In particle physics, the history of quantum
field theory starts with its creation by Paul
Dirac, when he attempted to quantize the electromagnetic
field in the late 1920s. Major advances in
the theory were made in the 1950s, and led
to the introduction of quantum electrodynamics
(QED). QED was so successful and accurately
predictive that efforts were made to apply
the same basic concepts for the other forces
of nature. By the late 1970s, these efforts
successfully utilized gauge theory in the
strong nuclear force and weak nuclear force,
producing the modern standard model of particle
physics.
Efforts to describe gravity using the same
techniques have, to date, failed. The study
of quantum field theory is still flourishing,
as are applications of its methods to many
physical problems. It remains one of the most
vital areas of theoretical physics today,
providing a common language to several different
branches of physics.
== Early developments ==
Quantum field theory originated in the 1920s
from the problem of creating a quantum mechanical
theory of the electromagnetic field. In particular,
de Broglie in 1924 introduced the idea of
a wave description of elementary systems in
the following way: "we proceed in this work
from the assumption of the existence of a
certain periodic phenomenon of a yet to be
determined character, which is to be attributed
to each and every isolated energy parcel".In
1925, Werner Heisenberg, Max Born, and Pascual
Jordan constructed just such a theory by expressing
the field's internal degrees of freedom as
an infinite set of harmonic oscillators, and
by then utilizing the canonical quantization
procedure to these oscillators; their paper
was published in 1926. This theory assumed
that no electric charges or currents were
present and today would be called a free field
theory.
The first reasonably complete theory of quantum
electrodynamics, which included both the electromagnetic
field and electrically charged matter (specifically,
electrons) as quantum mechanical objects,
was created by Paul Dirac in 1927. This quantum
field theory could be used to model important
processes such as the emission of a photon
by an electron dropping into a quantum state
of lower energy, a process in which the number
of particles changes—one atom in the initial
state becomes an atom plus a photon in the
final state. It is now understood that the
ability to describe such processes is one
of the most important features of quantum
field theory.
The final crucial step was Enrico Fermi's
theory of β-decay (1934). In it, fermion
species nonconservation was shown to follow
from second quantization: creation and annihilation
of fermions came to the fore and quantum field
theory was seen to describe particle decays.
(Fermi's breakthrough was somewhat foreshadowed
in the abstract studies of Soviet physicists,
Viktor Ambartsumian and Dmitri Ivanenko, in
particular the Ambarzumian–Ivanenko hypothesis
of creation of massive particles (1930). The
idea was that not only the quanta of the electromagnetic
field, photons, but also other particles might
emerge and disappear as a result of their
interaction with other particles.)
== Incorporating special relativity ==
It was evident from the beginning that a proper
quantum treatment of the electromagnetic field
had to somehow incorporate Einstein's relativity
theory, which had grown out of the study of
classical electromagnetism. This need to put
together relativity and quantum mechanics
was the second major motivation in the development
of quantum field theory. Pascual Jordan and
Wolfgang Pauli showed in 1928 that quantum
fields could be made to behave in the way
predicted by special relativity during coordinate
transformations (specifically, they showed
that the field commutators were Lorentz invariant).
A further boost for quantum field theory came
with the discovery of the Dirac equation,
which was originally formulated and interpreted
as a single-particle equation analogous to
the Schrödinger equation, but unlike the
Schrödinger equation, the Dirac equation
satisfies both the Lorentz invariance, that
is, the requirements of special relativity,
and the rules of quantum mechanics.
The Dirac equation accommodated the spin-1/2
value of the electron and accounted for its
magnetic moment as well as giving accurate
predictions for the spectra of hydrogen.
The attempted interpretation of the Dirac
equation as a single-particle equation could
not be maintained long, however, and finally
it was shown that several of its undesirable
properties (such as negative-energy states)
could be made sense of by reformulating and
reinterpreting the Dirac equation as a true
field equation, in this case for the quantized
"Dirac field" or the "electron field", with
the "negative-energy solutions" pointing to
the existence of anti-particles. This work
was performed first by Dirac himself with
the invention of hole theory in 1930 and by
Wendell Furry, Robert Oppenheimer, Vladimir
Fock, and others. Schrödinger, during the
same period that he discovered his famous
equation in 1926, also independently found
the relativistic generalization of it known
as the Klein–Gordon equation but dismissed
it since, without spin, it predicted impossible
properties for the hydrogen spectrum. (See
Oskar Klein and Walter Gordon.) All relativistic
wave equations that describe spin-zero particles
are said to be of the Klein–Gordon type.
== Uncertainty, again ==
A subtle and careful analysis in 1933 by Niels
Bohr and Léon Rosenfeld showed that there
is a fundamental limitation on the ability
to simultaneously measure the electric and
magnetic field strengths that enter into the
description of charges in interaction with
radiation, imposed by the uncertainty principle,
which must apply to all canonically conjugate
quantities. This limitation is crucial for
the successful formulation and interpretation
of a quantum field theory of photons and electrons
(quantum electrodynamics), and indeed, any
perturbative quantum field theory. The analysis
of Bohr and Rosenfeld explains fluctuations
in the values of the electromagnetic field
that differ from the classically "allowed"
values distant from the sources of the field.
Their analysis was crucial to showing that
the limitations and physical implications
of the uncertainty principle apply to all
dynamical systems, whether fields or material
particles. Their analysis also convinced most
physicists that any notion of returning to
a fundamental description of nature based
on classical field theory, such as what Einstein
aimed at with his numerous and failed attempts
at a classical unified field theory, was simply
out of the question. Fields had to be quantized.
== Second quantization ==
The third thread in the development of quantum
field theory was the need to handle the statistics
of many-particle systems consistently and
with ease. In 1927, Pascual Jordan tried to
extend the canonical quantization of fields
to the many-body wave functions of identical
particles using a formalism which is known
as statistical transformation theory; this
procedure is now sometimes called second quantization.
In 1928, Jordan and Eugene Wigner found that
the quantum field describing electrons, or
other fermions, had to be expanded using anti-commuting
creation and annihilation operators due to
the Pauli exclusion principle (see Jordan–Wigner
transformation). This thread of development
was incorporated into many-body theory and
strongly influenced condensed matter physics
and nuclear physics.
== The problem of infinities ==
Despite its early successes quantum field
theory was plagued by several serious theoretical
difficulties. Basic physical quantities, such
as the self-energy of the electron, the energy
shift of electron states due to the presence
of the electromagnetic field, gave infinite,
divergent contributions—a nonsensical result—when
computed using the perturbative techniques
available in the 1930s and most of the 1940s.
The electron self-energy problem was already
a serious issue in the classical electromagnetic
field theory, where the attempt to attribute
to the electron a finite size or extent (the
classical electron-radius) led immediately
to the question of what non-electromagnetic
stresses would need to be invoked, which would
presumably hold the electron together against
the Coulomb repulsion of its finite-sized
"parts". The situation was dire, and had certain
features that reminded many of the "Rayleigh–Jeans
catastrophe". What made the situation in the
1940s so desperate and gloomy, however, was
the fact that the correct ingredients (the
second-quantized Maxwell–Dirac field equations)
for the theoretical description of interacting
photons and electrons were well in place,
and no major conceptual change was needed
analogous to that which was necessitated by
a finite and physically sensible account of
the radiative behavior of hot objects, as
provided by the Planck radiation law.
== Renormalization procedures ==
This "divergence problem" was solved in the
case of quantum electrodynamics through the
procedure known as renormalization in 1947–49
by Hans Kramers, Hans Bethe, Julian Schwinger,
Richard Feynman, and Shin'ichiro Tomonaga;
the procedure was systematized by Freeman
Dyson in 1949. Great progress was made after
realizing that all infinities in quantum electrodynamics
are related to two effects: the self-energy
of the electron/positron, and vacuum polarization.
Renormalization requires paying very careful
attention to just what is meant by, for example,
the very concepts "charge" and "mass" as they
occur in the pure, non-interacting field-equations.
The "vacuum" is itself polarizable and, hence,
populated by virtual particle (on shell and
off shell) pairs, and, hence, is a seething
and busy dynamical system in its own right.
This was a critical step in identifying the
source of "infinities" and "divergences".
The "bare mass" and the "bare charge" of a
particle, the values that appear in the free-field
equations (non-interacting case), are abstractions
that are simply not realized in experiment
(in interaction). What we measure, and hence,
what we must take account of with our equations,
and what the solutions must account for, are
the "renormalized mass" and the "renormalized
charge" of a particle. That is to say, the
"shifted" or "dressed" values these quantities
must have when due systematic care is taken
to include all deviations from their "bare
values" is dictated by the very nature of
quantum fields themselves.
== Gauge invariance ==
The first approach that bore fruit is known
as the "interaction representation" (see the
article Interaction picture), a Lorentz-covariant
and gauge-invariant generalization of time-dependent
perturbation theory used in ordinary quantum
mechanics, and developed by Tomonaga and Schwinger,
generalizing earlier efforts of Dirac, Fock
and Podolsky. Tomonaga and Schwinger invented
a relativistically covariant scheme for representing
field commutators and field operators intermediate
between the two main representations of a
quantum system, the Schrödinger and the Heisenberg
representations. Within this scheme, field
commutators at separated points can be evaluated
in terms of "bare" field creation and annihilation
operators. This allows for keeping track of
the time-evolution of both the "bare" and
"renormalized", or perturbed, values of the
Hamiltonian and expresses everything in terms
of the coupled, gauge invariant "bare" field-equations.
Schwinger gave the most elegant formulation
of this approach. The next and most famous
development is due to Feynman, who, with his
brilliant rules for assigning a "graph"/"diagram"
to the terms in the scattering matrix (see
S-matrix and Feynman diagrams). These directly
corresponded (through the Schwinger-Dyson
equation) to the measurable physical processes
(cross sections, probability amplitudes, decay
widths and lifetimes of excited states) one
needs to be able to calculate. This revolutionized
how quantum field theory calculations are
carried-out in practice.
Two classic text-books from the 1960s, James
D. Bjorken, Sidney David Drell, Relativistic
Quantum Mechanics (1964) and J. J. Sakurai,
Advanced Quantum Mechanics (1967), thoroughly
developed the Feynman graph expansion techniques
using physically intuitive and practical methods
following from the correspondence principle,
without worrying about the technicalities
involved in deriving the Feynman rules from
the superstructure of quantum field theory
itself. Although both Feynman's heuristic
and pictorial style of dealing with the infinities,
as well as the formal methods of Tomonaga
and Schwinger, worked extremely well, and
gave spectacularly accurate answers, the true
analytical nature of the question of "renormalizability",
that is, whether ANY theory formulated as
a "quantum field theory" would give finite
answers, was not worked-out until much later,
when the urgency of trying to formulate finite
theories for the strong and electro-weak (and
gravitational interactions) demanded its solution.
Renormalization in the case of QED was largely
fortuitous due to the smallness of the coupling
constant, the fact that the coupling has no
dimensions involving mass, the so-called fine
structure constant, and also the zero-mass
of the gauge boson involved, the photon, rendered
the small-distance/high-energy behavior of
QED manageable. Also, electromagnetic processes
are very "clean" in the sense that they are
not badly suppressed/damped and/or hidden
by the other gauge interactions. By 1965 James
D. Bjorken and Sidney David Drell observed:
"Quantum electrodynamics (QED) has achieved
a status of peaceful coexistence with its
divergences ...".The unification of the electromagnetic
force with the weak force encountered initial
difficulties due to the lack of accelerator
energies high enough to reveal processes beyond
the Fermi interaction range. Additionally,
a satisfactory theoretical understanding of
hadron substructure had to be developed, culminating
in the quark model.
== Non-abelian gauge theory ==
Thanks to the somewhat brute-force, ad hoc
and heuristic early methods of Feynman, and
the abstract methods of Tomonaga and Schwinger,
elegantly synthesized by Freeman Dyson, from
the period of early renormalization, the modern
theory of quantum electrodynamics (QED) has
established itself. It is still the most accurate
physical theory known, the prototype of a
successful quantum field theory. Quantum electrodynamics
is the most famous example of what is known
as an Abelian gauge theory. It relies on the
symmetry group U(1) and has one massless gauge
field, the U(1) gauge symmetry, dictating
the form of the interactions involving the
electromagnetic field, with the photon being
the gauge boson.
Beginning in the 1950s with the work of Yang
and Mills, following the previous lead of
Weyl and Pauli, deep explorations illuminated
the types of symmetries and invariances any
field theory must satisfy. QED, and indeed,
all field theories, were generalized to a
class of quantum field theories known as gauge
theories. That symmetries dictate, limit and
necessitate the form of interaction between
particles is the essence of the "gauge theory
revolution". Yang and Mills formulated the
first explicit example of a non-abelian gauge
theory, Yang–Mills theory, with an attempted
explanation of the strong interactions in
mind. The strong interactions were then (incorrectly)
understood in the mid-1950s, to be mediated
by the pi-mesons, the particles predicted
by Hideki Yukawa in 1935, based on his profound
reflections concerning the reciprocal connection
between the mass of any force-mediating particle
and the range of the force it mediates. This
was allowed by the uncertainty principle.
In the absence of dynamical information, Murray
Gell-Mann pioneered the extraction of physical
predictions from sheer non-abelian symmetry
considerations, and introduced non-abelian
Lie groups to current algebra and so the gauge
theories that came to supersede it.
The 1960s and 1970s saw the formulation of
a gauge theory now known as the Standard Model
of particle physics, which systematically
describes the elementary particles and the
interactions between them. The strong interactions
are described by quantum chromodynamics (QCD),
based on "color" SU(3). The weak interactions
require the additional feature of spontaneous
symmetry breaking, elucidated by Yoichiro
Nambu and the adjunct Higgs mechanism, considered
next.
== Electroweak unification ==
The electroweak interaction part of the standard
model was formulated by Sheldon Glashow, Abdus
Salam, and John Clive Ward in 1959 with their
discovery of the SU(2)xU(1) group structure
of the theory. In 1967, Steven Weinberg brilliantly
invoked the Higgs mechanism for the generation
of the W and Z masses (the intermediate vector
bosons responsible for the weak interactions
and neutral-currents) and keeping the mass
of the photon zero. The Goldstone and Higgs
idea for generating mass in gauge theories
was sparked in the late 1950s and early 1960s
when a number of theoreticians (including
Yoichiro Nambu, Steven Weinberg, Jeffrey Goldstone,
François Englert, Robert Brout, G. S. Guralnik,
C. R. Hagen, Tom Kibble and Philip Warren
Anderson) noticed a possibly useful analogy
to the (spontaneous) breaking of the U(1)
symmetry of electromagnetism in the formation
of the BCS ground-state of a superconductor.
The gauge boson involved in this situation,
the photon, behaves as though it has acquired
a finite mass.
There is a further possibility that the physical
vacuum (ground-state) does not respect the
symmetries implied by the "unbroken" electroweak
Lagrangian from which one arrives at the field
equations (see the article Electroweak interaction
for more details). The electroweak theory
of Weinberg and Salam was shown to be renormalizable
(finite) and hence consistent by Gerardus
't Hooft and Martinus Veltman. The Glashow–Weinberg–Salam
theory (GWS theory) is a triumph and, in certain
applications, gives an accuracy on a par with
quantum electrodynamics.
== Quantum chromodynamics ==
In the case of the strong interactions, progress
concerning their short-distance/high-energy
behavior was much slower and more frustrating.
For strong interactions with the electro-weak
fields, there were difficult issues regarding
the strength of coupling, the mass generation
of the force carriers as well as their non-linear,
self interactions. Although there has been
theoretical progress toward a grand unified
quantum field theory incorporating the electro-magnetic
force, the weak force and the strong force,
empirical verification is still pending. Superunification,
incorporating the gravitational force, is
still very speculative, and is under intensive
investigation by many of the best minds in
contemporary theoretical physics. Gravitation
is a tensor field description of a spin-2
gauge-boson, the "graviton", and is further
discussed in the articles on general relativity
and quantum gravity.
== Quantum gravity ==
From the point of view of the techniques of
(four-dimensional) quantum field theory, and
as the numerous efforts to formulate a consistent
quantum gravity theory attests, gravitational
quantization has been the reigning champion
for bad behavior.There are technical problems
underlain by the fact that the gravitational
coupling constant has dimensions involving
inverse powers of mass, and, as a simple consequence,
it is plagued by perturbatively badly behaved
non-linear self-interactions. Gravity is itself
a source of gravity, analogously to gauge
theories (whose couplings, are, by contrast,
dimensionless) leading to uncontrollable divergences
at increasing orders of perturbation theory.
Moreover, gravity couples to all energy equally
strongly, as per the equivalence principle,
so this makes the notion of ever really "switching-off",
"cutting-off" or separating, the gravitational
interaction from other interactions ambiguous,
since, with gravitation, we are dealing with
the very structure of space-time itself.
Moreover, it has not been established that
a theory of quantum gravity is necessary (see
Quantum field theory in curved spacetime).
== Contemporary framework of renormalization
==
Parallel breakthroughs in the understanding
of phase transitions in condensed matter physics
led to novel insights based on the renormalization
group. They involved the work of Leo Kadanoff
(1966) and Kenneth Geddes Wilson & Michael
Fisher (1972)—extending the work of Ernst
Stueckelberg–André Petermann (1953) and
Murray Gell-Mann–Francis Low (1954)—which
led to the seminal reformulation of quantum
field theory by Kenneth Geddes Wilson in 1975.This
reformulation provided insights into the evolution
of effective field theories with scale, which
classified all field theories, renormalizable
or not (cf. subsequent section). The remarkable
conclusion is that, in general, most observables
are "irrelevant", i.e., the macroscopic physics
is dominated by only a few observables in
most systems.
During the same period, Kadanoff (1969) introduced
an operator algebra formalism for the two-dimensional
Ising model, a widely studied mathematical
model of ferromagnetism in statistical physics.
This development suggested that quantum field
theory describes its scaling limit. Later,
there developed the idea that a finite number
of generating operators could represent all
the correlation functions of the Ising model.
The renormalization group spans a set of ideas
and methods to monitor changes of the behavior
of the theory with scale, providing a deep
physical understanding which sparked what
has been called the "grand synthesis" of theoretical
physics, uniting the quantum field theoretical
techniques used in particle physics and condensed
matter physics into a single powerful theoretical
framework.
The gauge field theory of the strong interactions,
quantum chromodynamics, QCD, relies crucially
on this renormalization group for its distinguishing
characteristic features, asymptotic freedom
and color confinement.
== Modern developments ==
Algebraic quantum field theory
Axiomatic quantum field theory
Topological quantum field theory (TQFT)
== See also ==
History of quantum mechanics
History of string theory
QED vacuum
== 
Notes ==
== Further reading ==
Pais, Abraham; Inward Bound - Of Matter & Forces
in the Physical World, Oxford University Press
(1986) ISBN 0-19-851997-4 Written by a former
Einstein assistant at Princeton, this is a
beautiful detailed history of modern fundamental
physics, from 1895 (discovery of X-rays) to
1983 (discovery of vectors bosons at C.E.R.N.).
Richard Feynman; Lecture Notes in Physics.
Princeton University Press: Princeton, (1986).
Richard Feynman; QED. Princeton University
Press: Princeton, (1982).
Weinberg, Steven; The Quantum Theory of Fields
- Foundations (vol. I), Cambridge University
Press (1995) ISBN 0-521-55001-7 The first
chapter (pp. 1–40) of Weinberg's monumental
treatise gives a brief history of Q.F.T.,
p. 608.
Weinberg, Steven; The Quantum Theory of Fields
- Modern Applications (vol. II), Cambridge
University Press:Cambridge, U.K. (1996) ISBN
0-521-55001-7, pp. 489.
Weinberg, Steven; The Quantum Theory of Fields
– Supersymmetry (vol. III), Cambridge University
Press:Cambridge, U.K. (2000) ISBN 0-521-55002-5,
pp. 419.
Schweber, Silvan S.; QED and the men who made
it: Dyson, Feynman, Schwinger, and Tomonaga,
Princeton University Press (1994) ISBN 0-691-03327-7
Ynduráin, Francisco José; Quantum Chromodynamics:
An Introduction to the Theory of Quarks and
Gluons, Springer Verlag, New York, 1983. ISBN
0-387-11752-0
Miller, Arthur I.; Early Quantum Electrodynamics
: A Sourcebook, Cambridge University Press
(1995) ISBN 0-521-56891-9
Schwinger, Julian; Selected Papers on Quantum
Electrodynamics, Dover Publications, Inc.
(1958) ISBN 0-486-60444-6
O'Raifeartaigh, Lochlainn; The Dawning of
Gauge Theory, Princeton University Press (May
5, 1997) ISBN 0-691-02977-6
Cao, Tian Yu; Conceptual Developments of 20th
Century Field Theories, Cambridge University
Press (1997) ISBN 0-521-63420-2
Darrigol, Olivier; La genèse du concept de
champ quantique, Annales de Physique (France)
9 (1984) pp. 433–501. Text in French, adapted
from the author's Ph.D. thesis.
