Its development
began in the 1920s with the description of
interactions between light and electrons,
culminating in the first quantum field theory
— quantum electrodynamics. A major theoretical
obstacle soon followed with the appearance
and persistence of various infinities in perturbative
calculations, a problem only resolved in the
1950s with the invention of the renormalization
procedure. A second major barrier came with
QFT's apparent inability to describe the weak
and strong interactions, to the point where
some theorists called for the abandonment
of the field theoretic approach. The development
of gauge theory and the completion of the
Standard Model in the 1970s led to a renaissance
of quantum field theory.
=== Theoretical background ===
Quantum field theory is the result of the
combination of classical field theory, quantum
mechanics, and special relativity. A brief
overview of these theoretical precursors is
in order.
The earliest successful classical field theory
is one that emerged from Newton's law of universal
gravitation, despite the complete absence
of the concept of fields from his 1687 treatise
Philosophiæ Naturalis Principia Mathematica.
The force of gravity as described by Newton
is an "action at a distance" — its effects
on faraway objects are instantaneous, no matter
the distance. In an exchange of letters with
Richard Bentley, however, Newton stated that
"it is inconceivable that inanimate brute
matter should, without the mediation of something
else which is not material, operate upon and
affect other matter without mutual contact."
It was not until the 18th century that mathematical
physicists discovered a convenient description
of gravity based on fields — a numerical
quantity (a vector (mathematics and physics))
assigned to every point in space indicating
the action of gravity on any particle at that
point. However, this was considered merely
a mathematical trick.Fields began to take
on an existence of their own with the development
of electromagnetism in the 19th century. Michael
Faraday coined the English term "field" in
1845. He introduced fields as properties of
space (even when it is devoid of matter) having
physical effects. He argued against "action
at a distance", and proposed that interactions
between objects occur via space-filling "lines
of force". This description of fields remains
to this day.The theory of classical electromagnetism
was completed in 1862 with Maxwell's equations,
which described the relationship between the
electric field, the magnetic field, electric
current, and electric charge. Maxwell's equations
implied the existence of electromagnetic waves,
a phenomenon whereby electric and magnetic
fields propagate from one spatial point to
another at a finite speed, which turns out
to be the speed of light. Action-at-a-distance
was thus conclusively refuted.Despite the
enormous success of classical electromagnetism,
it was unable to account for the discrete
lines in atomic spectra, nor for the distribution
of blackbody radiation in different wavelengths.
Max Planck's study of blackbody radiation
marked the beginning of quantum mechanics.
He treated atoms, which absorb and emit electromagnetic
radiation, as tiny oscillators with the crucial
property that their energies can only take
on a series of discrete, rather than continuous,
values. These are known as quantum harmonic
oscillators. This process of restricting energies
to discrete values is called quantization.
Building on this idea, Albert Einstein proposed
in 1905 an explanation for the photoelectric
effect, that light is composed of individual
packets of energy called photons (the quanta
of light). This implied that the electromagnetic
radiation, while being waves in the classical
electromagnetic field, also exists in the
form of particles.In 1913, Niels Bohr introduced
the Bohr model of atomic structure, wherein
electrons within atoms can only take on a
series of discrete, rather than continuous,
energies. This is another example of quantization.
The Bohr model successfully explained the
discrete nature of atomic spectral lines.
In 1924, Louis de Broglie proposed the hypothesis
of wave-particle duality, that microscopic
particles exhibit both wave-like and particle-like
properties under different circumstances.
Uniting these scattered ideas, a coherent
discipline, quantum mechanics, was formulated
between 1925 and 1926, with important contributions
from de Broglie, Werner Heisenberg, Max Born,
Erwin Schrödinger, Paul Dirac, and Wolfgang
Pauli.:22-23In the same year as his paper
on the photoelectric effect, Einstein published
his theory of special relativity, built on
Maxwell's electromagnetism. New rules, called
Lorentz transformation, were given for the
way time and space coordinates of an event
change under changes in the observer's velocity,
and the distinction between time and space
was blurred.:19 It was proposed that all physical
laws must be the same for observers at different
velocities, i.e. that physical laws be invariant
under Lorentz transformations.
Two difficulties remained. Observationally,
the Schrödinger equation underlying quantum
mechanics could explain the stimulated emission
of radiation from atoms, where an electron
emits a new photon under the action of an
external electromagnetic field, but it was
unable to explain spontaneous emission, where
an electron spontaneously decreases in energy
and emits a photon even without the action
of an external electromagnetic field. Theoretically,
the Schrödinger equation could not describe
photons and was inconsistent with the principles
of special relativity — it treats time as
an ordinary number while promoting spatial
coordinates to linear operators.
=== Quantum electrodynamics ===
Quantum field theory naturally began with
the study of electromagnetic interactions,
as the electromagnetic field was the only
known classical field as of the 1920s.:1Through
the works of Born, Heisenberg, and Pascual
Jordan in 1925-1926, a quantum theory of the
free electromagnetic field (one with no interactions
with matter) was developed via canonical quantization
by treating the electromagnetic field as a
set of quantum harmonic oscillators.:1 With
the exclusion of interactions, however, such
a theory was yet incapable of making quantitative
predictions about the real world.:22In his
seminal 1927 paper The quantum theory of the
emission and absorption of radiation, Dirac
coined the term quantum electrodynamics (QED),
a theory that adds upon the terms describing
the free electromagnetic field an additional
interaction term between electric current
density and the electromagnetic vector potential.
Using first-order perturbation theory, he
successfully explained the phenomenon of spontaneous
emission. According to the uncertainty principle
in quantum mechanics, quantum harmonic oscillators
cannot remain stationary, but they have a
non-zero minimum energy and must always be
oscillating, even in the lowest energy state
(the ground state). Therefore, even in a perfect
vacuum, there remains an oscillating electromagnetic
field having zero-point energy. It is this
quantum fluctuation of electromagnetic fields
in the vacuum that "stimulates" the spontaneous
emission of radiation by electrons in atoms.
Dirac's theory was hugely successful in explaining
both the emission and absorption of radiation
by atoms; by applying second-order perturbation
theory, it was able to account for the scattering
of photons, resonance fluorescence, as well
as non-relativistic Compton scattering. Nonetheless,
the application of higher-order perturbation
theory was plagued with problematic infinities
in calculations.In 1928, Dirac wrote down
a wave equation that described relativistic
electrons — the Dirac equation. It had the
following important consequences: the spin
of an electron is 1/2; the electron g-factor
is 2; it led to the correct Sommerfeld formula
for the fine structure of the hydrogen atom;
and it could be used to derive the Klein-Nishina
formula for relativistic Compton scattering.
Although the results were fruitful, the theory
also apparently implied the existence of negative
energy states, which would cause atoms to
be unstable, since they could always decay
to lower energy states by the emission of
radiation.The prevailing view at the time
was that the world was composed of two very
different ingredients: material particles
(such as electrons) and quantum fields (such
as photons). Material particles were considered
to be eternal, with their physical state described
by the probabilities of finding each particle
in any given region of space or range of velocities.
On the other hand photons were considered
merely the excited states of the underlying
quantized electromagnetic field, and could
be freely created or destroyed. It was between
1928 and 1930 that Jordan, Eugene Wigner,
Heisenberg, Pauli, and Enrico Fermi discovered
that material particles could also be seen
as excited states of quantum fields. Just
as photons are excited states of the quantized
electromagnetic field, so each type of particle
had its corresponding quantum field: an electron
field, a proton field, etc. Given enough energy,
it would now be possible to create material
particles. Building on this idea, Fermi proposed
in 1932 an explanation for β decay known
as Fermi's interaction. Atomic nuclei do not
contain electrons per se, but in the process
of decay, an electron is created out of the
surrounding electron field, analogous to the
photon created from the surrounding electromagnetic
field in the radiative decay of an excited
atom.:22-23It was realized in 1929 by Dirac
and others that negative energy states implied
by the Dirac equation could be removed by
assuming the existence of particles with the
same mass as electrons but opposite electric
charge. This not only ensured the stability
of atoms, but it was also the first proposal
of the existence of antimatter. Indeed, the
evidence for positrons was discovered in 1932
by Carl David Anderson in cosmic rays. With
enough energy, such as by absorbing a photon,
an electron-positron pair could be created,
a process called pair production; the reverse
process, annihilation, could also occur with
the emission of a photon. This showed that
particle numbers need not be fixed during
an interaction. Historically, however, positrons
were at first thought of as "holes" in an
infinite electron sea, rather than a new kind
of particle, and this theory was referred
to as the Dirac hole theory.:23 QFT naturally
incorporated antiparticles in its formalism.:24
=== Infinities and renormalization ===
Robert Oppenheimer showed in 1930 that higher-order
perturbative calculations in QED always resulted
in infinite quantities, such as the electron
self-energy and the vacuum zero-point energy
of the electron and photon fields, suggesting
that the computational methods at the time
could not properly deal with interactions
involving photons with extremely high momenta.:25
It was not until 20 years later that a systematic
approach to remove such infinities was developed.
A series of papers were published between
1934 and 1938 by Ernst Stueckelberg that established
a relativistically invariant formulation of
QFT. In 1947, Stueckelberg also independently
developed a complete renormalization procedure.
Unfortunately, such achievements were not
understood and recognized by the theoretical
community.Faced with these infinities, John
Archibald Wheeler and Heisenberg proposed,
in 1937 and 1943 respectively, to supplant
the problematic QFT with the so-called S-matrix
theory. Since the specific details of microscopic
interactions are inaccessible to observations,
the theory should only attempt to describe
the relationships between a small number of
observables (e.g. the energy of an atom) in
an interaction, rather than be concerned with
the microscopic minutiae of the interaction.
In 1945, Richard Feynman and Wheeler daringly
suggested abandoning QFT altogether and proposed
action-at-a-distance as the mechanism of particle
interactions.:26In 1947, Willis Lamb and Robert
Retherford measured the minute difference
in the 2S1/2 and 2P1/2 energy levels of the
hydrogen atom, also called the Lamb shift.
By ignoring the contribution of photons whose
energy exceeds the electron mass, Hans Bethe
successfully estimated the numerical value
of the Lamb shift.:28 Subsequently, Norman
Myles Kroll, Lamb, James Bruce French, and
Victor Weisskopf again confirmed this value
using an approach in which infinities cancelled
other infinities to result in finite quantities.
However, this method was clumsy and unreliable
and could not be generalized to other calculations.The
breakthrough eventually came around 1950 when
a more robust method for eliminating infinities
was developed Julian Schwinger, Feynman, Freeman
Dyson, and Shinichiro Tomonaga. The main idea
is to replace the initial, so-called "bare",
parameters (mass, electric charge, etc.),
which have no physical meaning, by their finite
measured values. To cancel the apparently
infinite parameters, one has to introduce
additional, infinite, "counterterms" into
the Lagrangian. This systematic computational
procedure is known as renormalization and
can be applied to arbitrary order in perturbation
theory.By applying the renormalization procedure,
calculations were finally made to explain
the electron's anomalous magnetic moment (the
deviation of the electron g-factor from 2)
and vacuum polarisation. These results agreed
with experimental measurements to a remarkable
degree, thus marking the end of a "war against
infinities".At the same time, Feynman introduced
the path integral formulation of quantum mechanics
and Feynman diagrams.:2 The latter can be
used to visually and intuitively organise
and to help compute terms in the perturbative
expansion. Each diagram can be interpreted
as paths of particles in an interaction, with
each vertex and line having a corresponding
mathematical expression, and the product of
these expressions gives the scattering amplitude
of the interaction represented by the diagram.:5It
was with the invention of the renormalization
procedure and Feynman diagrams that QFT finally
arose as a complete theoretical framework.:2
=== Non-renormalizability ===
Given the tremendous success of QED, many
theorists believed, in the few years after
1949, that QFT could soon provide an understanding
of all microscopic phenomena, not only the
interactions between photons, electrons, and
positrons. Contrary to this optimism, QFT
entered yet another period of depression that
lasted for almost two decades.:30The first
obstacle was the limited applicability of
the renormalization procedure. In perturbative
calculations in QED, all infinite quantities
could be eliminated by redefining a small
(finite) number of physical quantities (namely
the mass and charge of the electron). Dyson
proved in 1949 that this is only possible
for a small class of theories called "renormalizable
theories", of which QED is an example. However,
most theories, including the Fermi theory
of the weak interaction, are "non-renormalizable".
Any perturbative calculation in these theories
beyond the first order would result in infinities
that could not be removed by redefining a
finite number of physical quantities.:30The
second major problem stemmed from the limited
validity of the Feynman diagram method, which
are based on a series expansion in perturbation
theory. In order for the series to converge
and low-order calculations to be a good approximation,
the coupling constant, in which the series
is expanded, must be a sufficiently small
number. The coupling constant in QED is the
fine-structure constant α ≈ 1/137, which
is small enough that only the simplest, lowest
order, Feynman diagrams need to be considered
in realistic calculations. In contrast, the
coupling constant in the strong interaction
is roughly of the order of one, making complicated,
higher order, Feynman diagrams just as important
as simple ones. There was thus no way of deriving
reliable quantitative predictions for the
strong interaction using perturbative QFT
methods.:31With these difficulties looming,
many theorists began to turn away from QFT.
Some focused on symmetry principles and conservation
laws, while others picked up the old S-matrix
theory of Wheeler and Heisenberg. QFT was
used heuristically as guiding principles,
but not as a basis for quantitative calculations.:31
=== Standard Model ===
In 1954, Yang Chen-Ning and Robert Mills generalised
the local symmetry of QED, leading to non-Abelian
gauge theories (also known as Yang-Mills theories),
which are based on more complicated local
symmetry groups. In QED, (electrically) charged
particles interact via the exchange of photons,
while in non-Abelian gauge theory, particles
carrying a new type of "charge" interact via
the exchange of massless gauge bosons. Unlike
photons, these gauge bosons themselves carry
charge.:32Sheldon Glashow developed a non-Abelian
gauge theory that unified the electromagnetic
and weak interactions in 1960. In 1964, Abdul
Salam and John Clive Ward arrived at the same
theory through a different path. This theory,
nevertheless, was non-renormalizable.Peter
Higgs, Robert Brout, and François Englert
proposed in 1964 that the gauge symmetry in
Yang-Mills theories could be broken by a mechanism
called spontaneous symmetry breaking, through
which originally massless gauge bosons could
acquire mass.:5-6By combining the earlier
theory of Glashow, Salam, and Ward with the
idea of spontaneous symmetry breaking, Steven
Weinberg wrote down in 1967 a theory describing
electroweak interactions between all leptons
and the effects of the Higgs boson. His theory
was at first mostly ignored,:6 until it was
brought back to light in 1971 by Gerard 't
Hooft's proof that non-Abelian gauge theories
are renormalizable. The electroweak theory
of Weinberg and Salam was extended from leptons
to quarks in 1970 by Glashow, John Iliopoulos,
and Luciano Maiani, marking its completion.Harald
Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler
discovered in 1971 that certain phenomena
involving the strong interaction could also
be explained by non-Abelian gauge theory.
Quantum chromodynamics (QCD) was born. In
1973, David Gross, Frank Wilczek, and Hugh
David Politzer showed that non-Abelian gauge
theories are "asymptotically free", meaning
that under renormalization, the coupling constant
of the strong interaction decreases as the
interaction energy increases. (Similar discoveries
had been made numerous times prior, but they
had been largely ignored.) :11 Therefore,
at least in high-energy interactions, the
coupling constant in QCD becomes sufficiently
small to warrant a perturbative series expansion,
making quantitative predictions for the strong
interaction possible.:32These theoretical
breakthroughs brought about a renaissance
in QFT. The full theory, which includes the
electroweak theory and chromodynamics, is
referred to today as the Standard Model of
elementary particles. The Standard Model successfully
describes all fundamental interactions except
gravity, and its many predictions have been
met with remarkable experimental confirmation
in subsequent decades.:3 The Higgs boson,
central to the mechanism of spontaneous symmetry
breaking, was finally detected in 2012 at
CERN, marking the complete verification of
the existence of all constituents of the Standard
Model.
=== Other developments ===
The 1970s saw the development of non-perturbative
methods in non-Abelian gauge theories. The
't Hooft-Polyakov monopole was discovered
by 't Hooft and Alexander Polyakov, flux tubes
by Holger Bech Nielsen and Poul Olesen, and
instantons by Polyakov et al.. These objects
are inaccessible through perturbation theory.:4Supersymmetry
also appeared in the same period. The first
supersymmetric QFT in four dimensions was
built by Yuri Golfand and Evgeny Likhtman
in 1970, but their result failed to garner
widespread interest due to the Iron Curtain.
Supersymmetry only took off in the theoretical
community after the work of Julius Wess and
Bruno Zumino in 1973.:7Among the four fundamental
interactions, gravity remains the only one
that lacks a consistent QFT description. Various
attempts at a theory of quantum gravity led
to the development of string theory,:6 itself
a type of two-dimensional QFT with conformal
symmetry. Joël Scherk and John Schwarz first
proposed in 1974 that string theory could
be the quantum theory of gravity.
=== Condensed matter physics ===
Although quantum field theory arose from the
study of interactions between elementary particles,
it has been successfully applied to other
physical systems, particularly to many-body
systems in condensed matter physics.
Historically, the Higgs mechanism of spontaneous
symmetry breaking was a result of Yoichiro
Nambu's application of superconductor theory
to elementary particles, while the concept
of renormalization came out of the study of
second-order phase transitions in matter.Soon
after the introduction of photons, Einstein
performed the quantization procedure on vibrations
in a crystal, leading to the first quasiparticle
— phonons. Lev Landau claimed that low-energy
excitations in many condensed matter systems
could be described in terms of interactions
between a set of quasiparticles. The Feynman
diagram method of QFT was naturally well suited
to the analysis of various phenomena in condensed
matter systems.Gauge theory is used to describe
the quantization of magnetic flux in superconductors,
the resistivity in the quantum Hall effect,
as well as the relation between frequency
and voltage in the AC Josephson effect.
== Principles ==
For simplicity, natural units are used in
the following sections, in which the reduced
Planck constant ħ and the speed of light
c are both set to one.
=== Classical fields ===
A classical field is a function of spatial
and time coordinates. Examples include the
gravitational field in Newtonian gravity g(x,
t) and the electric field E(x, t) and magnetic
field B(x, t) in classical electromagnetism.
A classical field can be thought of as a numerical
quantity assigned to every point in space
that changes in time. Hence, it has infinite
degrees of freedom.Many phenomena exhibiting
quantum mechanical properties cannot be explained
by classical fields alone. Phenomena such
as the photoelectric effect are best explained
by discrete particles (photons), rather than
a spatially continuous field. The goal of
quantum field theory is to describe various
quantum mechanical phenomena using a modified
concept of fields.
Canonical quantisation and path integrals
are two common formulations of QFT. To motivate
the fundamentals of QFT, an overview of classical
field theory is in order.
The simplest classical field is a real scalar
field — a real number at every point in
space that changes in time. It is denoted
as ϕ(x, t), where x is the position vector,
and t is the time. Suppose the Lagrangian
of the field is
L
=
∫
d
3
x
L
=
∫
d
3
x
[
1
2
ϕ
˙
2
−
1
2
(
∇
ϕ
)
2
−
1
2
m
2
ϕ
2
]
,
{\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int
d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac
{1}{2}}(\nabla \phi )^{2}-{\frac {1}{2}}m^{2}\phi
^{2}\right],}
where
ϕ
˙
{\displaystyle {\dot {\phi }}}
is the time-derivative of the field, ∇ is
the divergence operator, and m is a real parameter
(the "mass" of the field). Applying the Euler–Lagrange
equation on the Lagrangian::16
∂
∂
t
[
∂
L
∂
(
∂
ϕ
/
∂
t
)
]
+
∑
i
=
1
3
∂
∂
x
i
[
∂
L
∂
(
∂
ϕ
/
∂
x
i
)
]
−
∂
L
∂
ϕ
=
0
,
{\displaystyle {\frac {\partial }{\partial
t}}\left[{\frac {\partial {\mathcal {L}}}{\partial
(\partial \phi /\partial t)}}\right]+\sum
_{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac
{\partial {\mathcal {L}}}{\partial (\partial
\phi /\partial x^{i})}}\right]-{\frac {\partial
{\mathcal {L}}}{\partial \phi }}=0,}
we obtain the equations of motion for the
field, which describe the way it varies in
time and space:
(
∂
2
∂
t
2
−
∇
2
+
m
2
)
ϕ
=
0.
{\displaystyle \left({\frac {\partial ^{2}}{\partial
t^{2}}}-\nabla ^{2}+m^{2}\right)\phi =0.}
This is known as the Klein–Gordon equation.:17The
Klein–Gordon equation is a wave equation,
so its solutions can be expressed as a sum
of normal modes (obtained via Fourier transform)
as follows:
ϕ
(
x
,
t
)
=
∫
d
3
p
(
2
π
)
3
1
2
ω
p
(
a
p
e
−
i
ω
p
t
+
i
p
⋅
x
+
a
p
∗
e
i
ω
p
t
−
i
p
⋅
x
)
,
{\displaystyle \phi (\mathbf {x} ,t)=\int
{\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt
{2\omega _{\mathbf {p} }}}}\left(a_{\mathbf
{p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf
{p} \cdot \mathbf {x} }+a_{\mathbf {p} }^{*}e^{i\omega
_{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf
{x} }\right),}
where a is a complex number (normalised by
convention), * denotes complex conjugation,
and ωp is the frequency of the normal mode:
ω
p
=
|
p
|
2
+
m
2
.
{\displaystyle \omega _{\mathbf {p} }={\sqrt
{|\mathbf {p} |^{2}+m^{2}}}.}
Thus each normal mode corresponding to a single
p can be seen as a classical harmonic oscillator
with frequency ωp.:21,26
=== Canonical quantisation ===
The quantisation procedure for the above classical
field is analogous to the promotion of a classical
harmonic oscillator to a quantum harmonic
oscillator.
The displacement of a classical harmonic oscillator
is described by
x
(
t
)
=
1
2
ω
a
e
−
i
ω
t
+
1
2
ω
a
∗
e
i
ω
t
,
{\displaystyle x(t)={\frac {1}{\sqrt {2\omega
}}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega
}}}a^{*}e^{i\omega t},}
where a is a complex number (normalised by
convention), and ω is the oscillator's frequency.
Note that x is the displacement of a particle
in simple harmonic motion from the equilibrium
position, which should not be confused with
the spatial label x of a field.
For a quantum harmonic oscillator, x(t) is
promoted to a linear operator
x
^
(
t
)
{\displaystyle {\hat {x}}(t)}
:
x
^
(
t
)
=
1
2
ω
a
^
e
−
i
ω
t
+
1
2
ω
a
^
†
e
i
ω
t
.
{\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt
{2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac
{1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger
}e^{i\omega t}.}
Complex numbers a and a* are replaced by the
annihilation operator
a
^
{\displaystyle {\hat {a}}}
and the creation operator
a
^
†
{\displaystyle {\hat {a}}^{\dagger }}
, respectively, where † denotes Hermitian
conjugation. The commutation relation between
the two is
[
a
^
,
a
^
†
]
=
1.
{\displaystyle [{\hat {a}},{\hat {a}}^{\dagger
}]=1.}
The vacuum state
|
0
⟩
{\displaystyle |0\rangle }
, which is the lowest energy state, is defined
by
a
^
|
0
⟩
=
0.
{\displaystyle {\hat {a}}|0\rangle =0.}
Any quantum state of a single harmonic oscillator
can be obtained from
|
0
⟩
{\displaystyle |0\rangle }
by successively applying the creation operator
a
^
†
{\displaystyle {\hat {a}}^{\dagger }}
::20
|
n
⟩
=
(
a
^
†
)
n
|
0
⟩
.
{\displaystyle |n\rangle =({\hat {a}}^{\dagger
})^{n}|0\rangle .}
By the same token, the aforementioned real
scalar field ϕ, which corresponds to x in
the single harmonic oscillator, is also promoted
to an operator
ϕ
^
{\displaystyle {\hat {\phi }}}
, while ap and ap* are replaced by the annihilation
operator
a
^
p
{\displaystyle {\hat {a}}_{\mathbf {p} }}
and the creation operator
a
^
p
†
{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger
}}
for a particular p, respectively:
ϕ
^
(
x
,
t
)
=
∫
d
3
p
(
2
π
)
3
1
2
ω
p
(
a
^
p
e
−
i
ω
p
t
+
i
p
⋅
x
+
a
^
p
†
e
i
ω
p
t
−
i
p
⋅
x
)
.
{\displaystyle {\hat {\phi }}(\mathbf {x}
,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac
{1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat
{a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p}
}t+i\mathbf {p} \cdot \mathbf {x} }+{\hat
{a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf
{p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}
Their commutation relations are::21
[
a
^
p
,
a
^
q
†
]
=
(
2
π
)
3
δ
(
p
−
q
)
,
[
a
^
p
,
a
^
q
]
=
[
a
^
p
†
,
a
^
q
†
]
=
0
,
{\displaystyle [{\hat {a}}_{\mathbf {p} },{\hat
{a}}_{\mathbf {q} }^{\dagger }]=(2\pi )^{3}\delta
(\mathbf {p} -\mathbf {q} ),\quad [{\hat {a}}_{\mathbf
{p} },{\hat {a}}_{\mathbf {q} }]=[{\hat {a}}_{\mathbf
{p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger
}]=0,}
where δ is the Dirac delta function. The
vacuum state
|
0
⟩
{\displaystyle |0\rangle }
is defined by
a
^
p
|
0
⟩
=
0
,
for all
p
.
{\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle
=0,\quad {\text{for all }}\mathbf {p} .}
Any quantum state of the field can be obtained
from
|
0
⟩
{\displaystyle |0\rangle }
by successively applying creation operators
a
^
p
†
{\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger
}}
, e.g.:22
(
a
^
p
3
†
)
3
a
^
p
2
†
(
a
^
p
1
†
)
2
|
0
⟩
.
{\displaystyle ({\hat {a}}_{\mathbf {p} _{3}}^{\dagger
})^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger
}({\hat {a}}_{\mathbf {p} _{1}}^{\dagger })^{2}|0\rangle
.}
Although the field appearing in the Lagrangian
is spatially continuous, the quantum states
of the field are discrete. While the state
space of a single quantum harmonic oscillator
contains all the discrete energy states of
one oscillating particle, the state space
of a quantum field contains the discrete energy
levels of an arbitrary number of particles.
The latter space is known as a Fock space,
which can account for the fact that particle
numbers are not fixed in relativistic quantum
systems. The process of quantising an arbitrary
number of particles instead of a single particle
is often also called second quantisation.:19The
preceding procedure is a direct application
of non-relativistic quantum mechanics and
can be used to quantise (complex) scalar fields,
Dirac fields,:52 vector fields (e.g. the electromagnetic
field), and even strings. However, creation
and annihilation operators are only well defined
in the simplest theories that contain no interactions
(so-called free theory). In the case of the
real scalar field, the existence of these
operators was a consequence of the decomposition
of solutions of the classical equations of
motion into a sum of normal modes. To perform
calculations on any realistic interacting
theory, perturbation theory would be necessary.
The Lagrangian of any quantum field in nature
would contain interaction terms in addition
to the free theory terms. For example, a quartic
interaction term could be introduced to the
Lagrangian of the real scalar field::77
L
=
1
2
(
∂
μ
ϕ
)
(
∂
μ
ϕ
)
−
1
2
m
2
ϕ
2
−
λ
4
!
ϕ
4
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial
_{\mu }\phi )(\partial ^{\mu }\phi )-{\frac
{1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi
^{4},}
where μ is a spacetime index,
∂
0
=
∂
/
∂
t
,
∂
1
=
∂
/
∂
x
1
{\displaystyle \partial _{0}=\partial /\partial
t,\ \partial _{1}=\partial /\partial x^{1}}
, etc. The summation over the index μ has
been omitted following the Einstein notation.
If the parameter λ is sufficiently small,
then the interacting theory described by the
above Lagrangian can be considered as a small
perturbation from the free theory.
=== Path integrals ===
The path integral formulation of QFT is concerned
with the direct computation of the scattering
amplitude of a certain interaction process,
rather than the establishment of operators
and state spaces. To calculate the probability
amplitude for a system to evolve from some
initial state
|
ϕ
I
⟩
{\displaystyle |\phi _{I}\rangle }
at time t = 0 to some final state
|
ϕ
F
⟩
{\displaystyle |\phi _{F}\rangle }
at t = T, the total time T is divided into
N small intervals. The overall amplitude is
the product of the amplitude of evolution
within each interval, integrated over all
intermediate states. Let H be the Hamiltonian
(i.e. generator of time evolution), then:10
⟨
ϕ
F
|
e
−
i
H
T
|
ϕ
I
⟩
=
∫
d
ϕ
1
∫
d
ϕ
2
⋯
∫
d
ϕ
N
−
1
⟨
ϕ
F
|
e
−
i
H
T
/
N
|
ϕ
N
−
1
⟩
⋯
⟨
ϕ
2
|
e
−
i
H
T
/
N
|
ϕ
1
⟩
⟨
ϕ
1
|
e
−
i
H
T
/
N
|
ϕ
I
⟩
.
{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi
_{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots
\int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi
_{N-1}\rangle \cdots \langle \phi _{2}|e^{-iHT/N}|\phi
_{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi
_{I}\rangle .}
Taking the limit N → ∞, the above product
of integrals becomes the Feynman path integral::282:12
⟨
ϕ
F
|
e
−
i
H
T
|
ϕ
I
⟩
=
∫
D
ϕ
(
t
)
exp
⁡
{
i
∫
0
T
d
t
L
}
,
{\displaystyle \langle \phi _{F}|e^{-iHT}|\phi
_{I}\rangle =\int {\mathcal {D}}\phi (t)\,\exp
\left\{i\int _{0}^{T}dt\,L\right\},}
where L is the Lagrangian involving ϕ and
its derivatives with respect to spatial and
time coordinates, obtained from the Hamiltonian
H via Legendre transform. The initial and
final conditions of the path integral are
respectively
ϕ
(
0
)
=
ϕ
I
,
ϕ
(
T
)
=
ϕ
F
.
{\displaystyle \phi (0)=\phi _{I},\quad \phi
(T)=\phi _{F}.}
In other words, the overall amplitude is the
sum over the amplitude of every possible path
between the initial and final states, where
the amplitude of a path is given by the exponential
in the integrand.
=== Two-point correlation function ===
Now we assume that the theory contains interactions
whose Lagrangian terms are a small perturbation
from the free theory.
In calculations, one often encounters such
expressions:
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
,
{\displaystyle \langle \Omega |T\{\phi (x)\phi
(y)\}|\Omega \rangle ,}
where x and y are position four-vectors, T
is the time ordering operator (namely, it
orders x and y according to their time-component,
later time on the left and earlier time on
the right), and
|
Ω
⟩
{\displaystyle |\Omega \rangle }
is the ground state (vacuum state) of the
interacting theory. This expression, known
as the two-point correlation function or the
two-point Green's function, represents the
probability amplitude for the field to propagate
from y to x.:82In canonical quantisation,
the two-point correlation function can be
written as::87
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
=
lim
T
→
∞
(
1
−
i
ϵ
)
⟨
0
|
T
{
ϕ
I
(
x
)
ϕ
I
(
y
)
exp
⁡
[
−
i
∫
−
T
T
d
t
H
I
(
t
)
]
}
|
0
⟩
⟨
0
|
T
{
exp
⁡
[
−
i
∫
−
T
T
d
t
H
I
(
t
)
]
}
|
0
⟩
,
{\displaystyle \langle \Omega |T\{\phi (x)\phi
(y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon
)}{\frac {\langle 0|T\left\{\phi _{I}(x)\phi
_{I}(y)\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}|0\rangle
}{\langle 0|T\left\{\exp \left[-i\int _{-T}^{T}dt\,H_{I}(t)\right]\right\}|0\rangle
}},}
where ε is an infinitesimal number, ϕI is
the field operator under the free theory,
and HI is the interaction Hamiltonian term.
For the ϕ4 theory, it is:84
H
I
(
t
)
=
∫
d
x
3
λ
4
!
ϕ
I
(
x
)
4
.
{\displaystyle H_{I}(t)=\int dx^{3}\,{\frac
{\lambda }{4!}}\phi _{I}(x)^{4}.}
Since λ is a small parameter, the exponential
function exp can be expanded into a Taylor
series in λ and computed term by term. This
equation is useful in that it expresses the
field operator and ground state in the interacting
theory, which are difficult to define, in
terms of their counterparts in the free theory,
which are well defined.
In the path integral formulation, the two-point
correlation function can be written as::284
⟨
Ω
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
Ω
⟩
=
lim
T
→
∞
(
1
−
i
ϵ
)
∫
D
ϕ
ϕ
(
x
)
ϕ
(
y
)
exp
⁡
[
i
∫
−
T
T
d
4
z
L
]
∫
D
ϕ
exp
⁡
[
i
∫
−
T
T
d
4
z
L
]
,
{\displaystyle \langle \Omega |T\{\phi (x)\phi
(y)\}|\Omega \rangle =\lim _{T\to \infty (1-i\epsilon
)}{\frac {\int {\mathcal {D}}\phi \,\phi (x)\phi
(y)\exp \left[i\int _{-T}^{T}d^{4}z\,{\mathcal
{L}}\right]}{\int {\mathcal {D}}\phi \,\exp
\left[i\int _{-T}^{T}d^{4}z\,{\mathcal {L}}\right]}},}
where
L
{\displaystyle {\mathcal {L}}}
is the Lagrangian density. As in 
the previous paragraph, the exponential factor
involving the interaction term can also be
expanded as a series in λ.
According to Wick's theorem, any n-point correlation
function in the free theory can be written
as a sum of products of two-point correlation
functions. For example,
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
2
)
ϕ
(
x
3
)
ϕ
(
x
4
)
}
|
0
⟩
=
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
2
)
}
|
0
⟩
⟨
0
|
T
{
ϕ
(
x
3
)
ϕ
(
x
4
)
}
|
0
⟩
+
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
3
)
}
|
0
⟩
⟨
0
|
T
{
ϕ
(
x
2
)
ϕ
(
x
4
)
}
|
0
⟩
+
⟨
0
|
T
{
ϕ
(
x
1
)
ϕ
(
x
4
)
}
|
0
⟩
⟨
0
|
T
{
ϕ
(
x
2
)
ϕ
(
x
3
)
}
|
0
⟩
.
{\displaystyle {\begin{aligned}\langle 0|T\{\phi
(x_{1})\phi (x_{2})\phi (x_{3})\phi (x_{4})\}|0\rangle
=&\langle 0|T\{\phi (x_{1})\phi (x_{2})\}|0\rangle
\langle 0|T\{\phi (x_{3})\phi (x_{4})\}|0\rangle
\\&+\langle 0|T\{\phi (x_{1})\phi (x_{3})\}|0\rangle
\langle 0|T\{\phi (x_{2})\phi (x_{4})\}|0\rangle
\\&+\langle 0|T\{\phi (x_{1})\phi (x_{4})\}|0\rangle
\langle 0|T\{\phi (x_{2})\phi (x_{3})\}|0\rangle
.\end{aligned}}}
Since correlation functions in the interacting
theory can be expressed in terms of those
in the free theory, only the latter need to
be evaluated in order to calculate all physical
quantities in the (perturbative) interacting
theory.:90Either through canonical quantisation
or path integrals, one can obtain:
D
F
(
x
−
y
)
≡
⟨
0
|
T
{
ϕ
(
x
)
ϕ
(
y
)
}
|
0
⟩
=
∫
d
4
p
(
2
π
)
4
i
p
μ
p
μ
−
m
2
+
i
ϵ
e
−
i
p
μ
(
x
μ
−
y
μ
)
.
{\displaystyle D_{F}(x-y)\equiv \langle 0|T\{\phi
(x)\phi (y)\}|0\rangle =\int {\frac {d^{4}p}{(2\pi
)^{4}}}{\frac {i}{p_{\mu }p^{\mu }-m^{2}+i\epsilon
}}e^{-ip_{\mu }(x^{\mu }-y^{\mu })}.}
This is known as the Feynman propagator for
the real scalar field.:31,288:23
=== Feynman diagram ===
Correlation functions in the interacting theory
can be written as a perturbation series. Each
term in the series is a product of Feynman
propagators in the free theory and can be
represented visually by a Feynman diagram.
For example, the λ1 term in the two-point
correlation function in the ϕ4 theory is
−
i
λ
4
!
⟨
0
|
T
{
ϕ
(
x
)
ϕ
(
y
)
∫
d
4
z
ϕ
(
z
)
ϕ
(
z
)
ϕ
(
z
)
ϕ
(
z
)
}
|
0
⟩
.
{\displaystyle {\frac {-i\lambda }{4!}}\langle
0|T\{\phi (x)\phi (y)\int d^{4}z\,\phi (z)\phi
(z)\phi (z)\phi (z)\}|0\rangle .}
After applying Wick's theorem, one of the
terms is
12
⋅
−
i
λ
4
!
∫
d
4
z
D
F
(
x
−
z
)
D
F
(
y
−
z
)
D
F
(
z
−
z
)
,
{\displaystyle 12\cdot {\frac {-i\lambda }{4!}}\int
d^{4}z\,D_{F}(x-z)D_{F}(y-z)D_{F}(z-z),}
whose corresponding Feynman diagram is
Every point corresponds to a single ϕ field
factor. Points labelled with x and y are called
external points, while those in the interior
are called internal points or vertices (there
is one in this diagram). The value of the
corresponding term can be obtained from the
diagram by following "Feynman rules": assign
−
i
λ
∫
d
4
z
{\displaystyle -i\lambda \int d^{4}z}
to every vertex and the Feynman propagator
D
F
(
x
1
−
x
2
)
{\displaystyle D_{F}(x_{1}-x_{2})}
to every line with end points x1 and x2. The
product of factors corresponding to every
element in the diagram, divided by the "symmetry
factor" (2 for this diagram), gives the expression
for the term in the perturbation series.:91-94In
order to compute the n-point correlation function
to the k-th order, list all valid Feynman
diagrams with n external points and k or fewer
vertices, and then use Feynman rules to obtain
the expression for each term. To be precise,
⟨
Ω
|
T
{
ϕ
(
x
1
)
⋯
ϕ
(
x
n
)
}
|
Ω
⟩
{\displaystyle \langle \Omega |T\{\phi (x_{1})\cdots
\phi (x_{n})\}|\Omega \rangle }
is equal to the sum of (expressions corresponding
to) all connected diagrams with n external
points. (Connected diagrams are those in which
every vertex is connected to an external point
through lines. Components that are totally
disconnected from external lines are sometimes
called "vacuum bubbles".) In the ϕ4 interaction
theory discussed above, every vertex must
have four legs.:98In realistic applications,
the scattering amplitude of a certain interaction
or the decay rate of a particle can be computed
from the S-matrix, which itself can be found
using the Feynman diagram method.:102-115Feynman
diagrams devoid of "loops" are called tree-level
diagrams, which describe the lowest-order
interaction processes; those containing n
loops are referred to as n-loop diagrams,
which describe higher-order contributions,
or radiative corrections, to the interaction.:44
Lines whose end points are vertices can be
thought of as the propagation of virtual particles.:31
=== Renormalisation ===
Feynman rules can be used to directly evaluate
tree-level diagrams. However, naïve computation
of loop diagrams such as the one shown above
will result in divergent momentum integrals,
which seems to imply that almost all terms
in the perturbative expansion are infinite.
The renormalisation procedure is a systematic
process for removing such infinities.
Parameters appearing in the Lagrangian, such
as the mass m and the coupling constant λ,
have no physical meaning — m, λ, and the
field strength ϕ are not experimentally measurable
quantities and are referred to here as the
bare mass, bare coupling constant, and bare
field, respectively. The physical mass and
coupling constant are measured in some interaction
process and are generally different from the
bare quantities. While computing physical
quantities from this interaction process,
one may limit the domain of divergent momentum
integrals to be below some momentum cut-off
Λ, obtain expressions for the physical quantities,
and then take the limit Λ → ∞. This
is an example of regularisation, a class of
methods to treat divergences in QFT, with
Λ being the regulator.
The approach illustrated above is called bare
perturbation theory, as calculations involve
only the bare quantities such as mass and
coupling constant. A different approach, called
renormalised perturbation theory, is to use
physically meaningful quantities from the
very beginning. In the case of ϕ4 theory,
the field strength is first redefined:
ϕ
=
Z
1
/
2
ϕ
r
,
{\displaystyle \phi =Z^{1/2}\phi _{r},}
where ϕ is the bare field, ϕr is the renormalised
field, and Z is a constant to be determined.
The Lagrangian density becomes:
L
=
1
2
(
∂
μ
ϕ
r
)
(
∂
μ
ϕ
r
)
−
1
2
m
r
2
ϕ
r
2
−
λ
r
4
!
ϕ
r
4
+
1
2
δ
Z
(
∂
μ
ϕ
r
)
(
∂
μ
ϕ
r
)
−
1
2
δ
m
ϕ
r
2
−
δ
λ
4
!
ϕ
r
4
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial
_{\mu }\phi _{r})(\partial ^{\mu }\phi _{r})-{\frac
{1}{2}}m_{r}^{2}\phi _{r}^{2}-{\frac {\lambda
_{r}}{4!}}\phi _{r}^{4}+{\frac {1}{2}}\delta
_{Z}(\partial _{\mu }\phi _{r})(\partial ^{\mu
}\phi _{r})-{\frac {1}{2}}\delta _{m}\phi
_{r}^{2}-{\frac {\delta _{\lambda }}{4!}}\phi
_{r}^{4},}
where mr and λr are the experimentally measurable,
renormalised, mass and coupling constant,
respectively, and
δ
Z
=
Z
−
1
,
δ
m
=
m
2
Z
−
m
r
2
,
δ
λ
=
λ
Z
2
−
λ
r
{\displaystyle \delta _{Z}=Z-1,\quad \delta
_{m}=m^{2}Z-m_{r}^{2},\quad \delta _{\lambda
}=\lambda Z^{2}-\lambda _{r}}
are constants to be determined. The first
three terms are the ϕ4 Lagrangian density
written in terms of the renormalised quantities,
while the latter three terms are referred
to as "counterterms". As the Lagrangian now
contains more terms, so the Feynman diagrams
should include additional elements, each with
their own Feynman rules. The procedure is
outlined as follows. First select a regularisation
scheme (such as the cut-off regularisation
introduced above or dimensional regularization);
call the regulator Λ. Compute Feynman diagrams,
in which divergent terms will depend on Λ.
Then, define δZ, δm, and δλ such that
Feynman diagrams for the counterterms will
exactly cancel the divergent terms in the
normal Feynman diagrams when the limit Λ → ∞
is taken. In this way, meaningful finite quantities
are obtained.:323-326It is only possible to
eliminate all infinities to obtain a finite
result in renormalisable theories, whereas
in non-renormalisable theories infinities
cannot be removed by the redefinition of a
small number of parameters. The Standard Model
of elementary particles is a renormalisable
QFT,:719–727 while quantum gravity is non-renormalisable.:798:421
==== Renormalisation group ====
The renormalisation group, developed by Kenneth
Wilson, is a mathematical apparatus used to
study the changes in physical parameters (coefficients
in the Lagrangian) as the system is viewed
at different scales.:393 The way in which
each parameter changes with scale is described
by its β function.:417 Correlation functions,
which underlie quantitative physical predictions,
change with scale according to the Callan–Symanzik
equation.:410-411As an example, the coupling
constant in QED, namely the elementary charge
e, has the following β function:
β
(
e
)
≡
1
Λ
d
e
d
Λ
=
e
3
12
π
2
+
O
(
e
5
)
,
{\displaystyle \beta (e)\equiv {\frac {1}{\Lambda
}}{\frac {de}{d\Lambda }}={\frac {e^{3}}{12\pi
^{2}}}+O(e^{5}),}
where Λ is the energy scale under which the
measurement of e is performed. This differential
equation implies that the observed elementary
charge increases as the scale increases. The
renormalized coupling constant, which changes
with the energy scale, is also called the
running coupling constant.:420The coupling
constant g in quantum chromodynamics, a non-Abelian
gauge theory based on the symmetry group SU(3),
has the following β function:
β
(
g
)
≡
1
Λ
d
g
d
Λ
=
g
3
16
π
2
(
−
11
+
2
3
N
f
)
,
{\displaystyle \beta (g)\equiv {\frac {1}{\Lambda
}}{\frac {dg}{d\Lambda }}={\frac {g^{3}}{16\pi
^{2}}}\left(-11+{\frac {2}{3}}N_{f}\right),}
where Nf is the number of quark flavours.
In the case where Nf ≤ 16 (the Standard
Model has Nf = 6), the coupling constant g
decreases as the energy scale increases. Hence,
while the strong interaction is strong at
low energies, it becomes very weak in high-energy
interactions, a phenomenon known as asymptotic
freedom.:531Conformal field theories (CFTs)
are special QFTs that admit conformal symmetry.
They are insensitive to changes in the scale,
as all their coupling constants have vanishing
β function. (The converse is not true, however
— the vanishing of all β functions does
not imply conformal symmetry of the theory.)
Examples include string theory and N = 4 supersymmetric
Yang–Mills theory.According to Wilson's
picture, every QFT is fundamentally accompanied
by its energy cut-off Λ, i.e. that the theory
is no longer valid at energies higher than
Λ, and all degrees of freedom above the scale
Λ are to be omitted. For example, the cut-off
could be the inverse of the atomic spacing
in a condensed matter system, and in elementary
particle physics it could be associated with
the fundamental "graininess" of spacetime
caused by quantum fluctuations in gravity.
The cut-off scale of theories of particle
interactions lies far beyond current experiments.
Even if the theory were very complicated at
that scale, as long as its couplings are sufficiently
weak, it must be described at low energies
by a renormalisable effective field theory.:402-403
The difference between renormalisable and
non-renormalisable theories is that the former
are insensitive to details at high energies,
whereas the latter do depend of them.:2 According
to this view, non-renormalisable theories
are to be seen as low-energy effective theories
of a more fundamental theory. The failure
to remove the cut-off Λ from calculations
in such a theory merely indicates that new
physical phenomena appear at scales above
Λ, where a new theory is necessary.:156
=== Other theories ===
The quantisation and renormalisation procedures
outlined in the preceding sections are performed
for the free theory and ϕ4 theory of the
real scalar field. A similar process can be
done for other types of fields, including
the complex scalar field, the vector field,
and the Dirac field, as well as other types
of interaction terms, including the electromagnetic
interaction and the Yukawa interaction.
As an example, quantum electrodynamics contains
a Dirac field ψ representing the electron
field and a vector field Aμ representing
the electromagnetic field (photon field).
(Despite its name, the quantum electromagnetic
"field" actually corresponds to the classical
electromagnetic four-potential, rather than
the classical electric and magnetic fields.)
The full QED Lagrangian density is:
L
=
ψ
¯
(
i
γ
μ
∂
μ
−
m
)
ψ
−
1
4
F
μ
ν
F
μ
ν
−
e
ψ
¯
γ
μ
ψ
A
μ
,
{\displaystyle {\mathcal {L}}={\bar {\psi
}}(i\gamma ^{\mu }\partial _{\mu }-m)\psi
-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-e{\bar
{\psi }}\gamma ^{\mu }\psi A_{\mu },}
where γμ are Dirac matrices,
ψ
¯
=
ψ
†
γ
0
{\displaystyle {\bar {\psi }}=\psi ^{\dagger
}\gamma ^{0}}
, and
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F_{\mu \nu }=\partial _{\mu
}A_{\nu }-\partial _{\nu }A_{\mu }}
is the electromagnetic field strength. The
parameters in this theory are the (bare) electron
mass m and the (bare) elementary charge e.
The first and second terms in the Lagrangian
density correspond to the free Dirac field
and free vector fields, respectively. The
last term describes the interaction between
the electron and photon fields, which is treated
as a perturbation from the free theories.:78
Shown above is an example of a tree-level
Feynman diagram in QED. It describes an electron
and a positron annihilating, creating an off-shell
photon, and then decaying into a new pair
of electron and positron. Time runs from left
to right. Arrows pointing forward in time
represent the propagation of positrons, while
those pointing backward in time represent
the propagation of electrons. A wavy line
represents the propagation of a photon. Each
vertex in QED Feynman diagrams must have an
incoming and an outgoing fermion (positron/electron)
leg as well as a photon leg.
==== Gauge symmetry ====
If the following transformation to the fields
is performed at every spacetime point x (a
local transformation), then the QED Lagrangian
remains unchanged, or invariant:
ψ
(
x
)
→
e
i
α
(
x
)
ψ
(
x
)
,
A
μ
(
x
)
→
A
μ
(
x
)
+
i
e
−
1
e
−
i
α
(
x
)
∂
μ
e
i
α
(
x
)
,
{\displaystyle \psi (x)\to e^{i\alpha (x)}\psi
(x),\quad A_{\mu }(x)\to A_{\mu }(x)+ie^{-1}e^{-i\alpha
(x)}\partial _{\mu }e^{i\alpha (x)},}
where α(x) is any function of spacetime coordinates.
If a theory's Lagrangian (or more precisely
the action) is invariant under a certain local
transformation, then the transformation is
referred to as a gauge symmetry of the theory.:482–483
Gauge symmetries form a group at every spacetime
point. In the case of QED, the successive
application of two different local symmetry
transformations
e
i
α
(
x
)
{\displaystyle e^{i\alpha (x)}}
and
e
i
α
′
(
x
)
{\displaystyle e^{i\alpha '(x)}}
is yet another symmetry transformation
e
i
[
α
(
x
)
+
α
′
(
x
)
]
{\displaystyle e^{i[\alpha (x)+\alpha '(x)]}}
. For any α(x),
e
i
α
(
x
)
{\displaystyle e^{i\alpha (x)}}
is an element of the U(1) group, thus QED
is said to have U(1) gauge symmetry.:496 The
photon field Aμ may be referred to as the
U(1) gauge boson.
U(1) is an Abelian group, meaning that the
result is the same regardless of the order
in which its elements are applied. QFTs can
also be built on non-Abelian groups, giving
rise to non-Abelian gauge theories (also known
as Yang–Mills theories).:489 Quantum chromodynamics,
which describes the strong interaction, is
a non-Abelian gauge theory with an SU(3) gauge
symmetry. It contains three Dirac fields ψi,
i = 1,2,3 representing quark fields as well
as eight vector fields Aa,μ, a = 1,...,8
representing gluon fields, which are the SU(3)
gauge bosons.:547 The QCD Lagrangian density
is::490-491
L
=
i
ψ
¯
i
γ
μ
(
D
μ
)
i
j
ψ
j
−
1
4
F
μ
ν
a
F
a
,
μ
ν
−
m
ψ
¯
i
ψ
i
,
{\displaystyle {\mathcal {L}}=i{\bar {\psi
}}^{i}\gamma ^{\mu }(D_{\mu })^{ij}\psi ^{j}-{\frac
{1}{4}}F_{\mu \nu }^{a}F^{a,\mu \nu }-m{\bar
{\psi }}^{i}\psi ^{i},}
where Dμ is the gauge covariant derivative:
D
μ
=
∂
μ
−
i
g
A
μ
a
t
a
,
{\displaystyle D_{\mu }=\partial _{\mu }-igA_{\mu
}^{a}t^{a},}
where g is the coupling constant, ta are the
eight generators of SU(3) in the fundamental
representation (3×3 matrices),
F
μ
ν
a
=
∂
μ
A
ν
a
−
∂
ν
A
μ
a
+
g
f
a
b
c
A
μ
b
A
ν
c
,
{\displaystyle F_{\mu \nu }^{a}=\partial _{\mu
}A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu
}^{b}A_{\nu }^{c},}
and fabc are the structure constants of SU(3).
Repeated indices i,j,a are implicitly summed
over following Einstein notation. This Lagrangian
is invariant under the transformation:
ψ
i
(
x
)
→
U
i
j
(
x
)
ψ
j
(
x
)
,
A
μ
a
(
x
)
t
a
→
U
(
x
)
[
A
μ
a
(
x
)
t
a
+
i
g
−
1
∂
μ
]
U
†
(
x
)
,
{\displaystyle \psi ^{i}(x)\to U^{ij}(x)\psi
^{j}(x),\quad A_{\mu }^{a}(x)t^{a}\to U(x)\left[A_{\mu
}^{a}(x)t^{a}+ig^{-1}\partial _{\mu }\right]U^{\dagger
}(x),}
where U(x) is an element of SU(3) at every
spacetime point x:
U
(
x
)
=
e
i
α
(
x
)
a
t
a
.
{\displaystyle U(x)=e^{i\alpha (x)^{a}t^{a}}.}
The preceding discussion of symmetries is
on the level of the Lagrangian. In other words,
these are "classical" symmetries. After quantisation,
some theories will no longer exhibit their
classical symmetries, a phenomenon called
anomaly. For instance, in the path integral
formulation, despite the invariance of the
Lagrangian density
L
[
ϕ
,
∂
μ
ϕ
]
{\displaystyle {\mathcal {L}}[\phi ,\partial
_{\mu }\phi ]}
under a certain local transformation of the
fields, the measure
∫
D
ϕ
{\displaystyle \int {\mathcal {D}}\phi }
of the path integral may change.:243 For a
theory describing nature to be consistent,
it must not contain any anomaly in its gauge
symmetry. The Standard Model of elementary
particles is a gauge theory based on the group
SU(3) × SU(2) × U(1), in which all anomalies
exactly cancel.:705-707The theoretical foundation
of general relativity, the equivalence principle,
can also be understood as a form of gauge
symmetry, making general relativity a gauge
theory based on the Lorentz group.Noether's
theorem states that every continuous symmetry,
i.e. the parameter in the symmetry transformation
being continuous rather than discrete, leads
to a corresponding conservation law.:17-18:73
For example, the U(1) symmetry of QED implies
charge conservation.Gauge transformations
do not relate distinct quantum states. Rather,
it relates two equivalent mathematical descriptions
of the same quantum state. As an example,
the photon field Aμ, being a four-vector,
has four apparent degrees of freedom, but
the actual state of a photon is described
by its two degrees of freedom corresponding
to the polarisation. The remaining two degrees
of freedom are said to be "redundant" — apparently
different ways of writing Aμ can be related
to each other by a gauge transformation and
in fact describe the same state of the photon
field. In this sense, gauge invariance is
not a "real" symmetry, but are a reflection
of the "redundancy" of the chosen mathematical
description.:168To account for the gauge redundancy
in the path integral formulation, one must
perform the so-called Faddeev–Popov gauge
fixing procedure. In non-Abelian gauge theories,
such a procedure introduces new fields called
"ghosts". Particles corresponding to the ghost
fields are called ghost particles, which cannot
be detected externally.:512-515 A more rigorous
generalisation of the Faddeev–Popov procedure
is given by BRST quantization.:517
==== Spontaneous symmetry breaking ====
Spontaneous symmetry breaking is a mechanism
whereby the symmetry of the Lagrangian is
violated by the system described by it.:347To
illustrate the mechanism, consider a linear
sigma model containing N real scalar fields,
described by the Lagrangian density:
L
=
1
2
(
∂
μ
ϕ
i
)
(
∂
μ
ϕ
i
)
+
1
2
μ
2
ϕ
i
ϕ
i
−
λ
4
(
ϕ
i
ϕ
i
)
2
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial
_{\mu }\phi ^{i})(\partial ^{\mu }\phi ^{i})+{\frac
{1}{2}}\mu ^{2}\phi ^{i}\phi ^{i}-{\frac {\lambda
}{4}}(\phi ^{i}\phi ^{i})^{2},}
where μ and λ are real parameters. The theory
admits an O(N) global symmetry:
ϕ
i
→
R
i
j
ϕ
j
,
R
∈
O
(
N
)
.
{\displaystyle \phi ^{i}\to R^{ij}\phi ^{j},\quad
R\in \mathrm {O} (N).}
The lowest energy state (ground state or vacuum
state) of the classical theory is any uniform
field ϕ0 satisfying
ϕ
0
i
ϕ
0
i
=
μ
2
λ
.
{\displaystyle \phi _{0}^{i}\phi _{0}^{i}={\frac
{\mu ^{2}}{\lambda }}.}
Without loss of generality, let the ground
state be in the N-th direction:
ϕ
0
i
=
(
0
,
⋯
,
0
,
μ
λ
)
.
{\displaystyle \phi _{0}^{i}=\left(0,\cdots
,0,{\frac {\mu }{\sqrt {\lambda }}}\right).}
The 
original N fields can be rewritten as:
ϕ
i
(
x
)
=
(
π
1
(
x
)
,
⋯
,
π
N
−
1
(
x
)
,
μ
λ
+
σ
(
x
)
)
,
{\displaystyle \phi ^{i}(x)=\left(\pi ^{1}(x),\cdots
,\pi ^{N-1}(x),{\frac {\mu }{\sqrt {\lambda
}}}+\sigma (x)\right),}
and the original Lagrangian density as:
L
=
1
2
(
∂
μ
π
k
)
(
∂
μ
π
k
)
+
1
2
(
∂
μ
σ
)
(
∂
μ
σ
)
−
1
2
(
2
μ
2
)
σ
2
−
λ
μ
σ
3
−
λ
μ
π
k
π
k
σ
−
λ
2
π
k
π
k
σ
2
−
λ
4
(
π
k
π
k
)
2
,
{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial
_{\mu }\pi ^{k})(\partial ^{\mu }\pi ^{k})+{\frac
{1}{2}}(\partial _{\mu }\sigma )(\partial
^{\mu }\sigma )-{\frac {1}{2}}(2\mu ^{2})\sigma
^{2}-{\sqrt {\lambda }}\mu \sigma ^{3}-{\sqrt
{\lambda }}\mu \pi ^{k}\pi ^{k}\sigma -{\frac
{\lambda }{2}}\pi ^{k}\pi ^{k}\sigma ^{2}-{\frac
{\lambda }{4}}(\pi ^{k}\pi ^{k})^{2},}
where k = 1,...,N-1. The original O(N) global
symmetry is no longer manifest, leaving only
the subgroup O(N-1). The larger symmetry before
spontaneous symmetry breaking is said to be
"hidden" or spontaneously broken.:349-350Goldstone's
theorem states that under spontaneous symmetry
breaking, every broken continuous global symmetry
leads to a massless field called the Goldstone
boson. In the above example, O(N) has N(N-1)/2
continuous symmetries (the dimension of its
Lie algebra), while O(N-1) has (N-1)(N-2)/2.
The number of broken symmetries is their difference,
N-1, which corresponds to the N-1 massless
fields πk.:351On the other hand, when a gauge
(as opposed to global) symmetry is spontaneously
broken, the resulting Goldstone boson is "eaten"
by the corresponding gauge boson by becoming
an additional degree of freedom for the gauge
boson. The Goldstone boson equivalence theorem
states that at high energy, the amplitude
for emission or absorption of a longitudinally
polarised massive gauge boson becomes equal
to the amplitude for emission or absorption
of the Goldstone boson that was eaten by the
gauge boson.:743-744In the QFT of ferromagnetism,
spontaneous symmetry breaking can explain
the alignment of magnetic dipoles at low temperatures.:199
In the Standard Model of elementary particles,
the W and Z bosons, which would otherwise
be massless as a result of gauge symmetry,
acquire mass through spontaneous symmetry
breaking of the Higgs boson, a process called
the Higgs mechanism.:690
==== Supersymmetry ====
All experimentally known symmetries in nature
relate bosons to bosons and fermions to fermions.
Theorists have hypothesised the existence
of a type of symmetry, called supersymmetry,
that relates bosons and fermions.:795:443The
Standard Model obeys Poincaré symmetry, whose
generators are spacetime translation Pμ and
Lorentz transformation Jμν. In addition
to these generators, supersymmetry in (3+1)-dimensions
includes additional generators Qα, called
supercharges, which themselves transform as
Weyl fermions.:795:444 The symmetry group
generated by all these generators is known
as the super-Poincaré group. In general there
can be more than one set of supersymmetry
generators, QαI, I = 1, ..., N, which generate
the corresponding N = 1 supersymmetry, N = 2
supersymmetry, and so on.:795:450 Supersymmetry
can also be constructed in other dimensions,
most notably in (1+1) dimensions for its application
in superstring theory.The Lagrangian of a
supersymmetric theory must be invariant under
the action of the super-Poincaré group.:448
Examples of such theories include: Minimal
Supersymmetric Standard Model (MSSM), N = 4
supersymmetric Yang–Mills theory,:450 and
superstring theory. In a supersymmetric theory,
every fermion has a bosonic superpartner and
vice versa.:444If supersymmetry is promoted
to a local symmetry, then the resultant gauge
theory is an extension of general relativity
called supergravity.Supersymmetry is a potential
solution to many current problems in physics.
For example, the hierarchy problem of the
Standard Model — why the mass of the Higgs
boson is not radiatively corrected (under
renormalisation) to a very high scale such
as the grand unified scale or the Planck scale
— can be resolved by relating the Higgs
field and its superpartner, the Higgsino.
Radiative corrections due to Higgs boson loops
in Feynman diagrams are cancelled by corresponding
Higgsino loops. Supersymmetry also offers
answers to the grand unification of all gauge
coupling constants in the Standard Model as
well as the nature of dark matter.:796-797Nevertheless,
as of 2018, experiments have yet to provide
evidence for the existence of supersymmetric
particles. If supersymmetry were a true symmetry
of nature, then it must be a broken symmetry,
and the energy of symmetry breaking must be
higher than those achievable by present-day
experiments.:797:443
==== Other spacetimes ====
The ϕ4 theory, QED, QCD, as well as the whole
Standard Model all assume a (3+1)-dimensional
Minkowski space (3 spatial and 1 time dimensions)
as the background on which the quantum fields
are defined. However, QFT a priori imposes
no restriction on the number of dimensions
nor the geometry of spacetime.
In condensed matter physics, QFT is used to
describe (2+1)-dimensional electron gases.
In high-energy physics, string theory is a
type of (1+1)-dimensional QFT,:452 while Kaluza–Klein
theory uses gravity in extra dimensions to
produce gauge theories in lower dimensions.:428-429In
Minkowski space, the flat metric ημν is
used to raise and lower spacetime indices
in the Lagrangian, e.g.
A
μ
A
μ
=
η
μ
ν
A
μ
A
ν
,
∂
μ
ϕ
∂
μ
ϕ
=
η
μ
ν
∂
μ
ϕ
∂
ν
ϕ
,
{\displaystyle A_{\mu }A^{\mu }=\eta _{\mu
\nu }A^{\mu }A^{\nu },\quad \partial _{\mu
}\phi \partial ^{\mu }\phi =\eta ^{\mu \nu
}\partial _{\mu }\phi \partial _{\nu }\phi
,}
where ημν is the inverse of ημν satisfying
ημρηρν = δμν. For QFTs in curved
spacetime on the other hand, a general metric
(such as the Schwarzschild metric describing
a black hole) is used:
A
μ
A
μ
=
g
μ
ν
A
μ
A
ν
,
∂
μ
ϕ
∂
μ
ϕ
=
g
μ
ν
∂
μ
ϕ
∂
ν
ϕ
,
{\displaystyle A_{\mu }A^{\mu }=g_{\mu \nu
}A^{\mu }A^{\nu },\quad \partial _{\mu }\phi
\partial ^{\mu }\phi =g^{\mu \nu }\partial
_{\mu }\phi \partial _{\nu }\phi ,}
where gμν is the inverse of gμν. For a
real scalar field, the Lagrangian density
in a general spacetime background is
L
=
|
g
|
(
1
2
g
μ
ν
∇
μ
ϕ
∇
ν
ϕ
−
1
2
m
2
ϕ
2
)
,
{\displaystyle {\mathcal {L}}={\sqrt {|g|}}\left({\frac
{1}{2}}g^{\mu \nu }\nabla _{\mu }\phi \nabla
_{\nu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right),}
where g = det(gμν), and ∇μ denotes the
covariant derivative. The Lagrangian of a
QFT, hence its calculational results and physical
predictions, depends on the geometry of the
spacetime background.
==== Topological quantum field theory ====
The correlation functions and physical predictions
of a QFT depend on the spacetime metric gμν.
For a special class of QFTs called topological
quantum field theories (TQFTs), all correlation
functions are independent of continuous changes
in the spacetime metric. QFTs in curved spacetime
generally change according to the geometry
(local structure) of the spacetime background,
while TQFTs are invariant under spacetime
diffeomorphisms but are sensitive to the topology
(global structure) of spacetime. This means
that all calculational results of TQFTs are
topological invariants of the underlying spacetime.
Chern–Simons theory is an example of TQFT.
Applications of TQFT include the fractional
quantum Hall effect and topological quantum
computers.
=== Perturbative and non-perturbative methods
===
Using perturbation theory, the total effect
of a small interaction term can be approximated
order by order by a series expansion in the
number of virtual particles participating
in the interaction. Every term in the expansion
may be understood as one possible way for
(physical) particles to interact with each
other via virtual particles, expressed visually
using a Feynman diagram. The electromagnetic
force between two electrons in QED is represented
(to first order in perturbation theory) by
the propagation of a virtual photon. In a
similar manner, the W and Z bosons carry the
weak interaction, while gluons carry the strong
interaction. The interpretation of an interaction
as a sum of intermediate states involving
the exchange of various virtual particles
only makes sense in the framework of perturbation
theory. In contrast, non-perturbative methods
in QFT treat the interacting Lagrangian as
a whole without any series expansion. Instead
of particles that carry interactions, these
methods have spawned such concepts as 't Hooft–Polyakov
monopole, domain wall, flux tube, and instanton.
== Mathematical rigour ==
In spite of its overwhelming success in particle
physics and condensed matter physics, QFT
itself lacks a formal mathematical foundation.
For example, according to Haag's theorem,
there does not exist a well-defined interaction
picture for QFT, which implies that perturbation
theory of QFT, which underlies the entire
Feynman diagram method, is fundamentally not
rigorous.Since the 1950s, theoretical physicists
and mathematicians have attempted to organise
all QFTs into a set of axioms, in order to
establish the existence of concrete models
of relativistic QFT in a mathematically rigorous
way and to study their properties. This line
of study is called constructive quantum field
theory, a subfield of mathematical physics,
which has led to such results as CPT theorem,
spin-statistics theorem, and Goldstone's theorem.Compared
to ordinary QFT, topological quantum field
theory and conformal field theory are better
supported mathematically — both can be classified
in the framework of representations of cobordisms.Algebraic
quantum field theory is another approach to
the axiomatisation of QFT, in which the fundamental
objects are local operators and the algebraic
relations between them. Axiomatic systems
following this approach include Wightman axioms
and Haag-Kastler axioms.:2-3 One way to construct
theories satisfying Wightman axioms is to
use Osterwalder-Schrader axioms, which give
the necessary and sufficient conditions for
a real time theory to be obtained from an
imaginary time theory by analytic continuation
(Wick rotation).:10Yang-Mills existence and
mass gap, one of the Millenium Prize Problems,
concerns the well-defined existence of Yang-Mills
theories as set out by the above axioms. The
full problem statement is as follows.
== See also
